Physics 161

Lab Manual

Revised December 2017

This publication was developed with the support of a Hispanic Serving Institution STEM &

Articulation grant from the U.S. Department of Education.

Rule Why?

Plan on staying the entire 3 hours every time. It happens every time; if you leave I count you

absent.

Get to lab on time. Lateness = absence, 2nd absence reduces course

grade.

Don’t interrupt during lecture. This saves time and keeps people focused.

Don’t use your cell. It distracts and slows down lab.

Don’t leave the room to text/call people. You leave your lab group hanging and irritate me.

Rotate responsibilities (measuring, using Excel, etc). Everyone will need all basic lab skills in future

classes and on lab exams.

Use Excel formulas to compute your data. Mistakes are quick to correct; less time in the long

run; good job skill.

Don’t tear down apparatus until the end of class. I will check your data for errors and ask you to show

me how you did the experiment.

Check print preview before printing. Saves wasting paper.

Print out a single data sheet for me to look over. Ensures credit and prevents mistakes.

Make sure you understand the conclusion questions. So it is easier to write up later.

Make sure you were checked in & out before

leaving.

Leaving early is the same as an absence!

Write up the lab the same day you acquire data.

Turn in the lab at the appropriate time.

It is fresh in your mind & you get it over with.

Checklist for data tables:

• Times New Roman, 10 or 11 pt, variables are italicized, units are not italicized

• the δ symbol, the ∆ symbol, and parentheses are not italicized

• numeric subscripts are not italicized while letters in subscripts are italicized (m1 or mexp)

• the %precision and %difference do not have units (everything else does have units)!

• the cells which will contain data are centered with borders (Dom has more strict guidelines here)

• constants (the same for all experiments) are written at the top in a separate table

• prefixes, subscripts, and superscripts are appropriately used

• the number of sig figs is appropriate for all cells (based on measuring device, calculations, etc)

• error calculations only have one sig fig (I use two for numbers over whose first digit starts with a 1)

Checklist for graphs:

• gridlines removed

• graph has title using variables (in italics) but no

units in title

• axis labels have variables in italics with units

not italicized

• graph fills the entire field (graph size should be

about 1/3 to 1/2 a page)

• no legend for single set of data (a legend is only

used for multiple data sets on a single graph)

• if trendline is shown there is an equation with R2

value on the chart

• show the data as points only (no connecting

smooth line)

• for graphs with both theoretical & experimental data: theory is a smooth line (no points) while experiment is a

only points (no line)

• the graph and data table are sized such that it fits on a single sheet of paper (not always possible)

• everything except the legend has appropriate subscripts and superscripts

• title is y-axis label versus x-axis label

1

Table of Contents

Error & Excel 3

Introduction to Uncertainty 13

Math Modeling of 1D Motion 23

1st Oral Presentation Ideas:

1D Motion with Tracker 37

Freefall 53

Vectors 57

Projectiles 61

Newton’s Second Law Part I 65

Newton’s Second Law Part II 67

Phriction Phreaque-Out Part I 69

Phriction Phreaque-Out Part II 71

2nd Oral Presentation Ideas:

Newton’s 2nd, Friction, Circular 79

Circular Motion 87

Work & Energy Conservation 89

Inelastic Collisions 93

Elastic Collisions 95

Ballistic Pendulum 97

Rotation 99

3rd Oral Presentation:

Rotation Madness 103

Angular Momentum 123

Buoyant Force 127

Countour Plots & Stepping It Up a Level 129

2

3

Error Estimation and Using Excel

Apparatus: desktop or laptop for each student with Excel

Goal: You are to practice the following topics relating to significant figures, percent errors, propagation of errors,

standard deviation, Excel formulas, and graphing with Excel.

Checklist to Turn in:

• Use engineering paper. Only use one side of the paper (the side without the lines on it).

• Do not cram work into the side or bottom of the page. Write neatly and use plenty of space to make it easy

to follow your steps.

• Do not work side to side on the paper. Each step should come below the previous step to make your work

easy to follow.

• Put a box around your final answers.

• Be sure to always include an unrounded answer AND a rounded final answer (with units) for all

numerical results. Also, be sure to show several steps of work for long calculations like the propagation

of error and standard deviation.

• Show work and solutions for the numbered questions.

• Completed data table with all sections completed. This should include not only the blank cells filled in but

also the two graphs. Erase all red text and check print preview before printing. Try to make everything fit

on one page for today.

To determine the number of sig figs in a number and write it in scientific notation:

Rule1: All non-zero digits are significant.

Rule2: Zeros between other significant figures (bounded zeros) are significant.

Rule3: Leading zeros are left of a decimal point and to the left of all non-zero numbers. They are never

significant.

Rule4: Trailing zeros are right of all non-zero numbers. Trailing zeros are only significant when at least one of

them comes to the right the decimal point!

Examples: 1) The number 1234.5 has five sig figs.

2) The number 0.0445 has three sig figs. The two zeros are called leading zeros.

3) The number 0.0405 also has three sig figs. The first two zero are leading but not the zero between the 4

and the 5. By rule 2 it is significant.

4) The number 435.00 has 5 sig figs. The last two zeros are called trailing zeros. Since they are trailing zeros

with at least one is right of the decimal (rule 4).

5) The number 43500 has three sig figs. Here the two zeros are still trailing zeros but none of them is right of

the decimal point.

6) The number 4350.0 has five sig figs. The trailing zeros are again significant since at least one of them

comes after the decimal.

Exception: The number 4000 appears to have only one sig fig. Often in a physics class this ambiguity is resolved

by assuming such a number has at least three sig figs unless otherwise mentioned. It is best to indicate the sig fig

using and under bar to clarify this.

For example, the number 4000 has an underbar indicating the middle zero is significant. This implies the number

has three sig figs. The middle zero is significant as noted by the underbar, the left zero is significant by rule 2, and

the right zero is not significant by rule 4.

4

Math with sig figs

• When mult/div the smaller number of sig figs is kept.

• When add/sub the least significiant column of sig figs is kept. Watch out: the number of sig figs can

change with add/sub! I recommend writing out add/sub problems vertically to keep track of sig figs.

• In scientific notation, all numbers are always significant!

• Usually you keep at least one extra sig fig for all math work and then round to the appropriate number of

sig figs in the last step. Trig functions can be a notable exception to this rule. When using trig functions,

keep an extra sig fig just in case.

• Keep track of your sig figs using order of operations.

Examples:

Suppose you have 5.68 × 6. 2 = 35. 216 = 35. Notice that 5.68 has three sig figs while 6.2 only had two sig figs.

The smaller number of sig figs is two. When multiplying or dividing two numbers, your final answer will use the

smaller number of sig figs…in this case two. That’s how I knew to round in the ones column for this result.

Suppose you have 4.3+7.1? First write it vertically and put in the sig fig markers like this: 4. 3 +7. 1 11. 4

Notice that the two input numbers each have only two sig figs while the answer actually has three!

Percent Errors

Every measurement has error. This error can be quantified using absolute error or percent error. For example

suppose a length is measured with a meterstick to be

� = 0.504m = 504mm

If you have ever used a meterstick, you know that typically the smallest increment on the meterstick is 1mm. To

some this implies an error of at most ½ of the smallest increment. In this lab, to make error calculations a bit faster,

we will assume that we never really do any better than the smallest increment.

This implies the absolute error in the measurement is

�� = 0.001m = 1mm

To save space while clearly indicating the error we often write this as

� = 0.504 ± 0.001m

Sometimes it is more convenient to express the error as a percent. We say the percent error in L is given by

%������ = ��� = 0.001m0.504m = 0.00198 ≈ 0.002 = 0.2%

When the percent error is used we can then say

� = 0.504m ± 0.2%

5

Propagation of Errors

The back of your lab manual lists a bunch of formulas regarding absolute errors and percent errors. These formulas

can be used to determine how errors in initial measurements carry though, or propagate, to your final calculated

values. It is simply a more trustworthy way of determining sig figs.

Suppose we use kinematics to compute g (the magnitude of acceleration of gravity near earth). We use the

kinematics formula

� = 12���

Someone obtains the following data

To find the value of g we should first rearrange the equation algebraically. True, for a single row of data it is faster

to plug in the numbers right away. However, in most labs, you’ll have a whole bunch of rows and it helps to have a

formula so you can have the computer do the calculations for you. Therefore we first obtain

� = 2���

Example of Propagation of Errors with Sig Fig Rules

� = 2��� = 2(2.00) 0.58!� = 11. 89 = 12m/s�

Notice that the number 2 from the kinematics formula has no sig figs indicated. It is assumed to be a perfect number

with infinite sig figs. Similarly, π in πr2 is assumed to have infinite sig figs.

y (m) δy (m) t (s) δt (s)

2.00 0.05 0.58 0.05

6

Example of Propagation using Absolute Errors

In our example we have the formula � = �$%& . I look on the back of the lab manual to find a propagation formula that

look similar. I see there is a formula for products/quotients with exponents that looks like this:

If q is a product/quotient with powers (say ' = ()/&$*+,*-/./0) the above formula is modified to:

122 = 345� 1(( 6� + 49 1$$ 6� + 478 1,, 6� + 44 1// 6�.

Here q is what is being computed so I identify that with g. Notice that we are interested in how much error g has.

That means we are looking for δg (or δq in the formula). The number 2 is assumed to have no error since it has

infinite sig figs.

This gives me

��� = 9:1 ��� ;� + :2 ��� ;�

Notice that 1$$ is simply the percent error in y. Similarly, this formula is computing the percent error in g! In this

case I want to find the absolute error in g so I will first cross multiply by g to make the formula

�� = �9:1 ��� ;� + :2 ��� ;�

WATCH OUT! Many students forget to cross-multiply that g out front! Notice that without doing this step you are

finding the percent error in g, not the absolute error in g. Finally, plug in your experimental values (including the

experimental value of g). Notice I plug in the unrounded experimental value of g to avoid possible intermediate

rounding errors.

�� = 11.899:1 0.052.00;� + :2 0.050.58;

� = 2.07 ≈ 2m

We express this result as � = 12 ± 2ms�

7

Interesting example of sig fig rules giving contradiction

On occasion the sig fig rules give answers that contradict each other based on the order of operation! This serves as

a reminder to us that sig fig rules are merely an estimate of precision. By using the sig fig rules we will be in the

right ballpark but might not always have the best estimate of precision.

Example: Suppose you are given two spheres with radii 9.6 cm and 5.2 cm respectively. You, a friend, and Susie

McPercento are asked to find the combined volume of the two spheres. You are each told to keep track of the sig

figs.

You decide to the problem as follows:

<%=% = <5 + <� = 43>�58 + 43>��8 = 4.189 × 9. 68 + 4.189 × 5. 28 = 3706 + 589 = 4295 = 4.3 × 108cm8

Your friend decides to do the problem by first factoring out the constants like this:

<%=% = <5 + <� = 43>�58 + 43>��8 = 43>(�58 + ��8) = 4.189 9. 68 + 5. 28! = 4.189 885.7 + 140.6! = 4.189 1025.3! = 4295 = 4.30 × 108cm8

Finally, Susie McPercento determines an estimate of precision based on percent errors. She first makes the

assumption that when someone measures 9.6 cm they are implying 9.6±0.05cm. Similarly she assumes that the

other measurement was 5.2±0.05cm. This implies the percent error on each measurement is given by:

%errin�5 = 0.059.6 × 100% = 0.52%

%errin�� = 0.055.2 × 100% = 0.96%

Since each radius was cubed, Susie expects the error for each radius to contribute three times. Then Susie further

takes the cautious approach and assumes the worst case scenario; Susie assumes that the error in measuring one

radius will not cancel out the error in measuring the other radius. For example, Susie is assuming that either both

radii were measured too large or too small (not one of each). This gives a total error as follows: %err = 3(%errin�5) + 3(%errin��) = 3(0.52%) + 3(0.96%) = 1.56% + 2.88% = 4.44% ≈ 4%

Using this method, Susie decides that the total volume is given by 4295 ± 4.44% = 4295 ± 190 ≈ 4295 ± 200

Susie infers the 3rd digit isn’t significant and the result should be written as 4.3 × 108cm8.

8

Sig Fig Questions

Compute the following using sig fig rules.

Write your final answers with correct sig figs and scientific notation.

1) 30.0 + 80.0 =

2) 4.0(30.00)0.002 =

3) (2.198 − 2.207)80.010� =

Practicing with % errors and fractional errors

4) A dog gets on a scale at the vet. It keeps wiggling and causes the scale reading to fluctuate. While it is

fluctuating, the lowest scale reading is 9.7 kg while the highest is 10.3 kg. What is the best value to record

for mass of the dog? Write your answer as m = # ± # kg

5) Abbey measures my height as 74±1 inches. Billy measures my height as 188±1 cm. We know 1 inch =

2.54 cm. What is the % error for each person’s measurement?

6) Susie measures a current in her circuit as 1.234 A. She states the % error in measuring the current is 0.3%.

This means there is a range of possible values she measures for the current. What is this range of values?

Hint: use the % error to get a ± number and use that to find a range of values.

Practicing Propagation of Errors:

Suppose you are given the formula

E = 9 F88F��F5 ��

You are also given the following information:

m1 (kg) m2 (kg) m3 (kg) g (m/s2) L (m)

1.00 2.00 3.00 9.81 1.00

δm1 (kg) δm2 (kg) δm3 (kg) δg (m/s2) δL (m)

0.03 0.05 0.07 0.01 0.02

7) Use the first row of data to determine an experimental value for v (with units). Be sure to keep track of sig

figs. Clearly list your unrounded and rounded final answers with units.

8) Use propagation of error formulas to compute the percent error in this experimental value of v. Be sure to

show each painful step. Include both an unrounded and rounded final answer.

9) Also determine the error in your experimental value of v. Be sure to use the unrounded answer from your

previous step to avoid intermediate rounding error.

10) If you did the previous step correctly, you will find that δv = 0.4 m/s. This means your experimental value

of v should be rounded to the tenth’s column. Think about it: errors in the hundredths column are

unimportant compared to errors in the tenths column. We shouldn’t include any digits beyond the tenths

column!

11) Write your final result as v = ?.? ± 0.4 m/s.

9

Compare sig fig rules to full propagation of error for previous example

12) Suppose the accepted value of v for the previous problem is 7.60 m/s. Quantify the percent

difference/discrepancy using a formula from the back of your lab manual. Show your work on paper. Do

not force the result to be positive.

13) Read about target diagrams in your lab manual. There should be information just inside the back cover of

the manual. Draw a target diagram based on the % error from the propagation of error calculation done

above. Be sure the diagram is drawn to scale, at least ¼ page in size, neat, and well-labeled. Hint: to

draw your circles to scale, use the gridlines of the engineering paper. Let each block of the paper represent

1%. That means the 5% ring has a five block radius and the 10% has a ten block radius. Hint2: if you have

a negative percent difference the bullet is located below the bulls-eye.

14) Consider your result in question 4 using sig fig rules only. You should have found that v = 8.14 m/s. This

implies an error in the hundredths column. We assume then that sig fig rules imply an error of ±0.01m/s.

If this error was used (instead of the result of question 6), how would the target diagram change? Would

the bullet be farther, closer, or the same distance from the center? Would the bullet be larger, smaller, or

the same radius?

10

Comparing Standard Deviation to Propagation of Error

Suppose you have the following data table. Here is a case where, instead of using a single data

point, an experiment has been repeated several times. This allows us to compute the error in

different ways. Fortunately, most of the time we will use the computer to do the bulk of our

computations.

The basic idea: We compute the error in the average value of y in two ways. The first method

uses the results derived from our propagation of error formulas. The second method uses the

statistical spread of the measured values. One of these methods is usually significantly larger than the other one.

We see which error result is larger and use that as our conservative estimate.

Computing the error using method 1:

15) Determine the average value of y. Keep at least four sig figs just in case and think about the units!

16) Determine the absolute error in the average using the formula ��G = 1$√I where N is the number of

measurements. Think: does it have units?

17) Determine the % error for �G using your result from the previous step. Think: does this have units?

Verify the final result is �JKL = �G = 1.310F ± 0.3%.

Computing the error using method 2: 18) Write down the average value of y from you found in a previous step.

19) Determine the standard deviation of y. The symbol for standard deviation of y is σy. Use the formula on

the back of the lab manual near the bottom. Show all steps by hand this time.

20) The best statistical estimate for error of each individual measurement is σy. To get the standard error in the

average, we must divide σy by √M.

21) Determine the percent error based on standard error using %��� = NO √IP$G × 100%.

Verify the final result is �G = 1.30m ± 1.6% = 1.30m ± 2%. Note: for errors starting with the number 1,

sometimes people keep a second sig fig.

Evaluate which method to use:

22) Look at the above results and it is easy to see that the second result is more conservative. If we use the first

result, we are underestimating our errors and implying greater precision than appropriate. Clearly state

which case which error estimate we should use for this example.

y (m) δy (m)

1.34 0.01

1.25 0.01

1.37 0.01

1.30 0.01

1.29 0.01

11

Practice with Excel

Your instructor will provide you the location of an Excel worksheet to use with today’s lab. Open this file then

follow the instructions below.

Start with Section 1 which has the same data table we used in the previous problem.

• Make a formula that computes all of the percent errors in the appropriate column. Ask your instructor for

help if you don’t know how to do this.

• Use the “AVERAGE” command to compute the average of the 5 y values.

• Use the “STDEV” command to compute the standard deviation of the 5 y values. WATCH OUT: be sure

to only include the 5 data points and not the average.

• Repeat the same procedure on the second data table. Notice the numbers appear to be less random.

Notice when you get essentially the same measurement several times, the error in the average is determined using

δy/√M. When your measurements have a spread greater than the error of the measuring device, the error in the

average is determined using σy/√M. You can express your average result as �G ± Q RS√I or �G ± Q TU√I.

Go to Section 2 which has a data table with of v and t.

• Make a plot of v versus t. I am calling this a Type I graph. Look in the table of contents of your lab

manual for Sample Graph Type I. Follow those instructions.

Go to Section 3 which has values of F and rexp.

• The top two rows of this section include the constants G, m, and M.

• First use an Excel to compute the theoretical column of data for rth. Use the theoretical function

�%V = 9WFXY

• Use the propagation of error formulas to compute the error for each rth. Notice you are told all errors

except δF are negligible.

• Then make a column showing the percent difference between each rexp and rth. Please include any minus

signs so I can quickly tell if rexp is too high or too low. The formula to use is on the back cover of your lab

manual.

• Make a Type 2 graph that shows r vs F. Be sure to read the table of contents to find the section in your lab

manual called Sample Type II Graph.

12

13

Introduction to Experimental Uncertainty

Apparatus: tape measure and/or metersticks, stopwatch, golf ball, ping pong ball, plum bobs

Theory: Error analysis or experimental uncertainty is the study and evaluation of uncertainty in measurement.

Experience has shown that no measurement, however carefully made, can be completely free of uncertainties. Since

the whole structure and application of science depends on measurements, it is extremely important to be able to

evaluate these uncertainties and to keep them to a minimum. In science, the word “error” does not carry the usual

connotations of “mistake” or “blunder.” “Error” in a scientific measurement means the inevitable uncertainty that

exists in all measurements. As such, errors are not mistakes; you cannot avoid them by being very careful. The best

you can hope to do is to ensure that the experimental errors are as small as reasonably possible, and to have some

reliable estimate of the experimental errors. We shall use the term “error” exclusively in the sense of “uncertainty,”

and treat the two words as being interchangeable.

Random and Systematic Errors

Not all types of experimental uncertainties can be assessed by statistical analysis based on repeated

measurements. For this reason, uncertainties are classified into two groups: the random uncertainties, which can be

treated statistically; and the systematic uncertainties, which cannot. Experimental uncertainties that can be revealed

by repeating the measurements are called random errors; those that cannot be revealed in this way are called

systematic errors. To illustrate this distinction, let us consider an example. Suppose first that we time a revolution of

a steadily rotating turntable. One source of error will be our reaction time in starting and stopping the watch. If our

reaction time were always exactly the same, these two delays would cancel one another. In practice, however, our

reaction time will vary. We may delay more in starting, and so underestimate the time of a revolution; or we may

delay more in stopping, and so overestimate the time. Since either possibility is equally likely, the sign of the effect

is random. If we repeat the measurement several times, we will sometimes overestimate and sometimes

underestimate. Thus our variable reaction time will show up as a variation of the answers found. By analyzing the

spread in result statistically, we can get a very reliable estimate of this kind of error. On the other hand, if our

stopwatch is running consistently slow, then all our times will be underestimates, and no amount of repetition (with

the same watch) will reveal this source of error. This kind of error is called systematic, because it always pushes our

result in the same direction. Systematic errors cannot be discovered by the kind of statistical analysis that we will be

discussing below.

The treatment of random errors is quite different from that of systematic errors. The statistical methods

described in the following sections give a reliable estimate of the random uncertainties, and, as we shall see, provide

a well-defined procedure for reducing them. On the other hand, systematic uncertainties are hard to evaluate, and

even to detect. Unfortunately, in introductory physics laboratory courses, such evaluations are rarely possible; so

the treatment of systematic errors is often awkward. For now, we will discuss experiments in which all sources of

systematic errors have been identified and made much smaller than the required precision.

14

The Mean and Stand Deviation

Suppose we need to measure some quantity x, and have identified all sources of systematic error and

reduced them to a negligible level. Since all remaining sources of uncertainty are random, we should be able to

detect them by repeating the measurement several times. We might, for example, make the measurement five times

and find the results

71, 72, 72, 73, 71

(where, for convenience, we have omitted any units). The first question that we address is as follows: given the five

measured values, what should we take for our best estimate xbest of the quantity x? It seems reasonable that our best

estimate would be the average or mean x of the five values. Thus

71 72 72 73 7171.8

5bestx x

+ + + += = =

More generally, suppose we make N measurements of the quantity x, and find the N values 1 2, ,..., .Nx x x The best

value or average is

i

best

xx x

N= =

∑

The concept of the average is almost certainly familiar to most students. Our next concept, that of the

standard deviation, is probably less so. The standard deviation of the measurements 1 2, ,..., Nx x x is an estimate

of the average uncertainty of the measurements 1 2, ,..., ,Nx x x and is arrived at as follows.

Given that the average x is our best estimate of the quantity x, it is natural to consider the difference

i ix x d− = . This difference, often called the deviation (or variance) of xi from x , tells us how much the ith

measurement xi differs from the average x . If the deviations are all very small, then our measurements are all close

together and are presumably very precise. If some of the deviations are large, then our measurements are obviously

not so precise.

To be sure we understand the idea of the deviation, let us calculate the deviations for the set of five

measurements reported in the table below.

Table 1. Calculation of Deviations

Trial number, i Measured value, ix

Deviation,

i id x x= −

1 71 -0.8

2 72 0.2

3 72 0.2

4 73 1.2

5 71 -0.8

71.8x = 0.0d =

Notice that the deviations are not (of course) all the same size; id is small if the ith measurement xi

happens to be close to x , but id is large if xi is far from x . Notice also that some of id are positive and some

negative, since some of the xi are bound to be higher than the average x , and some are bound to be lower. To

15

estimate the average reliability of the measurements 1 5,..., ,x x we might naturally try averaging the deviations id .

Unfortunately, as a glace at Table 1 shows, the average of the deviations is zero. In fact, this will be the case for any

set of measurements 1 2, ,..., ,Nx x x since the definitions of the average x ensures that id is sometimes positive and

sometimes negative in just such a way that d is zero. Obviously, then, the average of the deviations is not a useful

way to characterize the reliability of the measurements 1 2, ,..., Nx x x .

The best way to avoid this annoyance is to square all the deviations, which will create a set of positive

numbers, and then average these numbers. If we then take the square root of the result, we obtain a quantity with the

same units as x itself. This is called the standard deviation of 1 2, ,..., Nx x x , and is denoted by xσ :

2

1

1( )

N

x i

i

dN

σ=

= ∑

With this definition, the standard deviation can be described as the root mean square (RMS) deviation of the

measurements 1 2, ,..., Nx x x . Unfortunately, there is an alternative definition of the standard deviation. There are

theoretical arguments for replacing the factor N by (N-1) and defining the standard deviation xσ of 1 2, ,..., Nx x x

as

2 2

1 1

1 1( ) ( )

1 1

N N

x i i

i i

d x xN N

σ= =

= = −− −∑ ∑

For the five measurements of Table 1, we calculate xσ :

52 2

1 1

1 1 1( ) ( ) (0.64 0.04 0.04 1.44 0.64) 0.84

1 5 1 4

N

x i i

i i

d x xN

σ= =

= = − = + + + + ≈− −∑ ∑

Thus the average uncertainty of the five measurements is about 0.84.

16

Suppose that we obtain the values 1 2, ,..., Nx x x and compute x and xσ . If we then make one more

measurement (using the same equipment), there is a 68 percent probability that the new measurement will be within

xσ of x . Now, if the original number of measurements N was large, then x should be a very reliable estimate for

the actual value of x. Therefore we can say that there is a 68 percent probability that a single measurement will be

within xσ of the actual value. Clearly xσ means exactly what we have used the term “uncertainty” to mean in the

preceding sections. If we make one measurement of x, then the uncertainty associated with this measurement can be

taken to be xσ ; and with this choice we are 68 percent confident that our measurement is within xσ of the correct

answer.

68%

1.5 87%

2

x best

x best

x

The number x will overlap the best estimate x approximately of the time

The number x will overlap the best estimate x approximately of the time

The number x will ove

σ

σ

σ

±

±

± 95% best

rlap the best estimate x approximately of the time

In the previous example of Table 1, our best estimate at one standard deviation is

71.8 0.84 71.8 0.8

71.0 72.6

best x

best

x x

x

σ= ± = ± ≅ ±

⇒ ≤ ≤

If we make one more measurement, one has a 68% confident level that it will be between 71.0 and 72.6.

Interpretation of the Standard Deviation

The first problem in discussing measurements that are repeated many times is to find a way to handle and

display the many values obtained. One convenient method is to use a distribution or histogram. Suppose, for

instance, that we were to make ten measurements of some length x. We might obtain the values (all in cm)

26, 24, 26, 28, 23, 24, 25, 24, 26, 25

Written in this way, these ten numbers convey fairly little information; and if we were to record many more

measurements in this, the result would be a confusing jungle of numbers. Obviously a better system is called for.

As a first step we can reorganize the numbers in ascending order,

23, 24, 24, 24, 25, 25, 26, 26, 26, 28.

Then we can record the different values of x obtained, together with the number of times each value was found in a

table.

Table 2

Different values 23 24 25 26 27 28

Number of times found 1 3 2 3 0 1

17

The distribution of our measurements can be graphically displayed in a histogram. This is just a plot of the

frequency f(x) against xi, with different measured values xi potted along the horizontal axis, and the frequency that

each xi was obtained indicated the height of the vertical bar drawn above xi.

Figure 1

If one increases the number of measurements, then the histogram begins to take on a bell-shaped curve that is called

Gaussian or Normal.

Figure 2

If the measurement under consideration is very precise, then all the values obtain will be very close to the

actual value of x; so the histogram of results will be narrowly peaked, like the solid curve in Fig. 3. If the

measurement is of low precision, then the values found will be widely spread out, and the distribution will be broad

and low, like the dashed curve in Fig. 3.

Figure 3

18

The mathematical function that describes the bell-shaped curve is called the normal distribution or

Gaussian function. The general form of this function is

2 2/ 2( ) ,xf x e σ−∝

where σ is a fixed parameter that we will call the width parameter. When x = 0, the Gaussian function is equal to

one. The function is symmetric about x = 0, since it has the same value for x and – x. Note that as x moves away

from zero, the bell shape is wide if σ is large and narrow if σ is small:

Figure 4

To obtain a bell-shaped curve centered on some other point, say x x= , we merely replace x by x x− in the

Gaussian function:

2 2( ) / 2( ) .x xf x e σ− −∝

The Gaussian function f(x) for measurement of some quantity x tells us the probability of obtaining any given value

of x. Specially, the integral ( )b

af x dx∫ is the probability that any one measurement gives an answer in the range

.a x b≤ ≤ In the figure below, the shaded area between xx σ± is the probability of a measurement falling within

one standard deviation of x . (Note that we could have also found the probability for an answer falling within 2 xσ

of x , within 3 xσ of x , etc.)

Figure 5

19

Standard Error (SE): the best estimate of the uncertainty of the mean

Let’s summarize what we have done up to now: If x1, x2…, xN are the results of N measurements of the

same quantity x, then, our best estimate for the quantity x is their mean, x . The standard deviation xσ characterizes

the average uncertainty of the separate measurements x1, x2…, xN. The standard deviation tells us the spread or

uncertainty in our measured values, but it does not tell us how close our mean is likely to be to the true mean. To

understand this, consider the following observation: if many different lab groups performed the same set of

measurements, each group would obtain a different mean, and a different standard deviation. However (and this is

the crucial point), the spread between the means of the different groups should be less than the spread between the

measurements of each individual group. For example, suppose that five different lab groups obtained the following

means ( v ) and standard deviations ( vσ ) for a set of measurements of the speed of a fired projectile:

Group 1: v = 10.5 m/s; vσ = 0.5 m/s

Group 2: v = 10.6 m/s; vσ = 0.4 m/s

Group 3: v = 10.8 m/s; vσ = 0.6 m/s

Group 4: v = 10.3 m/s; vσ = 0.3 m/s

Group 5: v = 10.2 m/s; vσ = 0.5 m/s

The standard deviation of the means of the five different lab groups is 0.2 m/s, which is less than the spread

(standard deviation) of the measurements of any individual group.

The standard deviation of the set of means is called the standard error (SE) and is the best estimate of the

uncertainty of the mean from any individual group. Graphically, the frequency distribution of such a set of means

would nearly always be a bell-shaped normal distribution.

Statistical theory tells us that even when we have only a single set of measurements, we can estimate the

uncertainty in the mean by calculating the standard error (SE), defined as:

SE = uncertainty of the mean = N

σ

Therefore, we can state our final answer for the measurement of some quantity x as

( ) x

bestN

value of x x xSEσ

= ± = ±

As an example, we can consider the 5 measurements reported in Table 1. We saw that these measurements had a

mean of x = 71.8 and the standard deviation is σx = 0.84. Therefore, the standard error is

/ 5 0.38xSE σ= =

and our final answer, based on these 5 measurements, would be that the lengths are

71.8 0.4

71.4 72.2best

xx x

x

σ= ± = ±

⇒ ≤ ≤

An important feature of the standard error is the factor √N in the denominator. The standard deviation σx

represents the average uncertainty in the individual measurements x1, x2…, xN. If we were to make some more

20

measurements using the same technique, we expect that the standard deviation σx would not change appreciably. On

the other hand, the standard error σx/√N would slowly decrease as we increase N. This is just what we would expect.

If we make more measurements before computing an average, we should naturally expect the final result to be more

reliable, and this is just what the denominator √N guarantees. This provides one obvious way to improve the

precision of our measurements.

Summary:

If we measure a quantity x several times, then our best estimate for x is the mean x , and the standard error,

SE, is our best measure of the uncertainty in the mean.

In lab, we can use the following criteria to determine whether an experimental measurement of x is

consistent with the theoretical prediction, xthy:

If 2 2thyx SE x x SE− ≤ ≤ + , then the experimental result and the

theoretical prediction are said to be consistent. That is, if the theoretical

prediction falls within two standard errors of the mean experimental value, then

the experimental result is considered consistent with the model.

21

Procedure: Form groups of 3-4 people. Each person in your group will make each measurement. Write down your

measurements and do not let your partners see your result. Try to make your own measurements independent – that

is, forget what you measured the first time when you make your next measurement. When everyone has recorded all

measurements, then put your data together on a single Excel spreadsheet.

1) Predict the time for a ball dropped from rest to fall a distance of 2.00m. You may assume there is no air

resistance.

2) Drop a golf ball from a 2.00 m height and carefully time how long the ball is in the air. Repeat this 15

times and record each time. Each member of the group should do this step independently. For

example, a 4 person group should have 60 measurements at this stage.

3) Organize your data and enter it in Excel.

4) Determine the average time �̅ for all of your group’s measurements. Hint: use the AVERAGE function in

Excel. Write this value on the whiteboard to share with the class.

5) Determine the standard deviation [% for all of your group’s measurements. Hint: use the STDEV function

in Excel.

6) The standard error can be determined from the standard deviation of the set of class averages. Class this

value SEclass. In this calculation N is the number of average values on the white board.

7) For a single set of measurements (i.e. only your group’s data), the standard error is given by N\√I. Here N

represents the number of measurements your group obtained. Call this value SEgroup.

8) Repeat the above steps for a ping pong ball.

Questions:

1) Regarding only your group’s data, we expect approximately 68% of your measurements to fall within one

standard deviation of your mean. This means 68% of your measured values should be between the values

of �̅ − [% and �̅ + [%. What percentage of your measured values actually fall in this range? Does your

percentage agree well with the 68% figure we expect?

2) Approximately 95% of your measurements to fall within two standard deviations of your mean. This

means 95% of your measured values should be between the values of �̅ − 2[% and �̅ + 2[%. What

percentage of your measured values actually fall in this range? Does your percentage agree well with the

95% figure we expect?

3) How does SEclass compare to SEgroup? Are the two values within 1% of each other? 10%? Quantify your

comparsion.

4) For each type of ball, are your predicted times in good agreement with the theoretical value? Said another

way, does the theoretical result fall within the 2SE confidence interval? Said yet another way, is �̅ − 2]^ < �%V < �̅ + 2]^

for each type of ball?

22

23

Mathematical Modeling of 1D Motion

Apparatus: desktop or laptop for each student

Goal: Today’s goal is to use 1D motion equations to make plots and analyze the features of those plots. I also hope

you will learn about using Excel formulas to do simple tasks such as numerical differentiation. Each task has

activities/questions to submit or show to your instructor.

Task 1: Generate a simple data table and plot y vs t and vy vs t. Open the Excel table from the class website.

Notice in the bottom, left corner of the screen there are several tabs named “Task1”, “Task2”, etc. Verify you are on

the tab for Task1.

The table on the Task1 tab shows a row of constants. These constants represent the initial position, initial velocity,

and constant acceleration of an object. Also, you see data columns for elapsed time (t), position (y), and velocity

(vy). From kinematics we know these variables are all related by

� = �` + E`$� + 12 a$�� (1)

E$ = E`$ + a$� (2)

Use these equations to generate theoretical data. Be sure to properly reference the constants using Excel formulas as

this will be important later.

• After generating the data, create plots of y vs t and vy vs t.

• For Task 1, do not worry about axis titles, labels,

gridlines, etc.

• You should still add an appropriate trendline to each.

• Also, delete the legends for each graph to economize

space.

• Finally, do your best to make sure the time increments of

each axis line up as closely as possible.

Once completed, you should have work that looks

similar to this:

• Does the slope of the vy vs t graph make

sense?

• Does the concavity of the y vs t graph make

sense?

DOUBLE CHECK: Did you remember to put the

appropriate trendlines on each graph AND display the

equations? This is needed to answer some questions

later.

Tip: To make the graph go to the insert tab

and look for the scatter plot (dots only).

Never use a line graph in this course.

24

Now comes the fun part. If you’ve done your work properly, we should be able to change the row of constants and

see how the graphs are affected immediately. Use this page to check if your equations and graphs are working.

Do this quickly then move on to the questions for Task 1 on the next page.

Leave the initial position as 0.0 but change

the initial velocity to +4.9. All the data

will automatically update as well as the

graphs by changing only the one number!

You should see the screen change to

something that looks like this:

• At what time does the object

reach max height?

• What do you notice about the

vertical velocity at that same

point in time?

• Does the slope of the vy vs t

graph make sense?

• Does the concavity of the y vs t

graph make sense?

Suppose someone is falling down but

gets bounced upwards by a trampoline.

Perhaps she started 1.0 m above the

ground, was initially moving downwards

at a rate of 2.0 m/s but had an

approximately constant upwards

acceleration of 5.0 m/s.

In this case, after properly fixing the

constants, you should see:

• Here does she reach a max or a

min height?

• Notice that again the velocity is

zero when she changes

direction.

• Does the slope of the vy vs t

graph make sense?

• Does the concavity of the y vs t

graph make sense?

25

Task 1 Questions/Activities:

T1.1: Now we are going to switch from up and down to right and left. You can still use y vs t…just imagine you

rotated your coordinate system. Consider a situation where a person is initially moving right and slowing down, but

by 1.0 s the person is moving left and speeding up. The initial position is negative, the person reaches a positive

position before turning around, then returns to a negative position before 1.0 s has elapsed. Determine an

appropriate set of initial constants that causes such a path to occur. Verify these constants work by plugging them

into your mathematical model in XL. Write down a set of three constants that work (you will need to show these to

your instructor later for credit).

T1.2: Consider a situation where an object is always moving left. The object never changes speed but within in the

first 1.0 seconds the position is sometimes negative and sometimes positive. Write down a set of constants that

model such a situation.

Finally, consider an object on a table 50 cm above the ground with an initial speed is 6.3 m/s.

You are told the object rises in the air then falls back to earth under the influence of gravity.

You are told that air resistance is negligible.

Make sure you have the trendlines on your graph.

Check print preview to ensure your works fits on a single page.

Hint: if you click print preview and then close it out the page margins will appear on your Excel worksheet.

Make your graphs as large as possible while still fitting within the page margins and not obscuring the data table.

Print out the y vs t andvy vs t plots.

Use the graphs (slopes, areas, values, etc) to answer the following questions.

You shouldn’t be using the kinematic equations.

T1.3a: How far above the ground is the particle at 1.0 s? Read it off the graph, don’t compute.

T1.3b: Is the particle’s displacement after 1.0 s? Use the area under one of the graphs!

T1.3c: What total distance has the particle traveled in 1.0 s? I estimated that v = 0 at t = 0.64 s.

T1.3d: What max height above the ground was reached by the particle? Read it off the graph, don’t compute.

T1.3e: Write down the trendline equation from the position versus time graph. Explain how each of the coefficients

relates to the constants of the model (which one is yi, vi, …watch out for ay!).

T1.3f: Write down the trendline from the velocity versus time graph and explain how each of the coefficients

relates to the constants of the model.

T1.3g: For this model, does ay = g or ay = -g? Define g using words and state its value with units for credit.

26

Task 2: Numerical differentiation

Go to the bottom, left corner of the screen and select the Task2 tab.

The data table should already have a set of t’s and y’s obtained by a scientist in an experiment. Let’s pretend she

used video footage of a hummingbird flying back and forth. In the video footage, the film is comprised of still

images that each last for 1/20th of a seconds (0.050 s). By noting the bird’s location in the images, an estimate of

position for each time was made.

You know that velocity is the derivative of position. Unfortunately, we don’t have a mathematical function so we

can’t take a derivative. We can, however, get an estimate using the following formula:

E$ = b�b� ≈ Δ�Δ�

To understand how to use this formula we need to be able

to discuss individual elements of the data table. See the

table at right. The first number in each column will be

denoted with a 0-subscript. To be clear, the first time is t0

and the first position would thus be y0. As you go down in

rows, the subscripts increase accordingly. Verify this

makes sense by considering t5, v6, etc in the figure.

In order to use E$ ≈ d$d% , we want to be clever. Suppose, for instance, you are trying to determine v6. You might

think to try Ee ≈ d$d% = $fg$-%fg%-…this is not always the best choice. You could similarly try a forward-looking

difference formula given by Ee ≈ d$d% = $hg$f%hg%f . Finally, you could consider the difference between the point just after

6 and just before 6. This results in the equation Ee ≈ d$d% = $hg$-%hg%- . In more general terms we say:

Ei ≈ Δ�Δ� = �ij5 − �ig5�ij5 − �ig5

This is the formula we will use today to determine each value in the vy-column. To make this work in the Excel

spreadsheet in cell C10 we should type “=(B11-B9)/(A11-A9)”. WATCH OUT: When using this formula, we are

unable to obtain a reasonable value for the first and last vy. That is discussed later.

Similarly, a set of data can be obtained for the ay-column using

ai ≈ ΔEΔ� = Eij5 − Eig5�ij5 − �ig5

This is t0

This is t5

This is t7

This is y5

This is y7

This is v6

27

• Create the values for vy and ay using the

appropriate XL formulas.

• Make graphs of y vs t, vy vs t and ay vs t.

• Notice I tweaked the yt-plot axis so the

graphs line up better.

• Delete the legends but don’t bother with

trendlines.

• Do your best to line up the time

increments as well.

You should end up with something similar to the

figure shown at right.

Task 2 Questions/Activities:

T2.1: Explain why it doesn’t make sense to compute v0, a0, or a1 with the formula we used.

T2.2: Describe the motion of that particle at time t = 0.200s. Is it moving up, down or sitting still? Is it speeding

up, slowing down, or traveling with constant speed? Defend your answer by discussing the signs, slopes, and/or

concavities of the y vs t and vy vs t graphs. You can check the signs in the table as well!

T2.3: Describe the motion of that particle at time t = 0.900s. Is it moving up, down or sitting still? Is it speeding

up, slowing down, or traveling with constant speed? Defend your answer by discussing the signs, slopes, and/or

concavities of the y vs t and vy vs t graphs. You can check the signs in the table as well!

T2.4: Describe the motion of that particle between times t = 0.450s and 0.500s. You may assume the acceleration is

roughly constant on that time interval. To me, it looks like the acceleration is about 3g’s (three times the value of g).

Is it moving up, down or sitting still? Is it speeding up, slowing down, or traveling with constant speed? Defend

your answer by discussing the signs, slopes, and/or concavities of the y vs t and vy vs t graphs.

T2.5: Notice the acceleration changes wildly between 0.3 and 0.7 s. This seems quite improbable. Meanwhile, both

the y vs t and vy vs t graphs don’t seem that unusual. The ay graph appears to have much more error than the vy

graph even though they are both made from the same data table. This is because the calculation of ay is an

approximation on top of a previous approximation. By using a smaller time increment to record data, the ay vs t

graph would look a lot more sensible. Imagine how bad jerk versus time would look! HAH! You don’t have to

write anything down but at least I made you read this!

28

Task3: An acceleration that changes in time (piece-wise functions)

Suppose a car starts from rest and accelerates for 3.00 seconds at a rate of 6.00m/s2. Call this stage 1.

In stage 2 the car continues but with the smaller acceleration of 2.00 m/s2 for 2.00 additional seconds.

In stage 3 the car stops accelerating but continues on at constant speed for 4.00 s.

It would be easy to calculate by hand the total distance traveled by the car using kinematics formulas.

In today’s lab, however, I want you to plot the position, velocity, and acceleration versus time.

We can use the kinematics equations but we need to be a little clever when typing them into XL.

For the first part we simply use:

k5l = k5` + E5`(� + 12 a5(�� (1)

E5l( = E5`( + a5(� (2)

For the next piece of the function, we need to reset the time variable. To do this we simply subtract off the already

elapsed 3.00 s from each t in the equations. The initial velocity for the second piece of the function (v2ix) is equal to

the final velocity of the first part of the function (v1fx). Similarly y1f = y2i. Of course, the acceleration also changes.

This gives

k�l = k�` + E�`((� − 3) + 12 a�((� − 3)� = k5l + E5l((� − 3) + 12 a�((� − 3)� (1)

E�l( = E�`( + a�((� − 3) = E5l( + a�((� − 3) (2)

For the next piece of the function, we need to reset the clock by 5 seconds (the initial 3 seconds of stage 1 plus an

additional 2 seconds for stage 2). We need to determine the initial conditions for stage 3 (x3i and v3xi) in a manner

similar to the previous stage.

• Upon using the formulas, you should

be able to make a set of graphs that

look like those shown below.

• Delete the legends, ignore axis labels,

etc. Add vertical gridlines.

• Use data points connected by a smooth

line (no trendlines make sense here as

the function changes every few

seconds).

• Hit print preview.

• Make the three graphs as large as

possible while still fitting on one page.

• Print a copy.

29

Task 3: Questions/Activities

T3.1: Determine the total displacement of the car by looking at the graph. Don’t do any calculations.

T3.2: Use a straight edge to split up the vt graph into chunks

(see the figure). Figure out the area of each chunk. Add up the

areas. Think: how should this compare to your previous

answer?

T3.3: Notice that distance and displacement can be used

interchangeably in this problem. That is not always the case!

In fact, Task 2 showed an instance when displacement and

distance were not equal. Generally speaking, for which 1D

motion problems will distance not equal displacement?

A final note regarding piece-wise functions: in reality, the

acceleration would not change abruptly from 6.00 m/s2 to 2.00

m/s2. The acceleration would gradually change, perhaps over

an interval of 0.2 sec or so. This would smooth out all the graphs a little bit. As long as the transition time between

accelerations is small compared to the times of the duration of the stages, this piece-wise technique accurately

models the situation.

30

Task4: Plotting position (and velocity) versus time for two objects simultaneously

Suppose a truck (labeled 1 in the figure) drives at a constant

speed of 20.0 m/s.

At the instant it passes a motorcycle, the motorcycle accelerates

from rest with acceleration 5.0 m/s2. The motorcycle is labeled 2

in the figure.

The above information allows us to write down the equations of

position and velocity as a function of time for each object. For

example: k5(�) = k5l = 20.0� E5(�) = E5l = 20.0

• Determine the appropriate equations for k�(�) and E�(�). • Click on Task 4 tab in Excel.

• Use these equations to plot x vs t.

• Include both x1 and x2 on the same graph. If you need help getting two data series on the same graph, ask

your instructor for help.

• Do not delete the legend this time! When more than one data set is represented on a graph a legend

is appropriate.

• Plot only smooth lines, no data points.

• Do something similar for v vs. t.

• Your XL sheet should now

look like this:

2

1 Constant speed

Constant acceleration

starts from rest

31

Task 4: Questions/Activities

T4.1: In the top graph we see that one of the curves is concave up while the other is a straight line. Which line

represents the motor cycle? Explain.

T4.2: In the 2nd graph, one of the lines is upwardly sloping while the other is flat (no slope). Which line represents

the motor cycle? Explain.

T4.3: At what time does the motorcycle start to get closer to (rather than farther behind) the truck? Hint: You can

read this information off one of the graphs without any computation. Verify with the data table.

T4.4: Between 0 and 8 seconds, what is the maximum distance between the truck and the motorcycle (after the truck

passes the motorcycle)?

T4.5: At what time will the motorcyclist finally catch up to the truck? Hint: use one of the graphs and verify with

the data table.

T4.6: Determine the speed of the motorcyclist as he overtakes the truck. Hint: use the graph or the data table at the

appropriate time. Answer in miles per hour.

T4.7: How many g’s is the motorcyclist experiencing? This is determined by taking the ratio a/g. for example, if � = 10 mn& and a = 25 mn& then we could also say a = 2.5�′s.

T4.8: Based on your previous results, do you think this problem is realistic? Support your answer.

BONUS

A piece-wise function might be more appropriate for the motorcycle. Say it has an acceleration of 5.0 m/s2 for the

first 6 seconds and then maintains constant speed. The motorcycle now overtakes the truck at 9.0 s instead of 8.0 s.

Notice that it is very difficult to distinguish where the parabolic part of the graph ends and the linear part begins for

the xt graph of the motorcycle. We see the transition point much more clearly on the vt graph of the motorcycle.

This is typical. Notice that the cop only reaches 30 m/s ≈ 67 mph instead of 40 m/s ≈ 89 mph.

Note: the figure below is for the BONUS only. The figure on the previous page applies to the questions at the

top of this page.

Another optional element: Suppose it takes the cop some small reaction time ∆t before he starts the acceleration

phase. Incorporate this into your model and see how the graphs change as you gradually increase ∆t from 0.00 to

1.00 sec.

32

Task 5: A falling particle with air resistance

Suppose a tiny particle is released from rest from a very tall building. You are told the drag force increases linearly

versus time on the falling object. In certain cases the force equation for such an object is given by Yip% = F� − qE

Later in physics we learn that acceleration is given by

a = Yip%F

For our object that falls while experiencing air resistance we find

a = � − qFE

(1)

As the object falls its speed will increase. As speed increases, notice the acceleration will decrease. This is clearly

not a constant acceleration problem. Therefore we cannot use the constant acceleration equations of motion. We

can find the equations of y(t) and vy(t) using separation of variables!

As good practice for an exam, use separation of variables to show

�(�) = Er� − Er Fq :1 − �gst%;

which is also written as

�(�) = Er� − Er Fq :1 − exp(− qF �);

Hint: in Excel formulas “e” means “×10 to the” while “exp” means “e to the”

(2)

E(�) = Er :1 − exp(− qF �); (3)

wℎ���Er = F�q (4)

Note: in the equations above I have assumed that down is the positive direction, not up. I did this to reduce the

number of minus signs cluttering up the equations. I also set yi = 0. Furthermore, vT is called terminal velocity.

Open the Task 5 tab in the Excel spreadsheet.

Use the given constants and equations (2) through (4) to make initial yt and vt plots.

Use equation (1) and your column of calculated v’s to make an at plot.

The next page shows you what this should look like…

33

Here is the graph.

Task 5: Questions/Activities

T5.0: Figure out the correct units of the constant b and input those units into cell C1.

T5.1: Print out the graphs (preview first, make as large as possible while still fitting on one page).

T5.2: Use equation 4 and a calculator to determine a numerical value for the terminal velocity.

T5.3: Notice that in the vt-plot the graph has what appears to be a horizontal asymptote. Use a straight edge and

draw in that asymptote. Compare the value of the horizontal asymptote to vT obtained in the previous question.

T5.4: When I look at the yt-plot it looks extremely straight after about 8.00 s. What is the approximate acceleration

for that time interval? Explain how the straightness of the xt plot and the small acceleration makes sense together.

T5.5: Try adjusting the drag coefficient b and the mass m. Perhaps try out other planets or the moon by adjusting g.

Notice how in some cases the particle never seems to reach terminal velocity while in other cases it reaches it

extremely quickly. This makes sense; a feather reaches terminal velocity very quickly while a human takes a bit

more time…

34

Task 6: Damped Harmonic Motion

The position as a function of time for a damped harmonic oscillator is given by

k(�) = y�gz% cos(|�)

Here A is the initial amplitude of oscillation, α is the decay constant, and ω is called angular frequency. This topic

is explored in detail in PHYS 162 but we can use our kinematics and graphing knowledge to take a first look at this

exciting and ubiquitous function.

Your goal for task 6 make plots of position, velocity, and acceleration versus time for the damped oscillator. By

making these graphs, I hope you will get practice with the following:

• Taking derivatives

• Using Excel to model x(t), v(t), and a(t)

• Analyze when the object is speeding up or slowing down

• Get a feel for how the motion is affected as you vary the parameters of A, α, and ω.

Take derivatives to find v(t) and a(t). Show you obtain E(�) = −y�gz%}? cos(|�) +? sin(|�)� a(�) = y�gz%}(?�−?� ) cos(|�) + 2? ? sin(|�)� Here the ?’s indicate a variable I want you to figure out on your own.

T6.0: Determine the appropriate units for the parameters A, α, and ω. Hint: x(t) is position and has units of…

T6.1: Use an amplitude of 1.00, a decay constant of 0.100, and an angular frequency of 0.628. Make constant cells

for these parameters so we can change them easily later. Use a time step size of 0.10 seconds from 0.00 to 20.00

sec. Make plots of x(t), v(t), and a(t). Compare them to the ones seen on the next page as a check on your work.

Tip: to use Excel to calculate �g�� use ���(−��). Also: verify you are using 0.628 in the constants row to check

your data against the plots on the next page.

T6.2: During what time or time interval is the damped oscillator moving as follows (hint: use the graph to estimate

the times then look in the data table to make your answers more precise):

a) Moving forward and speeding up

b) Moving forward and slowing down

c) At rest

d) Moving backward and speeding up

e) Moving backward and slowing down

T6.3: Now try changing the parameter ω and see how increasing ω affects the graphs. Try out the values of 1.57,

3.14, and 6.28 and see what happens. Notice that increasing the angular frequency cause the position of the

oscillator to oscillate more rapidly.

T6.4: Set ω = 3.14. Now try gradually increasing α and seeing how it affects the graph. Try values 0.2, 0.3, 0.5,

and 1.0. Notice that as α increases the oscillations dies out more rapidly. The amplitude decays.

T6.5: Set ω = 3.14 and α = 0.300. Try increasing the amplitude A and see what happens. Try values of 2, 5, and 10.

T6.6: This function could model a child on a swing after they are no longer pushed (the oscillation dies out).

This function might model the motion of a plucked guitar string.

a) Compare the angular frequencies used for each model (swing vs string). Which model would likely use a

larger ω (or would they about the same)? Explain and defend your answer.

b) Compare the decay constants used for each model (swing vs string). Which model would likely use a

larger α (or would they about the same)? Explain and defend your answer.

c) Compare the amplitudes used for each model (swing vs string). Which model would likely use a larger α

(or would they about the same)? Explain and defend your answer.

35

Interesting Note: Try cranking up ω to about 20. Notice the graph appears distorted. This is occurring because at

that high of a frequency our time step is relatively large compared to an oscillation cycle. As such we sometimes

don’t get any points near the peak of the oscillation causing the graph to look funky…to fix this problem one would

need to use a smaller step size (say increment the time by 0.01 sec instead of 0.10). Try it and see if this fixes the

problem. This relates to aliasing and sampling rates if you are interested in going further.

36

37

Using Tracker Software to Analyze 1D motion

Apparatus: See table below

The goal of this experiment is analyze graphs of motion. In particular, you are to do the following:

1. Choose an option for a moving object

2. Make a video of a moving object (include ruler for scale and clean background)

3. Open the video in the Tracker software program

4. Make plots of position versus time, velocity versus time, and acceleration versus time

5. Analyze the plots to determine if the kinematics of constant acceleration apply

6. Prepare and present a PowerPoint presentation to the class summarizing your work

Each group must pick one of the following options with a different starting number (unless we run out of options).

If you have an idea for a variation (or totally unique idea), discuss it with your instructor as early as possible.

General presentation tips and option specific tips begin on the following pages.

Option Description

0

Film the fan cart on a track. Use the video to determine if the fan cart is best modeled as operating under

constant force or constant power. Questions 5.33 and 7.13e in the workbook may be useful. Watch out:

Make sure the track is level and the fan cart is released from rest.

1a Film someone running 30 m on a track. Start from rest and try to run as fast as you can. Can you

accelerate for the entire 30 m? Place two orange cones on the track 30 m apart for calibration purposes.

1b Film someone running backwards for 15 m on a track starting from rest. Can you accelerate for the entire

15 m? Place two orange cones on the track 15 m apart for calibration purposes.

2a Drop a coffee filter from rest. Have it fall for about 2.0 m of distance. Repeat the experiment with 2, 3, 4

& 5 coffee filters nested inside each other. What is terminal velocity? Which model of drag works best?

2b Fluff up a cotton ball and drop it from rest. Drop it from about 2.0 m using a ladder. Do this inside to

avoid wind issues. Experimentally determine the drag coefficient q and compare to my estimate.

3

Obtain a cart which has a spring at one end. Place the cart at the bottom on an inclined track resting

against a lead brick. Compress the cart’s spring against the brick and release it from rest. Record the

motion of the cart as it bounces several times off the brick. Determine the average spring force during

collisions and the COR. For times when the spring is not engaged, compare theoretical motion to

experimentally observed motion.

4a

Hang a mass on spring and let it come to equilibrium. Pull the mass down a small additional amount. Do

your best to ensure the mass oscillates entirely in the vertical direction. Compare theoretical motion to

experimentally observed motion.

4b

Cart attached to spring on incline. Start with the spring unstretched with the cart near the top of the ramp.

Start video when the cart is released from rest. Record two full oscillations. Compare theoretical motion

to experimentally observed motion.

4c

Find the bowling ball pendulum in the lab. Pull the ball a small distance to the right. The string should not

exceed an angle of 8° from the vertical. The problem is complicated if vertical motion of the ball is non-

negligible. Record two full oscillations. Compare theoretical motion to experimentally observed motion.

5

Throw a tennis ball up in the air. Include the throwing and catching of the ball (the times your hand is in

contact with the ball) in your video. Do your best to make the motion purely vertical. Try to start and end

with your hand in the same position. Determine the average force exerted by the hand while throwing and

catching. While the ball is in flight, compare the ball’s experimental acceleration to what freefall predicts.

6

Angle a metal track very slightly (approx. 1°) above the horizontal. Place a steel ball on the track at one

end and secure a powerful magnet to the other end. Release the ball from rest and film the entire travel

including the impact with the magnet. Compare experimental to predicted acceleration while far from

magnet. Determine the effective range of magnetic force and the average force exerted by the magnet.

7a/7b Create simulations of experiment 2b, 4a and 5. Start by completing the suggested tutorials. Create k�-, E�- & a�- plots in your simulations. Compare your plots to plots from the experimental groups.

38

Make a video of your situation. Ensure you have a clean backdrop that provides good contrast (usually a black or

white background works ok). Also, ensure you have a device of known length (typically a meterstick) clearly

visible in your video. Ideally you should position your camera such that it is centered on the middle your object’s

path of travel. Also, suppose you object of interest travels total distance �. If the distance � from the camera to the

midpoint is less than 2.5� you introduce significant errors. Try a web search for “parallax error” to learn all kinds

of neat stuff related to this type of error. Finally, consider making one student in your group figure out the Tracker

software while the rest of you make the video.

Load your video into the tracker software. You can Google “How to use tracker software” and find a YouTube

video that explains how to use the software. I liked the one by “Vector Shock” at this link because it is short:

https://www.youtube.com/watch?v=D3p-CzWhhY8. This one is also good because it shows you how to rotate the

coordinate system at about the 1:50 mark: https://www.youtube.com/watch?v=ibY1ASDOD8Y. If you want to

practice while your group is getting your video, open a test video and try fooling around with that.

Determine how you want to align your coordinate system at this point. For example, do you want down the

incline to be positive or negative…your call? If you throw a ball downwards, do you want to make downwards the

positive direction? I’m cool with that. Is the object in your video moving left instead of right? Perhaps you want to

call leftwards positive instead of rightwards? Adjust the coordinate system in Tracker to match your decision.

Get position, velocity, and acceleration versus time data. Use the software to tabulate �, k, E, and a. You may

need to hit the button labeled Table on the right side of the screen to ensure you are displaying all four columns of

data. Double check the numbers seem reasonable. For instance, to quickly convert m/s to miles/hr use 1 m/s ≈ 2.2

mph. Think: should your accelerations be greater than, less than or roughly equal to 10 m/s2. Double check the signs

on the numbers and make sure that match expectations based on your choice of coordinate system. When satisfied,

copy the data from the Tracker software and paste it into an Excel spreadsheet. Note: if your numbers seem

abnormally huge (or tiny), check your calibration stick…are you in meters or centimeters?

Make plots of position versus time, velocity versus time, and acceleration versus time IN EXCEL. Tracker

will make graphs for you but I want you to get the practice of making the graphs in Excel yourself. Just grab cut and

paste the data into Excel.

� = Distance

object travels � = Distance of camera from

midpoint… if � � 2.5� parallax errors

at ends of path are less than 10% camera

39

1D Motion Tips For everyone:

• Tell people what coordinate system you chose. If non-standard, explain why you chose it that way.

• When you make plots of k�, E� & a�, be careful with the time axis. Ensure the time axes are identical on

all three graphs. Right click on each time axis and click “format axis” to manually adjust the axis times.

• For formatting on your plots, read the Lab Manual Appendices and Sample Graph Type I or II as

appropriate.

• Think! Sometimes trendlines should not be used. For instance, only use a linear trendline if a graph should

be modeled by a linear equation! Example: don’t put a linear (or polynomial) trendline on data when we

expect the physical situation to be modeled by a sine function.

First read the detailed instructions specific to your project on the following pages. After you make a rough

draft of your talk, come back to this point. The items in the list below are some possible ideas of things to say to fill

time. The first talk is usually about 6-8 minutes. You don’t have to do all of these, but you will probably need to do

some of these. These ideas match the type of questions I ask about graphs on exam day.

• On your k� graph:

o Determine the signs of E and a for two different instants. Explain how you know the signs of a

and E at each instant to the class. Hint: concavity and slope.

o For those same instants, discuss if the object moving forward/backward/at rest & speeding

up/slowing down/constant speed. Hint: compare signs of a and E.

o Point out any instants where the object is instantaneously at rest. Point out times for which the

slope of the k�-graph is zero at those times. These are usually instants when the object reverses

direction.

o Get a number with units for the slope at each of your 3-4 different times. Compare each

calculation to the appropriate time on the E� graph.

• On your E� graph:

o Determine the sign of a at 3-4 different times. Hint: slope.

o For those same times, discuss if the object moving forward/backward/at rest & speeding

up/slowing down/constant speed.

o Determine the area under the E� graph (get a number with units) and compare it to your position

versus time graph. It may help to split the area into chunks. Watch out for both positive and

negative areas.

o Point out any spots where the object is instantaneously at rest.

o Get a number with units for the slope at each point and compare it to the a� graph.

• On your a� graph:

o Don’t worry if the graph looks extremely noisy. It usually looks terrible since the acceleration

data is obtained using multiple steps of approximations. In contrast, the velocity data is obtained

using only a single approximation and is much less noisy.

o Relate the sign of a to the concavity of the k� graph at different times.

o Relate the sign of a to the slope of the E� graph at different times.

o Does your experiment exhibit any acceleration trends?

� Is the acceleration roughly constant?

� Is the acceleration always increasing or decreasing?

� Is the acceleration oscillating?

� Note: it may be easiest to describe these trends using the slope of the E�-graph rather than

a very noisy a�-graph.

• Watch out for these pitfalls

o Remember that speed and velocity are different…one includes ± signs while the other does not!

o If you have a negative velocity that is increasing in magnitude, the speed is increasing

o When you are speaking about acceleration, be clear if you mean the magnitude of the acceleration

or the acceleration…one includes ± signs while the other does not!

40

Option 0 Notes (assumes fan cart starts form rest to simplify the math):

If power is constant the velocity as a function of time should obey the equation

E = 92��F

where � is the constant power in Watts. Notice we could say

E� = 2�F � Using separation of variables one can also show the position as a function of time is given by

k = :8�9F;5/� �8/�

If power is constant a plot of E� vs � should be linear and the power is determined using ����� = ��t . Furthermore,

a plot of plot k vs �8/� should be linear with ����� = 4���t65/�.

If force is constant (and thus acceleration) the velocity as a function of time should obey the equation E = a�

E = YF � The position should obey

k = 12 a��

k = Y2F ��

If force is constant a plot of E vs � should be linear and the force is determined using ����� = �t. A plot of k vs ��

should be linear with ����� = ��t.

Once you make a tracker video of the fan cart, you can make tables in Excel of position, velocity and acceleration

versus time. You can use that data to create three additional columns: one column forE�, one column for �8/�, and

one column for ��. Use this augmented data table to make all of the plots discussed above:

1. E� vs � 2. k vs �8/�

3. E vs � 4. k vs � 5. k vs ��

Formatting instructions can be found in the lab manual appendices. See “SAMPLE GRAPH TYPE I”. Be sure to

include a linear trendline showing both the equation and the �� coefficient. The plot with the largest �� coefficient

is said to be the most linear.

If plots 1 & 2 are linear, you know the fan cart is well modeled using a constant power model. Use the slope on

each of the first two plots to determine the constant power of the fan.

If plots 1 & 2 are non-linear, show this to the class and stress this point.

If plots 3 & 5 are linear, you know the fan cart is well modeled using a constant force model. Use the slope on

each of the first two plots to determine the constant force exerted by the fan.

If plots 3 & 5 are non-linear, show this to the class and stress this point.

Note: you can also fill times with commentary relating to the items discussed on the previous page.

41

Options 1a & 1b Notes: FBD is confusing at first. Discuss with your instructor after acquiring data.

First take a guess. Can you run with constant acceleration for the entire distance or not? It is tough to get tracker

positions accurately. Perhaps try to use the middle of the person’s torso. Record multiple people to compare?

Cut and paste your time, position, and velocity data from Tracker to Excel. Make plots of position versus time,

velocity versus time, and acceleration versus time. Expect your acceleration versus time plot to be nearly worthless

due to excessive noise.

Usually groups split the plot into two stages. Usually we see a runner gradually reaching max speed then running

at roughly constant speed for the rest of the time.

During the first part (getting up to max speed):

• We can assume the initial acceleration stage has, very roughly, constant acceleration. During the

acceleration stage

o k = 5� a�� = ��t ��

o E = a� = �t � o In these equations, Y is the magnitude of the average force the runner exerts on the ground F is

the mass (not weight) of the runner.

o Recall, by Newton’s third law, the force the runner exerts on the ground is equal in magnitude

(and opposite in direction) to the force the ground exerts on the runner.

o Remember, using mass in kg should give force in N.

• You might notice the velocity plot is very roughly linear. We can model this stage of the experiment as

constant acceleration. The slope of this approximately linear velocity plot gives you the average

acceleration of the runner as they get up to speed. Use a trendline to get the slope (see Lab Manual

Appendices, Sample Graph Type I). Determine the magnitude of the average acceleration and the average

force the runner exerts on the ground.

• The position graph should be quadratic. Use a polynomial order 2 trendline because the theoretical position

is k = 5�a��. Show the trendline equation (and �� value) on the chart. Think: the coefficient on the

squared term in the trendline is not acceleration but rather 5� a. Again determine the magnitude of the

average acceleration and the average force the runner exerts on the ground.

During the second part (@ max speed):

• The velocity should be approximately constant. A constant velocity should look like a horizontal line

(slope of zero). Notice the constant velocity section corresponds to zero slope on the E�-plot (zero

acceleration). Use a trendline to get the slope (see Lab Manual Appendices, Sample Graph Type I). Verify

it is at least close to zero. Discuss with your instructor as this part can be a bit subjective.

• The position graph should be linear. Think: after you reach max speed ∆k ≈ EtJ(�. Show a linear

trendline equation (and �� value) on the chart. The slope should be EtJ( . Hopefully this compares well

with the speed shown on the velocity versus time plot.

Put it all together (both stages on a single plot):

• Show the position versus time plot. Indicate to the class when stage 2 starts. Discuss when the plot should

be concave versus linear. Animate in the values you found for a (from slope of E�-plot stage 1) and EtJ(

(from slope of k�-plot stage 2) for the appropriate stages.

• Show the velocity versus time plot. Indicate to the class when stage 2 starts. Animate in the value you

found for a (from slope of E�-plot stage 1) and the horizontal line E = EtJ( using EtJ( from slope of k�-plot stage 2.

• You should now consider the acceleration plot. You could try to split the acceleration data into the same

two stages. It will look pretty noisy, but perhaps you can notice a fain trend. Initially stage the acceleration

should be slightly positive change to approximately zero (on average).

Summarize your results for EtJ(, a, and Y. Think we ignored air resistance; discuss if this is reasonable. Do a web

search for high performance track stars. Compare her or his numbers for a or EtJ( to yours. Perhaps you could

find a chart showing max speeds and/or accelerations of various animals, motorcycle, etc and properly cite?

42

Options 2a & 2b Notes:

Do several practice trials before filming. Ideally you want to have a clean release and watch your object fall straight

down. For example, with the coffee filters you might use two hands. For either case you might design an

electromagnetic release using a nail, some copper wire, a battery, and a washer. The washer could hold the item of

interest onto the electromagnet. Once you disconnect the wire for the magnet the washer will fall and thus allow the

item of interest to fall. Spend a little bit of time on this but not too much.

Get time, position, velocity, and acceleration data for your experiment. Cut and paste that date from Tracker to

Excel. Create an additional column of data for theoretical velocity. The rest of this page gives you background

information on air resistance theory. The next page gives practical information on how to make a theoretical model.

Air resistance theory: Drag being modeled by � = qE� is usually considered valid for high speeds. Drag being

modeled by � = �E usually applies for very small objects at very low speeds. While neither model is perfect for

our experiments, both models provide a rough approximation exhibiting the main qualitative feature of drag

(asymptotic approach to terminal velocity).

A dropped ball of mass m experiences drag given by � = qE�. An FBD showing the

forces and coordinate system are shown at right. The equation of motion is determined by F� − qE� = Fa

We know when a = 0 the ball has reached terminal velocity given by

�� = 3���

Separating the variables in the equation of motion gives qFb� = bEEr� − E�

Note: we expect E < Er for all � � 0.

I found

E(�) = Er4Er + E`Er − E`6 − �g�sK�t %4Er + E`Er − E`6 + �g�sK�t %

For a dropped ball use �� = �. This simplifies the previous result dramatically:

E(�) = Er tanh :qErF �;

Note that E � 0 for all �; this makes sense as I rotated my coordinates such that down was positive.

Since we are using a tracker video, the first frame of the video might not correspond to � = 0. It is sensible to

introduce a shift in the time coordinate giving

�(�) = �� �� ¡ ¢���� (� + ∆�)£ where ∆� is the delay time between the release of the ball and the first usable frame of the video.

Note, the term q is theoretically given by

q = 12�¤y

where � is a dimensionless number called the drag coefficient, ¤ is the density of the fluid the object is passing

through, and y is the cross-sectional area of the object.

For a cotton ball moving through air we expect the following: ¤ = ¤J`¥ = 1.2 ¦§m., a cotton ball (� ≈ 3cm) has

approximate cross-sectional area y = 3 × 10g8m�. A perfectly smooth sphere has a � = 0.5 while rougher

surfaces can have numbers as high as 2. I will assume � = 1.5 for a cotton ball as surface roughness is significant.

This gives

q = 12�¤y = 12 :1.5 × 1.2 kgm8 × 3 × 10g8m�; ≈ 3 × 10g8 kgm

For a single coffee filter a similar analysis can be done. My web research indicated q ≈ 9 × 10g8 ¦§m .

a

mg

bv2

43

Theoretical velocity details:

1. Create a table like the one

shown at right.

2. Input your best guess for b

at this point. See bottom

of the previous page for

best guess tips.

3. Use an Excel formula to

compute cell D3 from cells

A3, B3, & C3. Use the

first bold formula on the

previous page.

4. Use an Excel formula to

determine cell E3 from A3

and B3.

5. Cell C6 is computed using

the formula shown at the

bottom of the figure. This

is based on the second

bold formula on the

previous page.

6. Fill down the equation to

compute the theoretical column of data (vth).

7. Plot v vs t. Include both vexp and vth. See Lab Manual Appendices, Sample Graph Type II for formatting.

8. Adjust the values of both ∆t and b so the graph matches as close as possible.

o Adjusting ∆t should shift the graph left or right. It makes sense that you will need to do this since

your object is probably already moving in the first usable frame of your video.

o Adjusting b changes both the terminal velocity and the rate at which the object approaches

terminal velocity. This is essentially adjusting the height of the horizontal asymptote and the

sharpness of the bend of the curve.

o Make an additional column computing the square of the difference between vexp and vth. Sum this

column. By minimizing this sum, you are finding the best curve fit (best value for q)!

9. Notice the numerical value of the horizontal asymptote shown on the graph. Also look at the constants

listed above the data table. Which constant relates to the asymptote? Does this make sense? Explain.

Going Further: Now that you know a value of b, you could make similar theoretical lines for both x(t) and a(t)

using the above equations. Take the derivative of v(t) to obtain a(t). You would need to integrate of v(t) to obtain

x(t). See how close theory matches the experiments on each of those graphs by repeating the above procedure. It

might be challenging as well. You might need tables, Mathematica or Maple to do the math.

Last note: For a ball thrown upwards with � = qE� you do NOT obtain the same solution for v(t). The

equation of motion is determined by F� + qE� = Fa

Since one of the signs has changed it no longer gives the same result upon integration! This sign change

gives a tangent function instead of hyperbolic tangent. Also, this situation will not make sense if vi = 0.

Try it out and see what you get! The result is easily found with an internet search. I used “ball thrown

vertically with air resistance” as my search term. Furthermore, solutions to these types of problems are

even messier if the initial velocity is actually greater than terminal velocity…

a

mg + bv2

44

Alternative Coffee Filter Style (avoids the �� ¡ function but requires 5 tracker vids)

Do one coffee filter experiment as described previously. Plot position versus time, velocity versus time, and

acceleration versus time.

Instead of making the theoretical model described on the previous page, repeat the coffee filter experiment

several times. First repeat the experiment using two nested coffee filters. Repeat the experiment 3 more times using

3, 4, and 5 nested coffee filters.

Do not make plots for each extra trial. Instead determine terminal velocity (Er) for each extra trial.

Plot Er versus F.

Also plot and Er� versus F.

If drag is modeled by ª = ��« the force equation gives F� − qE� = Fa

We know when a = 0 the ball has reached terminal velocity given by

Er = 3F�q

��« = ���

If the plot of ��« versus � is linear we know � = qE� is a good model for drag. The slope is ts and q = t¬­=®p.

Compare the experimentally determined value of q to my prediction of q ≈ 9 × 10g8 ¦§m from web research using a

percent difference.

If drag is modeled by ª = ¯� the force equation gives F� − �E = Fa

We know when a = 0 the ball has reached terminal velocity given by �� = ��̄

If the plot of �� versus � is linear we know � = �E is a good model for drag. The slope is t° and � = t¬­=®p.

Determine the value of �.

Show the class both plots and tell us which model is better. Be sure to include a trendline. Tip: the �� value on a

trendline indicates which plot is more linear. The closer �� is to one, the better the fit. Also, you could use the

LINEST function in Excel to get an estimate of statistical error for your value of q or �. The Lab Manual

Appendices have a help file on how to use the LINEST function. I’m pretty stoked to see the results as I’ve never

done this one!

45

Option 3 Notes:

The cart should bounce several times. Note: I like to start with the spring compressed heading up the track. If you

do the experiment this way you know the spring will never bottom out.

The shapes of the position versus time and velocity versus time plots should look similar to the ones shown below.

Note: I made up numbers randomly so expect yours to be very different. The red dots indicate times when the

spring is in contact with the cart.

Once you have your real data from tracker, cut and paste time, position, velocity, and acceleration data into Excel.

Plot position vs time, velocity versus time, and acceleration versus time.

Now make an extra copy of the entire data set on a second sheet. Cut up the extra set of data into stages where the

spring is or is not in contact with the cart. Perhaps go frame by frame in your tracker video to determine the best

times to split the plot into stages? Or mouse over the data points in your plot?

For each stage make a E�-plot showing a linear trendline equation with �� coefficient. See Lab Manual Appendices,

Sample Graph Type I for formatting tips. Think about what each number in the trendline means. Do the numbers

seem reasonable? When you put your plots up in front of the class, that is what I will be thinking…

For each stage when the spring is NOT in contact make an k�-plot, show a quadratic trendline equation (polynomial,

order 2) with �� coefficient. Think about what each number in the trendline means. Do the numbers seem

reasonable?

More on the next page…

0

0.2

0.4

0.6

0 0.5 1 1.5 2 2.5 3

x (m)

t (s)

x vs t

-2

-1

0

1

2

0 0.5 1 1.5 2 2.5 3

v (m/s)

t (s)

v vs t

46

There are several major things you should be able to learn from your plots:

1) Determine the average force exerted by the spring while it is in contact with the cart.

a. Sketch an FBD while spring in contact. Force up the ramp (average spring force) minus force

down the ramp (F� sin ±) should equal Fa. (Ask your instructor for help here.)

b. Rearrange equation to solve for average spring force.

c. Get mass of cart from a balance and acceleration from slope of E�-plot. Be sure to use a stage

where the spring is in contact with the cart!

d. Plug in numbers to get spring force.

e. Expect average spring force to decrease with each bounce as spring compresses slightly less after

each stage due to energy losses in system.

2) Compare acceleration of cart not in contact with spring to theoretical prediction

a. Sketch FBD while not in contact. Ignoring friction and drag, the force down the ramp (F� sin ±)

should equal Fa. This gives a%V = � sin ±.

b. Get the angle of your ramp by measuring the

length of the ramp and the height of whatever

block you used to lift it. You’ll want to double

check this with an angle indicator or an app on

your phone.

c. Compare a%V to the slope of your E�-plots (for

stages not in contact with spring).

d. Think: should this be the same after each bounce or different after each bounce? If it should be

the same, get an average of your experimental a’s and compare to your theory value with a

percent difference.

3) Going Further: Determine the Coefficient of Restitution

a. Coefficient of restitution (COR) is given by

²³� = 9 energyaµ���collsionenergyq�µ���collsion = 9F�ℎ�F�ℎ5 = 9ℎ�ℎ5

In this equation ℎ5 is the max height of the cart before and ℎ� is max height after each collision

(when the spring on the cart is compressed).

b. Look at the position versus time plot for the entire time (not split into stages)

c. Determine the COR between each stage using the bold formula above.

d. Get an average value of the COR.

e. Do a web search for “ball bouncing in slow motion”. Show no more than two videos. Keep the

total time for videos as short as possible…less than a minute in total including these videos and

any tracker videos you show.

If you need more crap to fill time: Compare the area under the E�-plot to displacement on the k�-plot. Calculate the

area under each E�-plot triangle with units. Estimate the displacement from these areas. Be sure to include the

signs. Then, show how these displacement numbers match up to your k�-plot.

Since this talk has so many neat things it is crucial to finish your talk with a clear summary. Think carefully key

takeaways you wish to emphasize in your summary slide. Afterwards, revise your initial goals to match well with

your summary slide. If desired you can ask for an extra minute of time if you are doing a lot of cool stuff.

±

�

ℎ

47

Option 4a-4c notes: I USUALLY ALLOW THESE GROUPS AN EXTRA 2 MINUTEs TO TALK.

FBDs are optional for 4a-c, I’d rather you focus on the stuff mentioned in “1D Motion Tips” on about pg 39.

First part is for 4a and 4b only: Record the length of the unstretched spring before anything is attached to it. Next,

attach your object (hanging mass or cart on ramp) and allow the system to reach equilibrium. Record the distance

the spring stretches between equilibrium. WATCH OUT: the length of the stretched spring is not exactly the same

as the amount the spring stretches. The amount the spring stretches should be the difference between the stretched

length and the unstretched length. The forces balance at equilibrium. This information helps you determine the

theoretical period in the table shown below. Deriving these equations is not my intention for this lab. If you want to

see the derivations, discuss with your instructor after you have acquired data.

For 4a For 4b For 4c

F� = ·∆kp2

which implies

· = F�∆kp2

Use this value of · to determine the

theoretical period

��¸ = «¹3�º

F� sin ± = ·∆kp2

which implies

· = F� sin ±∆kp2

Use this value of · to determine the

theoretical period

��¸ = «¹3�º

Notice sin ± is NOT in the period equation!

kp2 = 0

Theoretical period is given by

��¸ = «¹9»�

where � is the length of your

pendulum (to the center of the ball).

Make a video that includes at least one full oscillation but preferably two full oscillations.

To simplify things, put the origin of your Tracker coordinate system at the equilibrium position of the ball.

Copy your time, position, velocity, and acceleration data from Tracker and paste it into Excel.

Make an extra columns of theoretical data for position, velocity, and acceleration. Practical instructions for this are

found on the next page. You should then be able to make a second set of k�, E�, and a� plots showing both

theoretical and experimental data at the same time for easy comparison. For formatting tips see Lab Manual

Appendices, Sample Graph Type II.

Suggested talking points are listed below. You do not need to use every talking point.

• Define of amplitude, period and angular frequency. Show how to determine y and ¼ from an k�-plot.

• At equilibrium forces balance (net force of zero). Describe which regions of motion should exhibit positive

net force (and thus acceleration). Relate this to the concavity on your k�-plots and the slopes on your E�-plots. Repeat all of this analysis for regions with negative net force. Tip: if using a spring, should the

spring force is dominate the force equation at small or large stretch?

• At max or min position the velocity is zero but force (and thus acceleration) magnitude is maximized.

Relate this to slope of k�-plot and value of E on E�-plot.

• At equilibrium position force (and acceleration) are zero but speed is a max (velocity is max or min). In

addition, you were told to set the equilibrium position at the origin. Show the class the max speed is a

points on the k�-plot by emphasizing the steepness of the slope each time the object passes equilibrium.

• Show the k�-plot. Ask the class to determine when the object is moving forwards and slowing down.

Explain one can get the sign of E from the sign of the slope and the sign of a from the concavity. Now ask,

“When is the object moving backwards and speeding up?”

• Show the E�-plot. Ask the class to determine when the object is moving forwards and speeding up. Show

them the answer and explain how one can tell using the sign of E and the slope of E�-plot. Repeat by

asking them when the object is moving backwards and slowing down. If they aren’t getting it, do it again.

• Show all three plots first without theory data then with theory data superimposed.

Think carefully about your summary slide. What are the key takeaways you want to emphasize to the class? Make

your summary slide then rework your goal slide so it matches up well with your summary. Because this particular

experiment is such a good teaching example, remind your instructor to give you an extra 2 minutes to talk and

point out this sentence to her or him. Remind them just before your talk as well.

48

Oscillations: SET THE ORIGIN OF YOUR TRACKER COORDINATE SYSTEM AT EQUILIBIRIUM.

Assuming you set the origin of your Tracker vid at the equilibrium position the theoretical equations of motion are k%V(�) = y cos(|� + ½) E%V(�) = −|y sin(|� + ½) a%V(�) = −|�y cos(|� + ½) where| = 2>¼%V

In these equations y is the amplitude, | is angular frequency (units are rad/sec but I think of them as RPMs), ½ is a

phase angle (shifts the starting point of the oscillation) and ¼%V is the theoretical period from the previous page.

The example shown below is for Option 4b. Row 1 indicates the data required to determine ¼%V (using equations

from the table on the previous page). Row 4 allows you to input model parameters for amplitude and phase angle. .

If you used option 4a, you would not need the angle ± in the first row.

If you used option 4c, your first row would instead include �, �, ¼%V and |.

For all three cases the rest of the analysis is nearly identical. The amplitude (y) is the amount you displaced the

object from equilibrium prior to release. For now assume the phase angle (½) is zero. Starting in row 8 you can see

my fake Tracker data in columns A through D. I then used the theoretical position equations (above on this page) to

generate columns E through G.

Finally, if needs be, you may need to adjust the phase angle. If you are wondering what the phase angle does,

consider what happens to k%V the when � = 0. k%V(0) = y cos(½) The phase angle determines the initial position of the theoretical oscillation!!! It is appropriate to adjust the phase

angle to ensure your theoretical equations line up as well as possible with the experimental data. This is an artifact

of missing a few frames of video or modifying your coordinate system in Tracker. It is not appropriate to adjust the

amplitude or period as those parameters are predicted by physics.

¼

2y

49

Option 5 Notes:

Throw the ball as close to perfectly vertical as possible.

Somehow try to release and catch the ball at the same height. Perhaps you could have your forearm bump into a

stationary 2 by 4 as you are about to release the ball. The idea is to leave your forearm touching the board until the

ball comes back down. Hopefully your hand is then in nearly the same spot as you release and catch the ball. I have

no clue if this will work…feel free to try other ideas.

Try making several videos and pick the best one after several attempts.

Copy and paste your data for time, position, velocity, and acceleration from Tracker into Excel.

Make ��, E� & a� plots. I expect they might look a bit like the ones shown below. Note: I totally faked this data

just to give you an idea of the shape of the plot. Do not expect your numbers to be similar to mine. Notice I chose

to color the dots differently for times the ball is in freefall versus touching the hand. Now read the next page.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5

y (m)

t (s)

y vs t

-6

-4

-2

0

2

4

6

8

0 0.5 1 1.5

v (m/s)

t (s)

v vs t

50

There are several major things you should be able to do with your plots:

1) Copy your data and split your plots into two stages

a. The first stage should include all times the hand is throwing the ball.

b. The second stage should include all times the ball is in freefall.

c. The third stage should include all times the hand is catching the ball.

d. If you are confused about the exact transition times, assume the hand is still touching the ball.

2) Compare acceleration of stage two to freefall theory

a. Use a linear trendline on the stage two E�-plot. Think about what each number in the trendline

means. Do the numbers seem reasonable? When you put your plots up in front of the class, that is

what I will be thinking…

b. Use a quadratic trendline on the stage two k�-plot. Again consider the numbers in the trendline:

are they reasonable?

c. Compare a%V (what you expect for freefall) to the slope of your E�-plots (ap(®)…watch the signs!

d. Use a percent difference to compare a%V to ap(® for this stage

3) Determine average force exerted by hand for stages 1 & 3

a. While the hand is touching the ball Newton’s 2nd Law tells us Y − F� = Fa. In this equation I

am assuming Y is the magnitude of the average force exerted by the hand.

b. Solving this equation for Y gives Y = F(� + a) c. Notice the force exerted by the hand is NOT simply given by Fa! Newton’s second law says net

force equals Fa.

d. Use trendline on stage 1 & 3 E�-plots to get values for a. Use a balance to get a value for F.

Estimate the force of the hand during the throw and during the catch. Are they approximately the

same? Did anything in the video indicate one should be larger than the other for any reason?

4) Use the entire experiment ��-plot (all three stages in single plot) to describe the motion

a. Ask the class “Over what time intervals is the ball slowing down?” Wait thirty seconds for them

to answer. Then explain the answer. The ball is slowing down whenever acceleration and

velocity have opposite signs. The sign of acceleration is determined by the concavity of the ��-

plot. The sign of the velocity is determined by the slope of the ��-plot.

b. Now ask them when the ball is at rest. Explain the answer.

5) Use the entire experiment E�-plot (all three stages in single plot) to describe the motion

a. Ask the class “Over what time intervals is the ball speeding up?” Wait thirty seconds for them to

answer. Then explain the answer. The sign of acceleration is determined by the slope of the E�-plot. The sign of the velocity is determined by the value on the E�-plot.

b. Now ask them when the ball is at rest. Explain the answer.

If you need more crap to fill time: Compare the area under the E�-plot to displacement on the k�-plot. Calculate the

area under each E�-plot triangle with units. Estimate the displacement from these areas. Be sure to include the

signs. Then, show how these displacement numbers match up to your k�-plot.

By the way: at some point you should give the definition of freefall. Remind students a ball thrown upwards

is just as much in freefall as a dropped ball (as long as air resistance is negligible).

Since this talk has so many neat things it is crucial to finish your talk with a clear summary. Think carefully key

takeaways you wish to emphasize in your summary slide. Afterwards, revise your initial goals to match well with

your summary slide. If desired you can ask for an extra minute of time if you are doing a lot of cool stuff.

51

Option 6 Notes:

Ask for help if you want to include an FBD to fill time but it won’t make any sense until chapter 10.

In theory a solid ball rolling on an incline has a%V = 7¿� sin ±. We will derive this in chapter 10.

Please note this differs from a block sliding (with negligible friction) where a = � sin ±.

From experience you want to use a very small angle for this experiment. If the angle is too large, the ball moves

quickly by the time it reaches the magnet. If the ball moves too quickly one cannot observe any effects caused by

the magnet.

Try to use the smallest angle possible. If you have the option, try to use slow motion. Note: if you use slow motion

video, release the ball much closer to the magnet to avoid excessive data and a painful tracking experience.

Whichever style you choose (slo-mo vs regular speed), ensure you start the ball far enough from the magnet such

that it travels for at least 20 frames of video before the magnet has any effect.

Copy and paste your data for time, position, velocity, and acceleration from Tracker into Excel.

Make k�, E� & a� plots. There are several major things you should be able to do with your plots:

1) Estimate the range of the magnet

a. At some time near the end of the motion the slope of the E�-plot should spike up. This should help

you identify the time at which the magnet starts to affect the ball’s motion.

b. Use this time information with your k�-plot to estimate the distance between the ball and the

magnet at this same instant in time. This is the max range of the magnet.

2) Copy your data and split your plots into two stages (graphs on option 5 are similar but not identical)

a. The first stage should include all times the magnet appears to have no effect

b. The second stage should include all times the magnet appear to have some effect

c. If you are uncertain about a point near the transition time, include it in the second stage.

3) Compare acceleration of stage 1 to theoretical prediction

a. Get the angle of your ramp by measuring the length

of the ramp and the height of whatever block you

used to lift it. Feel free to double check this with an

angle indicator or an app on your phone but to get

more than 1 sig fig you need to use the trig…

b. Compare a%V = 7¿� sin ± to the slope of your E�-plots for times before the magnet has any effect.

Use a percent difference to do the comparison. Usually this number is way off. I suspect it is easy

to make small errors in the angle which cause huge % differences…

4) Estimate the average force exerted by magnet

a. Once close the magnet the forces down the plane have magnitudes F� sin ± and Y. Here Y is the

magnitude of the average magnetic force.

b. It can be shown (using Chapter 10 methods) a constant magnetic force acting at the center of the

ball changes theoretical acceleration equation to a = 7¿ 4� sin ± + �t6.

c. Rearrange the previous to show Y = ¿7F(a − � sin ±). d. Get mass of ball from a balance and acceleration from slope of E�-plot. Be sure to use a stage

where magnet is actually close enough to have an effect!

e. Plug in numbers to get the magnitude of the average magnetic force.

Make a summary slide tying all this crap together. Think about what you most want students to learn. Once the

summary slide is made, revise your goal slide to match.

±

�

ℎ

52

53

Freefall

Apparatus: 3Tape timers operating at 40 Hz, tape, 3 right angle clamps, 3 medium rods, 3 small rods, 3 small

bases, 3 foam pieces, rulers, meter sticks.

Purpose: To test kinematic theory as it relates to freefall, and to measure the acceleration due to gravity.

Theory: If the only force acting on an object is gravity, then the object is said to be in freefall. Since the force of

gravity near the surface of the Earth is constant, the object’s acceleration is also constant. We will use the

assumption that the object’s acceleration is constant with magnitude equal to g. In addition we know acceleration is

dt

dva = . It can be shown with calculus that the speed of the object as a function of time is given by

v(t) = vi + ayt

The position as a function of time is given by

y(t) = yi + vit + ½ayt2

In your lab write-up you should be sure to indicate what each variable means.

Procedure: Place tape in the marker apparatus so that it passes under the carbon paper:

Figure 1: Marker apparatus. Note the position of the paper tape.

Fix a mass to the end sufficient to overcome the drag of friction and air resistance on the tape (up to 500 g). Also,

detach your strip from the spool and take precautions to minimize the effects of drag and friction. Set the apparatus

at 40 Hz and release the tape.

Figure 2: Markings on paper tape. Note that all distances are measured from the first data point (not the

distance between adjacent dots).

The tape marker marks the position of the tape (and also the falling mass) every 1/40th of a second (0.025s). Fix the

tape securely to a horizontal surface and identify the position marks. Choose the first clear mark and label it 0, and

each successive mark 1, 2, 3, etc. These data points correspond to the tape’s position at t = 0s, t = 1/40s, t= 2/40s,

etc. You will convert these times to decimals for your table.

54

Data: Using Excel, make a table of the data indicating units, labels, and errors. Remember: You can change the sig

figs of a cell by clicking these buttons . Notice that each y value is negative because the tape was moving

downward. Double check that each y value is measured from the starting point (not the distance between

adjacent dots). Note: the errors indicated in the picture below have deliberately been changed to ensure you figure

out the %errors yourselves!

Figure 3: Data table made using Microsoft Excel. Keep in mind that you should still be estimating the %error on

your measurements.

Recall that vavg = ∆y/∆t so on your table v1 = (y2 – y0) / (t2 – t0) and a3 = (v4 – v2) / (t4 – t2). Notice that to get the

velocity at point 1 you use the data from points 0 and 2. THINK: Why? Why are you unable to get v0? Will you

be able to determine the acceleration and velocity of your final point(s)? SPEED HINT! You can use excel to speed

up your calculations by entering a formula. See Figure 4 below and follow the examples given by your instructor in

class.

Figure 4: These figures show the steps when using excel to “fill down” a formula. In cell C3 type the formula

shown. Then put the mouse over the bottom right corner of the cell. Click on the mouse and drag it down to the

bottom of the column.

55

Now graph y vs. t and v vs. t.

To make your graphs, first try to follow the step by step instructions in the graphing appendix.

Look back at your graph and check the following things:

• remove the gridlines

• graph has title using variables (in italics) but no units are necessary

• axis labels have variables in italics with units not italicized

• the graph fills the entire field (graph size should be about 1/3 to 1/2 a page)

• no legend for single set of data (a legend is only used if more than one thing is on a single graph)

• if a trendline is shown there is an equation with R2 value on the chart

• show the data as points only (no connecting smooth line)

• for graphs with both theoretical & experimental data: theory is a smooth line (no points) while experiment is a

only points (no line)

• the graph and data table are sized such that it fits on a single sheet of paper (not always possible)

• everything except the legend has appropriate subscripts and superscripts

• title is y-axis label versus x-axis label

A graph of v vs. t will produce a line, the slope of which is the acceleration.

The magnitude of this acceleration is gexp, the experimental value of g.

A graph of y vs. t should make a parabola.

Use the LINEST command to determine the error in the slope (δslope) for your v vs. t graph (instructions are in the

error appendix somewhere).

The percent precision in today’s lab will be estimated from the error in the slope calculation. The %precision is

given by (δgexp/gexp)x100%=(δslope/|slope|)x100%.

Check print preview prior to printing to avoid wasting paper.

Conclusions:

1. According to the y-t trendline, what is the experimental value of g?

2. According to the v-t trendline, what is the experimental value of g?

3. According to the y-t trendline, what is the experimental value of the initial velocity in the y-direction?

4. According to the v-t trendline, what is the experimental value of the initial velocity in the y-direction?

5. Compare the %difference to the %precision. Was the gexp in good agreement with the accepted value?

Draw a target diagram representing your measurement.

6. Air resistance and the friction between the paper and the spark timer were neglected. Should this cause

your %difference to be more positive or more negative? Explain for credit.

56

57

Vectors

Apparatus: Force table, force table pulleys (plastic & metal), ring, slotted masses, mass hangers (5 gram), scissors,

strings, rulers, and protractors, pulley cord.

Purpose: The purpose of this lab is to practice vector addition using force vectors. Mathematically vector addition

can be performed as follows.

If we have 2 vectors, A and B, which can be written as

jAiAA yxˆˆ +=

r and jBiBB yx

ˆˆ +=r

then the sum of A and B is a resultant vector R,

jRiRR yxˆˆ +=

r

where

xxx BAR += and yyy BAR += .

Finally the magnitude of R and the direction of R (given by the angle φ can be determined using

22

yx RRR += and

= −

x

y

R

R1tanφ .

Experiment 1: We will use components (method 1) and graphical addition (theory method 2) to add vectors.

θ1

θ2

+ =

φ

where , , ,and .

Note: there is often confusion on the following point: the scalar components of the vector are Ax and Ay

(with no vector symbol or i-hat/j-hat). The vector components of the vector are and .

Rx

Ry

58

Since force relates linearly to mass we will cheat this week and say that the amount of grams applied to a string is

the force. Technically force is measured in Newton’s which we will learn about in subsequent lectures.

WATCH OUT! The little gray mass hangers have a mass of 5 grams.

VERSION A: use 30 grams at 0o and 40 grams at 30o.

VERSION B: use 30 grams at 0o and 40 grams at 45o.

VERSION C: use 80 grams at 0o and 60 grams at 30o.

VERSION D: use 80 grams at 0o and 60 grams at 45o.

Sketch a picture and figure out Ax, Bx, Ay, and By.

Don’t worry about drawing perfectly to scale for this picture.

Figure out Rx, Ry, R, and φ.

The resultant, in this case, means that your two masses at add up to an equivalent single mass of R grams pulling at

the angle φ.

Add or subtract 180 degrees to your angle.

This is now your first theoretical result for the mass (mth1) and angle (θth1) which should balance the two masses.

Now use the tail-to-tip graphical method to connect your vectors.

To do this, first give yourself some room on the graph paper.

Draw your vectors to scale. To do this:

1) Set a ruler along side your graph paper

2) For a large distance (10 cm or 5”), see if your graph paper lines up better with cm or inch markings.

3) Suppose you find there are 6 blocks per inch. You want each block to correspond to 5 grams. That

means 6 blocks = 30 grams = 1 inch. Your conversion will be different than this.

4) To draw a line that corresponds to 48 grams I use my conversion:

48grams 1inch30grams = 1.6inches

5) Unfortunately, inches are not measured in metric so I can convert the decimal portion to sixteenths by

doing the following: 0.6inches0.0625 = 9.616 inches = 1016 inches

Draw the first vector.

Be sure to draw a little arrow tip on the appropriate end of the arrow.

At the tip of the first vector, draw a new coordinate system (make it tiny so it won’t get in the way of your vectors).

Use your protractor first to mark the new angle of the second vector.

Convert your grams to inches (or cm) for the second vector.

Now draw the second vector with the appropriate length (given by your conversion) in the proper direction (given

by your protractor.

Be sure to draw a little arrowhead at the end of your second vector to indicate the tip of the second vector.

Determine the resultant:

Draw a line FROM THE TAIL OF THE FIRST VECTOR TO THE TIP OF THE LAST VECTOR.

Measure the length of this vector with a ruler and use your scale to convert that length to grams.

Measure the angle with a protractor.

Add or subtract 180 degrees from your angle.

This is your second theoretical result for the mass (mth2) and angle (θth2) which should balance the two masses.

59

Lastly, set up the force table with the appropriate mass at each angle.

Set up a third pulley at the predicted angle with the predicted mass.

Slightly adjust the angle and hanging mass until the ring is properly centered.

Determine the minimum mass that balances the ring.

Determine the maximum mass that balances the ring.

Adjust the angle slightly if necessary to ensure the ring is centered then record it in

your data as θexp.

The experimental value of the mass (mexp) is the average of your minimum and

maximum (see the example below if this is confusing you).

The error in recording your mass, δmexp, is determined by the range of values which balanced the ring.

For today’s lab, let us assume that the %precision is given by the percent error in measuring mexp

(δmexp/mexpx100%).

Record this as the precision in your notes.

Find the percent difference between the experimental value and each of the two theoretical methods.

Example: The first theoretical method predicted mth1=62g. The minimum and maximum balancing masses were 58

g and 64 g respectively. This gave mexp=61g with δmexp=3g. More succinctly this is stated as mexp=61±3g. The

percent difference is -2% with a percent precision of 5%. Since the percent precision was greater than the percent

difference the experiment was in good agreement with theory.

Notice that in the example I used third person, past tense. The sentences are short and complete.

I never used words like:

“Measure the mass”

“I/We did this”

Lastly, to get the 2% and 5% numbers, I did the following:

%b�µµ = �k� − �ℎ�ℎ × 100% = 61g − 62g62g × 100% = 1.63% = −2%

and

%���À����� = δFp(®Fp(® × 100% = 361 × 100% = 4.92% = 5%

Notice, in both cases, the units of grams drop out.

Notice, in both cases, the final answer is rounded to a single significant figure since they are both error estimates and

any extra significant figures wouldn’t be meaningful. Exception: if first digit is a 1, keep an extra sig fig.

Activity 2: Repeat the above procedure with different masses.

VERSION A: use 60 grams at 215o and 80 grams at 120o.

VERSION B: use 60 grams at 135o and 80 grams at 210o.

VERSION C: use 30 grams at 215o and 40 grams at 120o.

VERSION D: use 30 grams at 135o and 40 grams at 210o.

Summary:

• For each activity: show a small sketch (not to scale) of your math for the component-wise vector addition

using the Ax, Bx, sin, cos, etc (theoretical method 1).

• For each activity: Show a sketch to scale and your work for the tail-to-tip Graphical Method (theoretical

method 2).

• For each activity: record the actual mass and angle needed to balance the ring (experimental method).

• For activity 1, theory 1: calculate the %difference, the %precision, and draw a target diagram.

• For activity 1, theory 2: calculate the %difference, the %precision, and draw a target diagram.

• For activity 2, theory 1: calculate the %difference, the %precision, and draw a target diagram.

• For activity 2, theory 2: calculate the %difference, the %precision, and draw a target diagram.

• Answer conclusion questions (see below) in a numbered list. For a formal write-up this would be in

paragraph form.

60

CONCLUSIONS:

1. State if the experimental results of vector addition were in good agreement with the theory. You should

have four statements since you used two different experimental methods for two different experiments.

Use verbiage similar to that in the example. Remember, if any part of the bullet hits the bull's-eye, your

experiment is in good agreement with the theory!

2. How should friction in the pulleys affect your percent differences? Should it make the percent differences

more positive or more negative? Explain why it does or does not affect the experiment?

61

Projectiles

Apparatus: projectile launchers, plastic projectile spheres, rulers, projectile accessories, scissors, pulley cord,

meter sticks, paper, plumb bobs, large clamps

Purpose: In this lab we will use kinematic theory to find the muzzle velocity of a spring gun and then predict the

range of a projectile launched from an angle.

Intro: In today’s lab you will use the freefall equations of motion in two dimensions to analyze the flight of a

projectile. One version of the kinematics equations you can use is given below.

∆y = viy t + ½ay t2

∆x = vix t

viy=vi sinθ

vix=vi cosθ

Think: does ay = 9.8, or -9.8…� or -�…Watch out: is ∆y positive or negative? Think: What does each variable

mean?

Part 1: Finding the spring gun muzzle velocity.

Measure the height (∆y) from the floor to the ball’s center of

mass.

Launch a ball horizontally. A horizontal launch ensures that

the ball’s initial velocity (the muzzle velocity) has no vertical

components.

Measure the horizontal distance (∆x) from its center of mass

when launched to the point where it lands.

Since there will be variations, shoot the ball and measure ∆x five times. WATCH OUT! When horizontal the ball

has a tendency to roll out of the launcher just before shooting. This messes up your data significantly.

HINT: Put paper on the ground where the ball lands. It will leave an imprint clearly indicating where to measure!

Now derive a result for the initial speed of the ball (the muzzle velocity) vi in terms of ∆y, �, and ∆x.

Remember: when the launch is horizontal the launch angle is θ=0°. Also watch out for + or – signs. The variable �

is the MAGNITUDE of acceleration (�=+9.8m/s2). Think: if the ball goes downward will ∆y be + or -?

Calculate the muzzle velocity vi for each value of ∆x.

Determine the average value of vi.

Determine the standard deviation of your five values of vi using the following formula:

[ = ��a�ba�bb�E�a���� = 3∑ (Ãg(ÄÅÆ!&Ig5 .

Show your work (by hand) in your lab book for the standard deviation.

Note: the N-1 in the denominator indicates that you can’t really get a standard deviation of only one measurement.

Check your result with the computer using the AVERAGE command and STDEV command in Excel.

It turns out that the standard deviation of a set of numbers gives you the error (the ±#) for that set of numbers.

When random errors dominate the precision of equipment we use

%���À������µa�aE�Ea�Ç� = ��a�ba�bb�E�a�����µ�ℎ�F�a�Ç��F����aE��µ�ℎ�F�a�Ç��F���� = [ √MÈaE�

The figure at left shows a target diagram for this experiment. Notice that the theoretical

value is the bull’s-eye of the target. Since we took several measurements the target

looks like it was hit with a shotgun blast. The distance from the bull’s-eye to the center

of the shotgun blast is determined by the percent difference. The %precision is

essentially the radius of the shotgun blast. Notice that a few measurements actually lie

outside the standard deviation (σ). Since the %difference (distance from bull’s-eye) is

larger than the %precision (size of the blast) the experiment is not in good agreement

with the theory (the shotgun blast doesn’t touch the bull’s-eye).

∆y

∆x

5%

10%

theory

62

Part 2: Use the muzzle velocity vi from part 1. Predict how far (horizontally) from the launcher the ball travels.

First determine an algebraic expression for the range in terms of the launch angle, the launch height, and the

muzzle velocity. This derivation should be in your lab notes for credit.

To do this, first determine an algebraic expression for t from the ∆y = viy t + ½ay t2

equation. This can be done with the quadratic formula.

Then plug in this value of time into ∆x= vix t.

Lastly, don’t forget that vix and viy depend on the angle so you’ll need to put in sinθ and cosθ in some places.

Your final result for the theoretical range should look like (you need to figure out the terms with ?’s):

∆k%V = E` cos ± E` sin ± ± √?−2?�

VERSION A: Set the launcher on the table and angle

the launch at 40 degrees. Predict the position it will

land on the ground. Predict the location of the max

height (both x and y position of max height). Place a

bucket at the spot the ball should land and a hoop at the

max height. Once you have predicted the locations,

obtain a hoop and bucket from your instructor and show

them that the projectile goes through the hoop at the

max height and lands in the bucket. Is it repeatable?

VERSION B: Set the launcher on a table and angle the

spring gun to launch at 40, 45, & 50 degrees. Create an

Excel worksheet that computes the theoretical range for

each angle. Compare experiment to theory.

VERSION C: Set the launcher on a table and angle the

spring gun to launch at 39, 45, & 49 degrees. Create an Excel worksheet that computes the theoretical range for each

angle. Compare experiment to theory.

VERSION D (Challenging but really fun): Assume muzzle velocity and launch height are known. Your instructor

will give you (k, �) coordinates of a target to hit. It is your job to create an Excel worksheet which computes all

launch angles which will cause the ball to hit the target. Once your worksheet is done, consider checking it against a

simulation (Projectile Phet). Use ℎ = 10.0m, EÊ = 30.0mn , and target location (k, �) = (94.1m, 0).

To get experimental values of range:

Now actually launch the ball at each angle. For each launch angle, verify that the height ∆y has not changed!

Remember that the launch height is where the ball leaves the cannon, not at the height of the table. Place a piece of

paper on the floor where the ball is expected to land. Get 5 experimental values of the range for each angle. The

average of these values will be your experimental range (∆xexp). The standard deviation (over sqrt(N)) will give you

the error in the experimental range (δ∆xexp).

∆xexp

ymax ∆y

Max height only

needed for Version A.

63

For all versions your notes should include the following in your calculations section:

Derive the eqt’n for muzzle velocity vi algebraically

Work out ONE sample calculation of the muzzle velocity

Work out ONE calculation of σ by hand from part 1.

Derive the eqt’n for predicted range (for non-zero angles) ∆xth.

Work out ONE sample calculation of the predicted range (for a single non-zero angle).

Estimate your % precision and explain your reasoning.

Show a target diagram for the %difference and %precision for the smallest angle.

NOTE: To do a sample calc: write down the eqt’n, then plug in the numbers without doing any math, and finally

work out the problem to the answer on your page. Do NOT show a sample calculation for every angle.

Conclusions

1. Were the experimental results found to be in good agreement with theoretical predictions (how did

%precision compare to %difference)?

2. If air resistance is considered negligible, can today’s lab be considered as an example of freefall? Defend

your answer.

3. VERSION A use part a, VERSION B & C use part b

a. Use your %error and a calculated value of the max height to estimate the smallest hoop size you

could use and reasonably expect to pass through the hoop at max height 70% of the time.

b. Use your %error and a calculated value of the max range to estimate the smallest target you could

place at the max range and reasonably expect to hit the target 70% of the time.

4. In part I of the experiment air resistance was neglected in the theoretical calculation of the muzzle velocity.

However, air resistance was included in the experimental value obtained for ∆x. Is the actual experimental

muzzle velocity to be slightly higher or slightly less than our computed value?

5. Assume the following conditions: ∆y = -1.0 m, vi = 6.3 m/s, and g = 9.8 m/s2. Use your range formula to

show that a 39.3° launch angle flies 1.4% farther than a 45.0° launch angle. Notice that 45° will give

maximum range only when the launch height equals the impact height!

Going Further

Use MATLAB to make a contour plot showing the range of the projectile for initial heights ranging from 0.0 to 18.0

meters for all angles between 0.0° and 90.0°. I suspect you might try incrementing the initial height in 0.5 meter

increments and the angle in 5.0° increments. If you are feeling frisky, I suppose you could even consider angles

between 0.0° and 180.0°. Think: for launch angles greater than 90.0° why does it make sense to get negative values

for range?

64

65

Newton’s Second Law Part I

Apparatus: Air track glider (1 per track), air tracks, air supplies, air supply hoses & power cords, air track

accessories kits, scissors, pulley cord, stopwatches, balances, meter sticks.

Purpose: To gain additional practice drawing force diagrams and solving force problems using Newton's

2nd Law. To predict and measure the acceleration of a cart on a level track when pulled by a hanging mass.

Procedure:

1. Level track. Level the air track by using the adjustable legs. To be sure it is level place a cart on

the leveled air track at rest. If it starts moving it is not quite level. Check it both in the middle of

the track and at the ends of the track. Try to get air track as level as possible because small

differences can cause large percent errors in today’s lab.

• Remove the four 50 g masses (they look like short, shiny, cylinders with holes in the middle) from

the air track accessories kit and place two on each side of your glider.

• Tie a piece of string to the glider/cart so that m2 is touching the ground and the glider/cart is close

to (but not yet touching) the stopper closest to the pulley.

• To determine how far the system will travel (x), all you need to do is pull the glider a known

distance away from the pulley. Use a convenient distance like 0.500 m.

• The values of m2 you should use for this lab are about 4, 6, 8, 10, and 12 grams. Determine

the masses by using the balances in the classroom. Don’t forget to account for the mass of the

hanger (approx. 2 grams).

• Start with the smallest m2.

• Release the cart from rest when m2 is at the top and listen for the sound of m2 hitting the ground.

• Time how long it takes for m2 to hit the ground. That is also the time for the cart to travel a

distance x. The time t for the m2 to hit the ground should be on the order of seconds.

• For each m2, record the time for each of five trials (note: the distance is x for each trial). You

should now have 5×5=25 times recorded.

• Think: How should the times change as mass increases (longer, same, and shorter)? How should

your predicted value of acceleration change (bigger, same, and smaller)? What does the

experimental result show?

• Record the mass of the glider INCLUDING any attachments that were on it during the experiment.

Will this be m1 or m2?

• For your calculations section, use kinematics to determine an equation for the experimental

acceleration aexp from this x and t data. Write the formula in your notes and define each variable

in the introduction section. Solve for aexp algebraically then show an explicit calculation for one

case (you do not have to show all aexp’s by hand).

• Tabulate all aexp’s in your data table.

• Get an average value of aexp and the standard deviation of aexp.

• Determine the %precision of aexp. If the standard deviation is exceedingly use propagation of

errors of sig fig rules to determine your precision.

• In your calculations section, draw a free body diagram for the cart.

• Draw a separate free body diagram for the hanging mass.

• Write down the forces in both the x and y directions for both FBD’s.

Table

m2

m1

66

• Then, using Newton’s laws, determine an expression for the theoretical acceleration of the system.

You should end up with only the variables m1, m2, and the acceleration due to gravity g. Write the

formula in your notes and define each variable in the introduction section. Hint:?)(?

?

+=

gath

.

• Show an explicit calculation of ath for one m2 (not all).

• Calculate the %difference between the prediction (theory) value and the experimental value.

Show this %difference in your table as well.

• Make a plot of a versus m2. Show both the theoretical and experimental values of a. I call this a

Type 2 Graph; look in the table of contents for instructions on how to make this graph. Also, after

you think your graph is complete, there is a checklist in the front cover of the manual that

describes what a submitted graph should look like. Before asking the instructor to look over your

graph, be sure you look at the checklist and instructions.

In your calculation section you should have:

Derivation of aexp

Sample calc of aexp

Derivation of ath

Sample calc of ath

Sample calc of %prec of aexp

%difference calculations

Conclusions:

1. Were Newton’s laws in good agreement with kinematics? Compare the %precision to the %difference.

2. Which measurement was the leading contributor to your errors: m1, m2, t, or x? For example, when you

measured the masses you obtained many sig figs. This indicates the percent error is very small for the masses.

Note: the percent error in m1 is given by δm1/m1×100%. Improving your measurement of the masses will not

lead to results that are significantly more precise. That implies they are not the leading contributor to error.

3. Which measurements had errors that were negligible when compared to the largest contributor to error?

4. What techniques of measurement or methods of performing the experiment could you employ to reduce the

largest contributor to error? Assume you have ample budget to pay for technology required.

5. To make sure your formula for the theoretical acceleration is correct, consider the case of m1=0.0 kg while

m2=1.0 kg (and the opposite case: m1=1 while m2=0). Think: what should the acceleration be in those cases?

Write down the formula for ath in each case and explain why it makes sense.

Going Further: Create a contour plot in MATLAB showing the acceleration as F5 and F� both vary. Perhaps let F5 = 0to400g and F� = 0to40g.

67

Newton’s Second Law Part II

Apparatus: Air track glider (1 per track), air tracks, air supplies, air supply hoses & power cords, air track

accessories kit, wood blocks, scissors, photogate heads, photogate interface cables, photogate stands, pulley

cord, meter sticks, calipers, PASCO Science Workshop 750 Interface & Power Supplies

Purpose: To gain additional practice drawing force diagrams and solving force problems using Newton's 2nd Law.

To predict and measure the acceleration of a cart up an inclined track pulled by a hanging mass.

Procedure:

• Before raising one end of the air track, verify it is level. To be sure it is level place a cart on the leveled air

track at rest. If the cart starts moving it is not quite level.

• Place one of the black flags on the glider. Measure the length L of the black flag by placing up to the air

track (which has a ruler built into it).

• Record the mass of the glider as m1.

• Raise one end of the air track by placing a wooden block under the single leg of the air track.

• Measure the distance between the legs D and the height of the block h. Use calipers for h.

• It turns out that in your equations today, only sinθ appears. Rather than determining the angle θ used, we

can use D and h to determine sinθ directly! Draw a triangle labeling the sides (one should be D and one

should be h). Determine if sinθ is given by h/D or D/h.

• Prepare the track with a pulley at the raised end. The string that runs over the pulley should connect to both

a mass hanger and to the glider.

• Open the Data Studio program on the computer.

• Once you get to the interface set-up window add two photogates to the interface box. You will want to

measure the time in each gate and the velocity in each gate. The instructions for doing this are in the

appendices.

• Set-up two photogates above the air track. The photogates should be set up to have the black metal flag on

the glider pass through the photogate. You can verify this if the little red light on the photogate turns on as

the black flag crosses through the photogate.

• ALSO, the photogates should be set up in such a way that the glider passes through both photogates prior to

the mass hanger reaching the ground. Place them as far apart as possible while still allowing this to occur.

Try to pick a convenient distance like 0.500 m.

• Once you are certain that the photogates are properly set up, hang a mass on the mass hanger that is

substantial enough to cause the glider to slowly accelerate up the plane. With a single block (or two

blocks) you’ll probably need to use all the little masses in the air track accessories kit. In general, a mass

that is more than mglider sinθ (perhaps about 15 or 20 grams) will cause the block to slide up.

• The photogates record the time it takes for the black flag to cross through the first photogate t0 and the

second photogate t. Earlier you recorded the length of the black flag. You may assume the cart moves at

an approximately constant speed during the time the photogate activated. Also, you know that during the

time the photogate is activated the cart must travel the length of the flag. Use these numbers to calculate the

m2

x is not necessarily = D!

L

D

h θ

photogate

x

68

velocity of the cart at the first photogate (v0=L/t0) and at the second photogate (v= L/t). This should also be

recorded in your data by the DataStudio program. Check a few to be sure it is working and then you can

stop doing the calculation yourself.

• Now use the velocity of the first photogate (v0), the velocity at the second photogate (v), and the distance

between the photogates (x) to determine the experimental acceleration of the glider. Hint: Use the

appropriate 1D kinematics equations. In your calculations section, first solve the appropriate equation for

a. Then plug in the numbers and show one sample calculation.

• Repeat this experiment 10 times; 5 times each for two angles with the same hanging mass m2.

In your calculations section:

• Derive the expression which determines aexp in terms of the velocities and v0, v, & x.

• Draw a free body diagram (FBD) for both the glider and the mass hanger. Label the glider as m1 and the

mass hanger as m2. Assume the object will accelerate up the plane and that the air track is essentially

frictionless.

• Use Newton’s laws to derive a formula for the acceleration in terms of the masses, the angle, and the

acceleration due to gravity. This is the theoretical acceleration. Check the units.

• Calculate the theoretical acceleration by plugging numbers into the formula you derived from Newton’s

laws earlier.

• Recall that [JÌÍÎ √MP of aexp is essentially δaexp. Use this to estimate the %precision. Note: if the standard

deviation is zero then you must use another method to determine the error.

• Determine the percent difference for this experiment.

Conclusions:

1. Were Newton’s laws in good agreement with kinematics? Compare the %precision to the %difference.

2. Looking back on all the data you took, is there a single measurement that stands out as the largest contributor to

error? Defend your answer.

3. Suppose m2>>m1. What approximate acceleration should the blocks have?

4. Suppose the angle was 0°. How does the equation for the theoretical acceleration simplify? Write down the

simplified formula and compare it to the formula from last week’s lab…

5. What if the angle was 90 degrees? Write down the formula for ath in this case & compare your theoretical

acceleration to the Atwood’s Machine example in your textbook. Does it make sense at 90 degrees if the

masses are equal?

Comments on using standard deviation as error

The figure at left shows one possible target diagram for this experiment. Since we took

several measurements the target looks like it was hit with a shotgun blast. The distance from

the bull’s-eye to the center of the shotgun blast is determined by the percent difference. The

individual measurements are very good (perhaps 4 or 5 sig figs) but σ indicates only 2 or 3

sig figs. Choosing the worse of the two cases, the %precision is then given by [ √MÈ . This

%precision is essentially the radius of the shotgun blast. Notice that a few measurements

actually lie outside σ which is typical.

The figure at left shows another possible target diagram for this experiment. In this case σ is

very small even though the bullets are large. This is the case where the measurements have

perhaps two sig figs while σ indicates that the calculation has 4 or 5 sig figs. Considering

the worse of these two for our error, we choose to let the sig figs of the measurement

determine the sig figs (and thus precision) of the final calculation.

Going Further: Create a contour plot in MATLAB showing a for F� = 0to400g, ± = 0to90°, & F� = 200g.

5%

10%

theory

5%

10%

theory

69

Phriction Phreaque-Out Part I

Apparatus: board, light pulley, string, scissors, cleaning fluid, paper towels, hockey pucks with hooks, angle

indicators, tape, table clamps, mass hangers with slotted mass sets, method for raising and lowering board (could be

a human or a system of rods, bases and right angle clamps)

Goal: Determine the value of µs between a hockey puck and a board. By determining this in several different ways

you should get practice with a variety of FBD’s, force equations, and algebra.

Preparation: Thoroughly clean the board and puck surfaces and let them dry for at least 2 minutes. Do not to touch

the board or puck surfaces after cleaning. Handle the puck by the sidewall; do not touch the circular faces. If

the puck falls on the ground or a surface gets touched, simply clean it again and wait 2 more minutes.

Method 1:

Start with the board horizontal. Place the puck on the board (handle puck by the edges). As

slowly and smoothly as possible, raise the board. While you raise the board, monitor the angle

indicator. At the instant the puck starts to slip, note the angle. The angle at which slipping

onsets is called the critical angle (θc). Repeat this experiment for a total of ten trials to get an

average value of θc. Use this θc to determine a value of µs between the puck and the board.

Method 2a:

Now attach a string to the puck and run it over the pulley to a mass hanger as shown in the

2nd figure at right. Your set-up should allow you to set the angle and leave the apparatus

fixed at that angle for several minutes (see 3rd and 4th figures at right). You will use angles

of 0.0°, 10.0°, 20.0°, and 60.0°. Make sure your design can accommodate all of these

angles. In some cases it will help if the puck is allowed to travel as far as possible.

Once you have your appratus set-up, ensure that the string is parallel to the board by

making adjustments to the pulley. For each angle, determine the largest possible hanging

mass that can be used without causing the puck to slip up the plane. I recommend using

large masses (say 100 g) to first determine a rough value at which the puck starts to slip.

Then take off some mass and go by smaller increments (say 20g) to get a better

approximation. Finally, go by 1 g increments to record your most precise value for the m2

that causes the onset of slipping. Use the values of m2 that cause slipping onset to determine

a value of µs for each angle.

Method 2b:

This part uses exactly the same set-up as Method 2a. Using only the 60.0° angle, determine

the minimum m2 needed to prevent the puck from sliding down the plane. Use this value of

m2 to determine yet another value for µs.

Method 3: Now consider the similar apparatus shown in the 4th figure at right. Determine the minimum

m2 required to cause the puck to slide down the incline for θ = 15.0°. Use this value of m2 to

determine yet another value for µs.

θc

θ

θ

70

Conclusions:

1) In Method 2 you were asked to adjust the pulley to ensure the string runs parallel to the board. If the string was

angled, why does it complicate the force equations used to determine µs? In particular, discuss how the normal force

(and thus the maximum possible frictional force) is affected.

2) What angle on the Method 2a FBD corresponds to 15.0° in the Method 3 FBD?

3) As you increase the angle in Method 2a, do we predict that µs should increase, decrease, or remain constant?

Explain or defend your answer.

4) Why do we set the acceleration to zero in all of these FBD’s even though the block is sliding? Explain why the

approximation a ≈ 0 is appropriate.

For this week, your intro should start with a sketch of the apparatus and an FBD for method 1 (draw them side by

side and don’t skimp on the size).

Don’t forget to include a coordinate system so I can follow your work.

Then list the force equations centered on their own lines.

Finally, derive equation for µs from the force equations.

Repeat this so I have a sketch, FBD, force equations, and derivation for Methods 1, 2a, 2b, and 3.

I won’t be looking at grammar, just your diagrams, equations, and derivations.

Tabulate your data.

For each measurement, be sure to also tabulate your angles and, for Methods 2a, 2b, & 3, your masses.

In total you should have seven measurements of µs.

Determine the average and standard deviation of your eight measurements to determine your best estimate of µs and

the associated statistical error.

Obtain the averages from all other groups in the class. Determine the best estimate of µs for the class and the

associated statistical error.

Also, do the conclusions questions.

71

Phriction Phreaque-Out Part II

Apparatus: board, light pulley, string, scissors, cleaning fluid, paper towels, hockey pucks with hooks, angle

indicators, tape, table clamps, mass hangers with slotted mass sets, method for raising and lowering board (could be

a human or a system of rods, bases and right angle clamps)

Goal: Determine the value of µk between a hockey puck and a board. By determining this in several different ways

you should get practice with a variety of FBD’s, force equations, and algebra.

Preparation: Thoroughly clean the board and puck surfaces and let them dry for at least 2 minutes. Do not to touch

the board or puck surfaces after cleaning. Handle the puck by the sidewall; do not touch the circular faces. If

the puck falls on the ground or a surface gets touched, simply clean it again and wait 2 more minutes.

Method 1:

Start with the board horizontal. Place the puck on the board (handle puck by the edges). As

slowly and smoothly as possible, raise the board. While you raise the board, monitor the angle

indicator. At the instant the puck starts to slip, note the angle. The angle at which slipping

onsets is called the critical angle (θc). Select an angle at least 10.0° above the critical angle

while holding the puck at rest. Release the puck from and record the elapsed time to slide a

known distance. Use this to determine a value for µk.

Method 2:

Now attach a string to the puck and run it over the pulley to a mass hanger as shown in the

2nd figure at right. You set-up should allows you to set the angle and leave the apparatus

fixed at that angle for several minutes (see 3rd and 4th figures at right). You will use an angle

of 60.0°. Make sure your design can accommodate all of these angles.

Once you have your appratus set-up, ensure that the string is parallel to the board by

making adjustments to the pulley. For each angle, determine the largest possible hanging

mass that can be used without causing the puck to slip. Select a significatnly larger value of

m2. The blocks should slide. Release the system from rest and record the elapsed time for

the puck to travel a known distance. Use this information to determine a value of µk for each

angle. Note: if the puck is travelling quickly, try to use as long a distance as possible.

Method 3: Now lay the board flat. Select a hanging mass that is sufficent to cause the puck

to slide. Make a video of the puck sliding (released from rest). Try to shoot the video from

several feet away and have a plain background with no shadows. Keep your hand steady if

possible. Maybe try holding the camera steady on a spearate table level with the puck about

1.5 meters away? Put some white paper behind the puck? Or a poster board of some light

color?

Try to input the video into the tracker software.

Obtain a plot of x vs t (or v vs t ?)and determine the acceleration of the of puck.

Use that acceleration to determine a value for µk.

Average your values of µk.

Obtain the other group values as well.

Estimate the statistical error using a standard deviation.

Can you make an animation in tracker that models the puck?

Going Further: For case three you should be able to determine the minimum hanging mass that causes slipping.

Create plots of acceleration versus mass AND frictional force versus mass. Note: there should be a discontinuity in

each graph at the value of hanging mass which causes the onset of slipping.

θc

θ

72

73

SUGGESTED ORAL PRESENTATION IDEAS

First pick should go to group who did air resistance, then oscillations, then lottery. Option 2 is easiest.

OPTION 1: Atwood’s Machine

Test: Does Newton’s 2nd Law accurately predict the acceleration of

a three mass system?

Theory: Include an FBD of each object. List force equations for

each object, and the derive equation for ath. Explain what

assumptions are made when we assume all m’s have same a.

Explain what m’s make ath = 0. Explain what we expect for ath

when m1 or m2 is much bigger than all other masses in the problem.

Procedure: Use a smart pulley as one of the pulleys to get the data

done in a timely manner.

Get E�-data using data studio for each mass combination.

There will be 7 possible values for F5&F�.

Use 50, 70, 100, 120, 150, 170&200g.

Notice this gives a total of 49 data sets of velocity and time.

Note: sometimes the system will accelerate negatively.

Record all masses used in each trial with a balance.

For one trial, create a E�-plot to use in your talk (to explain how you determined acceleration).

For the rest get acceleration using the LINEST function (see lab manual appendices).

Make a contour plot of your experimental data (see the table of contents of this manual…last lab in this manual).

Make a theoretical contour plot as well.

Get a % precision estimate using 1¬­=®p¬­=®p from the LINEST function.

Get % difference for each data point.

Also get the absolute value of the percent difference for each data point.

Data/Graph:

• For single trial, show a E�-plot to explain how you obtained acceleration.

• For a single value of F5, show a Type II graph (theory as smooth line without data points, experiment as

points without line) of a vs F�. Change the data points from dots to cross-hairs using error bars (see the

help file and discuss with your instructor).

• Also compare the theoretical and experimental contour plots.

Conclusions: Tell the audience if the Newton’s 2nd Law is in good agreement with your experimental results that

used kinematics. This is probably true if your average % difference is less than your average % error.

Note: I plotted a%V vs F� for F5 � 1.5kg. Notice the plot is non-linear and

asymmetric. Notice the value of acceleration when m2 is much less than or much

more than m1!

Since the force equations are so easy on this one, add in a Contour plot in

MATLAB showing Ð�¸ for a wide range of both �Ñ&�« values.

m2

m1

h

x

y

a

x

y

a

74

OPTION 2: Newton’s Law Part I

Test: Does Newton’s 2nd Law accurately predict the acceleration of a two

mass system?

Theory: Include an FBD of each object, force equations for each object,

and the derive equation for a%V. Explain how a%V makes sense in the two

obvious special cases F� ≫ F5&F� ≪ F5.

Procedure: Adjust the feet of the track (or add shims) until it is level.

You’ll know it is level if a glider remains motionless.

For each mass combo, determine ap(® using a photogate pulley (Smart Pulley) to record E�-data.

Record F5&F� used in each trial with a balance. Don’t forget to include the mass hanger!

I expect a table of E�-data for 3 possible values of F5 with 11 values of F�.

I would use F5 � 200, 300, &400g with F� � 0, 2, 4, … 20g.

For one trial, create a E�-plot to use in your talk (to explain how you determined acceleration).

For the rest get acceleration using the LINEST function (see lab manual appendices).

Make a contour plot of your experimental data (see the table of contents of this manual…last lab in this manual).

Make a theoretical contour plot as well.

Get a % precision estimate using 1¬­=®p¬­=®p from the LINEST function.

Get % difference for each data point.

Also get the absolute value of the percent difference for each data point.

Data/Graph:

• For single trial, show a E�-plot to explain how you obtained acceleration.

• For a single value of F5, show a Type II graph (theory as smooth line without data points, experiment as

points without line) of a vs F�. Change the data points from dots to cross-hairs using error bars (see the

help file and discuss with your instructor).

• Also compare the theoretical and experimental contour plots.

Conclusions: Tell the audience if the Newton’s 2nd Law is in good agreement with your experimental results that

used kinematics. This is probably true if % difference is less than % error.

As check on your work I provided a theoretical plot of for m1 = 200 g. Notice the

acceleration asymptotically approaches g. Ideally you could generate two similar

plots using your two different values for m1. These could be shown in the theory

portion of your talk. If you make the plots, try to point out the difference in the

rate at which the curve approaches the asymptote. Whoops…notice that I missed

changing the horizontal axis label into the same font as everything else!

Since the force equations are so easy on this one, add in a Contour plot in

MATLAB showing Ð�¸ for a wide range of both �Ñ&�« values.

Table

m2

m1

75

OPTION 3: Newton’s Law Part II

Test: Does Newton’s 2nd Law accurately predict the acceleration of a two

mass system?

Theory: Include an FBD of each object, force equations for each object, and

the derive equation for a%V. Explain how a%V makes sense in the two obvious

special cases F� ≫ F5&F� ≪ F5. Explain why the size of F� relative to F5 sin ± indicates the direction of acceleration.

Procedure: Adjust the feet of the track (or add shims) until it is level.

You’ll know it is level if a glider remains motionless.

ONCE IT IS LEVELED, then begin to raise the angle.

Use several different angles in your experiment

We may need to cut some shims (or use some slotted masses).

Think, the height of the shims is given by sin ± V5.ÊÊm (see figure).

For each mass combo, determine ap(® using a photogate pulley (Smart Pulley) to record E�-data.

Record F5&F� used in each trial with a balance. Don’t forget to include the mass hanger!

Get E�-data for 6 possible angles (0°, 1°, … , 5°) using F5 � 200g with 10 values of F� (0, 2, 4, … 18g).

BE CAREFUL WITH THE LARGE ANGLES!

• Try to ensure the equipment survives for next year by catching the glider without it smashing into things.

• Watch for the hose bumping into the table when you start angling up!

For one trial, create a E�-plot to use in your talk (to explain how you determined acceleration).

For the rest get acceleration using the LINEST function (see lab manual appendices).

Make a contour plot of your experimental data (see the table of contents of this manual…last lab in this manual).

Make a theoretical contour plot as well.

Get a % precision estimate using 1¬­=®p¬­=®p from the LINEST function.

Get % difference for each data point.

Also get the absolute value of the percent difference for each data point.

Data/Graph:

• For single trial, show a E�-plot to explain how you obtained acceleration.

• For a single value of ±, show a Type II graph (theory as smooth line without data points, experiment as

points without line) of a vs F�. Change the data points from dots to cross-hairs using error bars (see the

help file and discuss with your instructor).

• Also compare the theoretical and experimental contour plots.

Conclusions: Tell the audience if the Newton’s 2nd Law is in good agreement with your experimental results that

used kinematics. This is probably true if % difference is less than % error.

As check on your work I provided a theoretical plot for h = 6.35 cm & m1 = 200 g.

Notice the acceleration asymptotically approaches g. Ideally you could generate a

similar plot using your values for m1 and h. This could be shown in the theory portion

of your talk. Whoops…notice that I missed changing the horizontal axis label into

the same font as everything else!

Since the force equations are so easy on this one, add in a Contour plot in

MATLAB showing Ð�¸ for a wide range of both �Ñ&�« values.

m2

1.00 m h

θ

m1

76

OPTION 4: Circular Motion Using CENCO Quantitative Centripetal Force Apparatus

Warning: This experiment can cause a sizeable mass to hit your face at high speeds. Discuss appropriate safety

precautions with your instructor prior to operation.

Test: Does Newton’s 2nd Law accurately model the acceleration of a body in circular motion?

Theory: For a mass m in uniform circular motion the net force towards the

center (Fc) is given by YÕ FE��

where v is the speed of the mass and r is the radius of the circular motion.

Consider first the system in Figure 1a which is at rest. The indicator rod is

set to a fixed position. Balancing mass mb is adjusted until the pointer mass

is directly above the indicator rod. When the pointer mass is in equilibrium

above the indicator rod, we know that the force exerted by the spring is

equivalent to the balancing weight (Fspring = mbg).

Now consider the system in Figure 1b which is rotating. The system is now

caused to rotate in such a way that the pointer mass remains directly above

the indicator rod. Furthermore, the spring force is the only force exerted

towards the center of circular motion. This gives Y¬®¥`iL F® E��

where mp is the pointer mass, r is the distance from the center of the shaft to

the indicator rod, and v is the speed at which mp moves. For uniform

circular motion with period T one finds E 2>�¼

Notice the spring in Figure 1b is stretched the same as in Figure 1a so one

still has Fspring = mbg. Verify that combining these facts gives Fs� F® E��

This gives a way to directly compare the required centripetal force (mbg) to

the required speed as predicted by Newton’s 2nd Law as applied to uniform

circular motion.

Procedure: Lock the indicator rod in place; measure and record r. Adjust

the nuts on the threaded hook such that almost none of the threaded hook

extends out of the center shaft on the side opposite the spring. Measure and

record the balancing mass mb required to align mp with the indicator rod.

Remove the balancing mass before rotating the shaft.

Now spin the center shaft in such a way as to keep mp directly above the indicator rod. You will likely have to give

the center shaft a small twist every revolution or two to ensure the period of rotation remains roughly constant.

Tip: when the pointer mass first passes directly over the indicator rod start counting from 0 to 10. This will give

you 10 orbits. Take the time for 10 orbits with a stopwatch. Divide by 10 to get the period. This average period

should have acceptable error. Note: we’ve tried doing the experiment with a photogate and errors were much worse.

Now adjust the nuts on the threaded hook such that a slightly greater mass mb is required to balance the pointer mass

above the indicator rod. Repeat the experiment to obtain both mb & T. Continue adjusting the nuts on the threaded

hook until you obtain mb and T data for at least 7-10 different nut positions on the threaded hook. You can then

make a plot several plots. WARNING: trying to measure velocity directly with the photogate often gives bad data.

Measure period with the photogate…not velocity.

Indicator

rod

Table

Counterbalance weight

mb

Figure 1a – Determining the spring

force required to keep the pointer above

the indicator rod.

Figure 1b – Determining the rotation

period required to keep the pointer

above the indicator rod.

Pointer

mass

Period = T

r

Threaded

hook

77

Data/Graph: On one plot we want to plot the raw data in the experiment versus the predicted data from theory. The

raw data is Fs on the k-axis and ¼ on the �-axis. To create the theoretical data, first rearrange Fs� F® K&¥ to

include the period instead of E. Hint: use E Õ`¥Õ,tlp¥piÕp®p¥`=Ö . Then, solve your new equation for ¼ on the left side of

the equation. You should end up with ¼%V (aqÇ�Àℎ�µÀ�a�)Fsg5/�

or something like that. I provided fake data on next page to give you a better feel for what I mean…

In my experience, speed is more intuitive to students than period when discussing circular motion. Because of this,

we want to use your experimentally determined periods to calculate the experimental velocity for each trial.

If we do this we could make a table of E� versus Fs. Here Fs is the independent variable and lies on the k-axis

once again. Rearranging our equation again gives E� Fs ��F® This should be a linear equation if E� is on the �-axis and Fs is on the k-axis. The slope should be given by ����� ��F®

Note: don’t forget to determine the units of the slope. Solving the above equation for � gives �p(® F®� (�����)

This gives us a test! If the value of �p(® obtained from the slope is close to the accepted value of 9.8 m/s2 then it

must be appropriate to use aÕ K&¥ in Newton’s 2nd Law problems involving uniform circular motion.

Summary of required plots:

1) Get one plot of raw data (¼ vs. Fs) OR (¼ vs. Fsg5/�) with theoretical line (smooth line, no points) and

experiment (points with error bars only).

2) Get one plot of E� vs. Fs. Use a trendline on this plot to get the slope. Use the slope to get �p(® and

compare to the accepted value. Include error bars on your experimental points and use LINEST to get the

error in the slope. Propagate this error through to �p(® so we know how crappy your data is (1%, 2%, etc).

Note: When getting data you get seven trials for each of two different fixed radii.

Conclusions: Compare % difference to % error for your value of �p(®. If the % difference is less than the % error

then the use of aÕ K&¥ and Newton’s 2nd Law accurately model uniform circular motion. State if Newton’s 2nd law

accurately models the acceleration of a body in circular motion. Discuss errors & limitations inherent in using this

device. Discuss ways one might minimize these errors or re-design the apparatus to produce better results.

Fake data on next page to give you a better feel for some different styles of analyzing the data…

78

r (m) mp (kg)

0.170 0.525

mb (kg) mb-1/2 (kg-1/2) Texp (s) Tth (s) vexp (m/s) vth (m/s) vexp

2 (m2/s2) vth2 (m2/s2)

0.200 2.24 1.67 1.34 0.64 0.80 0.408 0.63

0.400 1.58 1.05 0.95 1.02 1.13 1.037 1.27

0.600 1.29 0.88 0.77 1.22 1.38 1.479 1.90

0.800 1.12 0.72 0.67 1.49 1.59 2.214 2.54

1.000 1.00 0.67 0.60 1.59 1.78 2.534 3.17

1.200 0.91 0.57 0.55 1.89 1.95 3.567 3.81

0.00

0.50

1.00

1.50

2.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

T (s)

mb-1/2 (kg-1/2)

0.000

1.000

2.000

3.000

4.000

5.000

0.000 0.500 1.000 1.500

v2 (m2/s2)

mb (kg)

y = 3.0024x - 0.2284

R² = 0.9828

0.000

1.000

2.000

3.000

4.000

5.000

0.000 0.500 1.000 1.500

v2 (m2/s2)

mb (kg)

79

OPTION 5: Static Phriction Phreaque-Out

Test: Does Newton’s 2nd Law accurately predict the behavior of a two mass system?

Consider the figure shown at right. Mass m1 is a hockey puck while mass m2 is a mass hanger with

additional slotted weights. Determine the largest m2 that can be placed on the system while still allowing

the puck to remain at rest. Use angles of 0.0° up to 72.0° in 8.0° increments. Test the same angle at least

five different spots on the board to get an average value of m2 for each angle. Watch out! Make sure the

string is always parallel to the board by adjusting the pulley.

Assume the accepted value for µs=0.70.

Use that µs to predict theoretical values of m2 for each θ.

Plot m2 versus θ.

Also determine the experimental value of µs for each angle.

Determine the average experimental µs.

Theory/Procedure: Will definitely need more than one slide for all this…Show a photo of the apparatus. Draw an

FBD for each mass. Write the force equations. Solve the equations algebraically for m2 to show the class how m2

should change as θ changes. Also solve the force equation (algebraically) for µs. Then plug in numbers and get

values for µs for each trial. Also determine an average value of µs from all trials.

Describe how you determined the values of m2 for each trial. How many trials did you perform at each θ to get an

average? Determine error estimates for each method based on the range of values for m2 obtained for each θ.

Describe how your experimental procedure to measure the values of a.

Data/Graph: You should be able to make a plot of m2 versus θ using your data. You should also be able to come

up with theoretical values based on the force equation you found. Create a graph similar to the one in the lab

manual appendix under Sample Graph Type II. The theory should show a smoothed line with no points while the

experiment should have data points indicated by dots. Put error bars on your graph using the MS Excel help file or

by having a discussion with your instructor.

Conclusions: Does Newton’s 2nd law accurately predict the values of m2 required to cause the onset of slipping?

Does the average value obtained for µs agree with accepted values (hint: compare using an internet search)? Discuss

both your percent errors and percent differences to support the validity of your claims.

For reference I made a theoretical plot of m2 vs θ for a puck

mass of 165 g and assuming µs = 0.9. Something similar

might be useful in your talk to explain the theoretical

equations. Notice that on level ground m2 < m1; this is an

artifact of µ < 1. Also notice at 90° the two masses must

equal as friction doesn’t come into play when the board is

straight up and down!

θ

m1 m2

µ (m) m1 (kg)

0.9 0.165

θ (°) m2 (kg)

0 0.149

10 0.175

20 0.196

30 0.211

40 0.220

50 0.222

60 0.217

70 0.206

80 0.188

90 0.165

80

OPTION 6: Static Phriction Phreaque-Out

Test: Does Newton’s 2nd Law accurately predict the behavior of a two mass system?

Consider the figure at right. Mass m1 is a hockey puck while mass m2 is a mass hanger with

additional slotted weights. Determine the largest m2 that can be placed on the system while still

allowing the puck to remain at rest. Use angles of 0.0° up to θ crit in 5.0° increments. Note: the

critical angle can also be recorded as a data point since m2 = 0 for θ crit. The critical angle is

obtained by the method described in class. Test the same angle at least five different spots on the

board to get an average value of m2 for each angle. Watch out! Make sure the string is always

parallel to the board by adjusting the pulley.

Assume the accepted value for µs=0.70.

Use that µs to predict theoretical values of m2 for each θ.

Plot m2 versus θ.

Also determine the experimental value of µs for each angle.

Determine the average experimental µs.

Theory/Procedure: Will definitely need more than one slide for all this…Show a photo of the apparatus. Draw an

FBD for each mass. Write the force equations. Solve the equations algebraically for m2 to show the class how m2

should change as θ changes. Also solve the force equation (algebraically) for µs. Then plug in numbers and get

values for µs for each trial. Also determine an average value of µs from all trials.

Describe how you determined the values of m2 for each trial. How many trials did you perform at each θ to get an

average? Determine error estimates for each method based on the range of values for m2 obtained for each θ.

Describe how your experimental procedure to measure the values of a.

Data/Graph: You should be able to make a plot of m2 versus θ using your data. You should also be able to come

up with theoretical values based on the force equation you found. Create a graph similar to the one in the lab

manual appendix under Sample Graph Type II. The theory should show a smoothed line with no points while the

experiment should have data points indicated by dots. Put error bars on your graph using the MS Excel help file or

by having a discussion with your instructor.

Conclusions: Does Newton’s 2nd law accurately predict the values of m2 required to cause the onset of slipping?

Does the average value obtained for µs agree with accepted values (hint: compare using an internet search)? Discuss

both your percent errors and percent differences to support the validity of your claims.

For reference I made a theoretical plot of m2 vs θ for a puck

mass of 165 g while assuming µs = 0.9. Something similar

might be useful in your talk to explain the theoretical equations.

Notice that on level ground m2 < m1; this is an artifact of µ < 1.

Also notice at 42° m2 = 0. This corresponds to the critical angle

predicted by µ = 0.9! Finally, while not easily noticeable, the

slope of the line becomes slightly more negative as the angle

increases. This is not a straight line so do not use a linear

trendline!

θ

m1

m2

µ (m) m1 (kg)

0.9 0.165

θ (°) m2 (kg)

0 0.149

10 0.118

20 0.083

30 0.046

40 0.008

41 0.004

42 0.000

81

OPTION 7: Kinetic Phriction Phreaque-Out

(works best with encoder/smart pulley)

Consider the figure shown at right. Mass m1 is a hockey puck while mass m2 is a mass hanger with

additional slotted weights. Select m2 such that the system accelerates up the plane at a reasonable rate for a

wide range of angles. Consider using 25.0° to 70.0° in 5.0° increments. Again, before starting, find a

single value of m2 that works for the entire range of angles. Watch out! Make sure the string is always

parallel to the board by adjusting the pulley.

For each F�, determine ap(® using a photogate pulley (Smart Pulley) to record E�-data.

For one trial, create a E�-plot to use in your talk (to explain how you determined acceleration).

For the rest get acceleration using the LINEST function (see lab manual appendices).

Assume the accepted value for µk=0.40.

Use that µk to predict theoretical values of a for each θ.

Plot a versus θ.

Also determine the experimental value of µk for each angle.

Determine the average experimental µ k.

Test: Does Newton’s 2nd Law accurately predict the behavior of a two mass system?

Theory/Procedure: Will definitely need more than one slide for all this…Show a photo of the apparatus. Draw

FBDs for each mass. Write the force equations. Solve the equations algebraically for a to show the class how a

should change as θ changes. Also solve the force equation (algebraically) for µk. Also show the average value of µk

from all trials.

Describe how your experimental procedure to measure the values of a. How many trials did you perform at each θ

to get an average? Did you use Tracker, photogates, or stopwatch? Give a few details on your technique as not

everyone used the same technique. Determine error estimates for each method based on the range of values for m2

obtained for each θ. Determine an error estimate for your average value of µk.

Data/Graph: You should be able to make a plot of a versus θ using your data. You should also be able to come up

with theoretical values based on the equation you found. Create a graph similar to the one in the lab manual

appendix under Sample Graph Type II. The theory should show a smoothed line with no points while the

experiment should have data points indicated by dots. Put error bars on your graph using the MS Excel help file or

by having a discussion with your instructor.

Conclusions: Does Newton’s 2nd law accurately predict the values of a for each angle? Does the value obtained for

µk agree with accepted values (hint: compare using an internet

search)? Discuss both your percent errors and percent differences

to support the validity of your claims.

For reference I made a theoretical plot of a vs θ for m1 =165 g, m2

= 250 g and assuming µk = 0.4 or 0.9…I forget which.

Something similar might be useful in your talk to explain the

theoretical equations. At first I was shocked to discover the

acceleration first decreases then increases again for the same m2.

As you increase the angle the normal force decreases. Thus, as

the angle is increased, the frictional force down the plane is

decreasing. At the same time, as the angle increases the

component of m1g down the plane increases. These two factors

cause the unusual graph. Note: while it looks like a parabola it is

not; do not use a polynomial of order 2 trendline on this graph!

θ

m1 m2

µ (m) m1 (kg) m2 (kg)

0.9 0.165 0.25

θ (°) a (kg)

0 2.40

10 1.77

20 1.28

30 0.92

40 0.71

50 0.66

60 0.78

70 1.04

80 1.46

90 2.01

82

OPTION 8: Kinetic Phriction Phreaque-Out Option 2

(works best with encoder/smart pulley)

Consider the figure shown at right. Mass m1 is a hockey puck while mass m2 is a mass hanger with

additional slotted weights. Select m2 such that the system accelerates down the plane at a reasonable

rate for a wide range of angles. You’ll probably want to use small angles such as 0.0° to 16.0° in

2.0° degree increments. Before starting, find a single value of m2 that works for the entire range of

angles. Watch out! Make sure the string is always parallel to the board by adjusting the pulley.

For each F�, determine ap(® using a photogate pulley (Smart Pulley) to record E�-data.

For one trial, create a E�-plot to use in your talk (to explain how you determined acceleration).

For the rest get acceleration using the LINEST function (see lab manual appendices).

Assume the accepted value for ×Ø 0.4.

Use that µk to predict theoretical values of a for each θ.

Plot a versus θ.

Also determine the experimental value of µk for each angle.

Determine the average experimental µ k.

Test: Does Newton’s 2nd Law accurately predict the behavior of a two mass system?

Theory/Procedure: Will definitely need more than one slide for all this…Show a photo of the apparatus. Draw

FBDs for each mass. Write the force equations. Solve the equations algebraically for a to show the class how a

should change as θ changes. Also solve the force equation (algebraically) for µk. Then plug in numbers and get

values for µk for each trial. Also determine an average value of µk from all trials.

Describe how your experimental procedure to measure the values of a. How many trials did you perform at each θ

to get an average? Did you use LINEST? Give a few details on your technique as not everyone used the same

technique. Determine error estimates for each method based on the range of values for m2 obtained for each θ.

Determine an error estimate for your average value of µk.

Data/Graph: You should be able to make a plot of a versus θ using your data. You should also be able to come up

with theoretical values based on the equation you found. Create a graph similar to the one in the lab manual

appendix under Sample Graph Type II. The theory should show a smoothed line with no points while the

experiment should have data points indicated by dots. Put error bars on your graph using the MS Excel help file or

by having a discussion with your instructor.

Conclusions: Does Newton’s 2nd law accurately predict the values of a for each angle? Does the value obtained for

µk agree with accepted values (hint: compare using an internet search)? Discuss both your percent errors and

percent differences to support the validity of your claims.

For reference I made a theoretical plot of a vs θ for m1 =165 g,

m2 = 200 g and assuming µk = 0.9. Something similar might be

useful in your talk to explain the theoretical equations. This is

a challenging experiment because we see the acceleration is

extremely sensitive to a small change in angle. Also, while the

graph appears linear it is not. Do not use a linear trendline on

this graph.

θ

m1

m2

µ m1 (kg) m2 (kg)

0.9 0.165 0.2

θ (°) a (kg)

0 1.38

5 1.78

10 2.21

15 2.67

20 3.14

25 3.63

30 4.13

35 4.64

83

OPTION 9: Static & Kinetic Phriction Phreaque-Out

Consider the top figure shown at right. Mass m1 is a hockey puck while mass m2 is a mass hanger with

additional slotted weights.

Select a single angle that is significantly greater than the critical angle (perhaps 65.0°).

First determine what value of F� causes puts the system on the verge of slipping up the plane.

Use five different spots on the board to get an average value of this critical mass.

Next determine what value of F� causes puts the system on the verge of slipping down the plane.

Use five different spots on the board to get an average value of this critical mass.

Choose three small F�’s that cause the system to accelerate down the plane.

Hint: one value of F� could be zero.

For each F�, determine an experimental value for a using a photogate pulley (Smart Pulley) to record E�-data.

For one trial, create a E�-plot to use in your talk (to explain how you determined acceleration).

For the rest get acceleration using the LINEST function (see lab manual appendices).

Choose four large F�’s that cause the system to accelerate up the plane.

Record the acceleration for each large m2 using one of the methods described below.

Assume accepted values of ׬ 0.7&×Ø 0.4.

Use that ׬ to predict a theoretical value for the critical mass.

Use that ×Ø to predict theoretical values of a for each F�.

Plot aversus F� (showing both experimental and theoretical values).

In addition, determine the average experimental ׬&×Ø.

Test: Does Newton’s 2nd Law accurately predict the behavior of a two mass system?

Theory/Procedure: Will definitely need more than one slide for all this…Show a photo of the apparatus. Draw

FBDs for each case. You should have four cases: static with friction up the hill, static with friction down the hill,

kinetic with friction up the hill, and kinetic with friction down the hill. Write the force equations. Solve the

equations algebraically for a to show the class how a should change as m2 changes. Also solve the force equation

for each case (algebraically) for ׬ or ×Ø.

Describe your experimental procedure to measure the values of a. How many trials did you perform at each m2 to

get an average? Did you use LINEST? How did you determine the m2 that would balance the system? Did you try

it at the same spot on the board every time or a bunch of random spots? Give a few details on your technique as not

everyone used the same technique. Determine error estimates for each method. Determine an error estimate for

your average values of ׬&×Ø.

Data/Graph: You should be able to make a plot of a versus F� using your data. You should also be able to come

up with theoretical values based on the equation you found. Create a graph similar to the one in the lab manual

appendix under Sample Graph Type II. The theory should show a smoothed line with no points while the

experiment should have data points indicated by dots. Put error bars on your graph using the MS Excel help file or

by having a discussion with your instructor.

Conclusions: Does Newton’s 2nd law accurately predict the values of a for each F�? Do the values obtained for ׬&×Ø agree with accepted values? Discuss both your percent errors and percent differences to support the validity

of your claims.

Note: see the next page for a theoretical plot of what should be happening in your experiment.

θ

m1 m2

84

I made a theoretical plot for reference using ׬ 0.9, F5 0.165kg, and ± 65°. The plot of a vs F� is particularly devious because the theoretical equation for a changes twice!

Think: when the block is sliding you should use ×Ø instead of ׬…

The critical masses m2 = 0.087 kg and 0.212 kg appear notable in the figure.

For m2 masses between those values the system should remain at rest.

Below the lower limit the system accelerates negatively (down the incline) while above the upper limit the system

accelerates positively (up the incline).

Notice I made a mistake in my theory equations.

Once the block starts accelerating, I should’ve used ×Ø instead of ׬.

Problem 6.29 in the workbook gives a similar situation but has the correct plot in the solutions.

Going Further…a lot further. Try to create a contour plot for the puck’s acceleration as the coefficients of friction

and hanging mass both change. I think this is doable yet very challenging. Notice the critical mass to cause sliding

will vary as the coefficients change. You’ll have to make a reasonable assumption about the coefficients. If you can

handle this, you’re ready to write tax code.

׬ m1 (kg) θ (°)

0.9 0.165 65

m2 (kg) a (m/s2)

0.030 -2.85

0.050 -1.68

0.075 -0.48

0.086 -0.03

0.087 0.00

0.212 0.00

0.213 0.02

0.250 0.89

0.300 1.85

0.350 2.62

0.400 3.26

85

OPTION 10: Static & Kinetic Phriction Phreaque-Out

Consider the apparatus shown at right. Ensure the string is parallel to the board. Mass F5 is a

hockey puck while mass F� is a mass hanger with additional slotted weights.

Select an angle of about 15° and start with F� 0.

Gradually increase the mass until the onset of slipping occurs.

Check perhaps 10 different spots on the board.

The average value of mass required to cause slipping is the experimental critical mass.

In addition, choose seven different F�’s which cause the system to accelerate down the plane.

For each F�, determine an experimental value for a using a photogate pulley (Smart Pulley) to record E�-data.

Assume accepted values of ׬ 0.7&×Ø 0.4.

Use that ׬ to predict a theoretical value for the critical mass.

Use that ×Ø to predict theoretical values of a for each F�.

Plot aversus F� (showing both experimental and theoretical values).

In addition, determine the average experimental ׬&×Ø.

Test: Does Newton’s 2nd Law accurately predict the behavior of a two mass system?

Theory/Procedure: Will definitely need more than one slide for all this…Show a photo of the apparatus. Draw

FBDs for each case. You should have two cases: static friction directed up the plane (acceleration is zero) and

kinetic friction directed up the plane (acceleration down the plane). Write the force equations. Solve the equations

algebraically for a to show the class how a should change as m2 changes. Also solve the force equation for each

case (algebraically) for µ. Then plug in numbers and get values for µ for each trial.

Describe how your experimental procedure to measure the values of a. How many trials did you perform at each m2

to get an average? Did you use Tracker, photogates, or stopwatch? How did you determine the m2 that would

balance the system? Did you try it at the same spot on the board every time or a bunch of random spots? Give a

few details on your technique as not everyone used the same technique. Determine error estimates for each method.

Determine an error estimate for your average value of µk.

Data/Graph: You should be able to make a plot of a for each m2 using your data. You should also be able to come

up with theoretical values based on the equation you found. Create a graph similar to the one in the lab manual

appendix under Sample Graph Type II. The theory should show a smoothed line with no points while the

experiment should have data points indicated by dots. Put error bars on your graph using the MS Excel help file or

by having a discussion with your instructor.

Conclusions: Does Newton’s 2nd law accurately predict the values of a for each F�? Do the values obtained for ׬&×Ø agree with accepted values? Discuss both your percent errors and percent differences to support the validity

of your claims.

I made a plot of a versus m2 on the next page to give you an idea of how things might look.

θ

m1

m2

86

The critical mass m2 = 0.101 kg is notable in the figure.

For m2 masses below that value the system should remain at rest.

Notice I made a mistake in my theory equations.

Once the block starts accelerating, I should’ve used ×Ø instead of ׬.

Problem 6.29 in the workbook gives a similar situation but has a correct plot in the solutions.

׬ m1 (kg) θ (°) m2crit (kg)

0.9 0.165 15 0.101

m2 (kg) a (m/s2)

0.000 0.00

0.050 0.00

0.100 0.00

0.101 0.00

0.102 0.05

0.103 0.08

0.150 1.53

0.200 2.67

0.250 3.52

0.300 4.20

0.350 4.74

0.400 5.19

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.000 0.100 0.200 0.300 0.400

a (m/s2)

m2 (kg)

87

Circular Motion

Apparatus: hanging mass sets, braided string, scissors, uniform circular

motion apparatus, stopwatches, goggles.

Purpose: To analyze the forces acting on objects undergoing circular motion

or moving along a circular arc. To confirm that objects in circular motion

are accelerated toward the center of the circle and the magnitude of radial

acceleration is given by ar = v2/R. Note: radial and centripetal acceleration

are essentially the same thing (|ar| = |ac|) with the possible exception of a ±

sign (depends what book you read). We will do this by comparing the

theoretical velocity (from radial acceleration) to experimental velocity (from

timing the circular motion).

Theory: In your calculations section it is expected that you will draw the FBD’s for each object, determine the

force equations for each object, and solve algebraically for vth. This can be done by starting with Newton’s second

law. Newton’s 2nd law states that ΣF = ma. In this case, the tension in the string is constraining the stopper to move

in a circle in a horizontal plane (see figure). Newton’s 2nd law for the rotating stopper says

Tcosθ = msac where r=Lcosθ

If we make the assumption that θ < 15°we find that cosθ > 0.95. By completely ignoring the angle we see that

T = msac and r=L

which should be correct to within about 5%. The hanging mass is motionless (in the vertical direction) when both

the tension and the gravitational force act on it. So for the hanging mass Newton’s 2nd law says

T – mh g = 0 ==> T = mhg

These equations can be combined to solve for the theoretical velocity of the stopper using

ar = v2/r

You should find that

E%V 9? ��?

where the ?’s are terms I expect you to figure out (they relate to the masses).

Lastly, one can measure the time t it takes the object to complete 10 revolutions. Knowing the length of the string L

that extends from the tube to the center of the stopper one can determine the distance traveled in 10 revolutions.

Recall that one revolution is equivalent to the circumference of a circle with r=L. The experimental velocity can

then be obtained by Ep(® ¼��a�����a�À�¼��a�¼�F�

You will repeat this experiment for several different hanging masses.

Procedure:

• Before taking any measurements you will want to get used to the feel of the experiment. Try to find a length of

string that works for both the heaviest and lightest masses before taking data.

• Start with your lightest mass. Take the tube with the string and attach the hanging mass (mh) to one end and a

20-50-gram stopper (ms) to the other.

• Rotate the stopper in a circle over your head at a speed fast enough so the hanging mass is suspended at a fixed

location. Notice what happens to the speed of the stopper with a long radius versus a short radius (you will be

asked about this later).

• Now try to do the same tests with a 500 g hanging mass. Can you keep the 500 g mass from slipping with a

radius of about 100 cm? If not, ask your instructor for help.

ms

mh

L

θ

88

• In order to maintain a constant radius, you can try marking the string with a pen. Practice spinning the mass

and observing the mark on the string. Select a target radius between 90-120 cm. Try to use only this radius for

the rest of the day.

• Have your lab partner measure the time t for 10 revolutions.

• Measure the length (L) of the string from the center of the stopper to where it enters the tube. Don’t forget,

if you marked the string and tried to keep a constant radius that should help you determine L.

• Repeat the experiment for five different mh’s. Maintain fixed values of ms and L for all five experiments.

Though you cannot do this perfectly, try to keep your radii within a few cm of your target value of the radius.

• You should end up with 5 different values for the theoretical velocity (from the FBD’s and force equations) and

5 different values for the experimental velocity (from the measurements of t and L).

• Tabulate your data. Be sure to calculate the percent difference and precision.

• Make a graph of both vexp and vth versus mh. This is a Type 2 Graph; look up instructions on how to make this

type of graph in the table of contents.

• You can take this experiment a bit further by also graphing vexp2 versus mh. This is a Type 1 Graph; look up

instructions for making this graph in the table of contents.

• Make an FBD. Get an equation from your FBD and solve it for v2.

• Compare the equation of a line (y = mx + b) to your equation relating mh and v2. What variable is x? What

variable is y? What expression is equal to the slope?

• From that slope, and knowing your values of L and ms, figure out an experimental value of g. Hint: compare the

equation of a line (y = mx + b) to your equation relating mh and v2.

• Compare this to the known value of g with a percent difference.

• Use the LINEST function to determine the percent error associated with your slope. Recall, the percent error in

a measurement is simply the reading error over the measurement itself times 100%.

• Assume that the angle in the angle is about 5%. Estimate the remaining %errors for your measurements.

• Determine the precision of your experiment by using the error analysis appendix.

Conclusions:

1. Did vth match vexp? Defend your conclusions by comparing the %differences to the precision.

2. Did the experimental determination of g (graphing v2 versus mh) match up to the accepted value of g? Defend

your conclusions by comparing the %differences to the precision.

3. In this experiment we kept ms and L constant. As mh is increased, what should happen to the period and the

velocity? To answer this, don’t guess but rather solve for period in terms of ms, mh, g, and L (you have

already done this for v). Then let the equation guide your reasoning.

4. Suppose instead that as the hanging mass was increased we kept the velocity constant but allowed the length of

the string to change. What would happen to the radius of circular motion (approximately L) as mh

increased? Again, don’t guess but let the equation guide your reasoning. First solve for L in terms of the

velocity, ms, g, and mh. Does it make sense?

5. Suppose instead that as the hanging mass was increased we kept the period constant but allowed the length of

the string to change. What would happen to the radius of circular motion (approximately L of the string

as mh increased? Again, don’t guess but let the equation guide your reasoning. First solve for L in terms of

the period, ms, g, and mh. Does it make sense? Sketch a plausible graph of L versus mh data.

89

Work and Energy Conservation

Apparatus: small bases, right angle clamps, medium rods, small rods, scissors, calipers, pulley cord, aluminum

cylinders, spheres with hooks, Air track glider (1 per track), air tracks, air supply hoses & power cords, air track

accessories kit, adjustable end stops (1 per table), air track springs, air track spring connectors, wood blocks,

photogate heads, photogate interface cables, meter sticks, PASCO Science Workshop 750 Interface & Power

Supplies, hanging mass sets, pulleys

Don’t write any introduction, or procedure. A sketch of each activity will be needed to make sense of your data.

Include front and side views to clarify the pictures. It may also be useful to show before and after pictures. ALL

PICTURES SHOULD BE AT LEAST A HALF PAGE IN SIZE TO RECEIVE CREDIT. Include important

variables in your diagrams as well as the names of the different types of equipment. Label all equipment in your

diagrams down to the tiniest detail (C-clamp, right angle clamp, base, string, pulley, rod, photogate, connector

level).

Experiment A (40 minutes): Use a table clamp, a medium rod, a right angle clamp, a small rod, and a metal

cylinder attached to a string to create a simple pendulum. Use L= 0.50 m from the center of

mass of the cylinder to the point where the string attaches to the rod. Predict how fast the

cylinder should be moving at the bottom of its motion. Use trig to derive h=L(1-cosθ).

Use Ei + W = Ef to derive vth in terms of L, θ, and g.

Set a photogate such that the center of mass of the cylinder breaks the beam of the

photogate. Find some way to measure an initial angle of 60° from the vertical. Release the

pendulum from rest at this initial angle and record the experimental velocity at the bottom

of the swing vexp.

Conclusions Exp A:

1) What is your derived result for vth?

2) Does the mass of the pendulum affect vth?

3) Estimate your %errors in L, θ, and g. Add these %errors to estimate the %precision.

4) Compare this to your %difference and state if your experimental result is in good agreement with the theory

of the conservation of energy.

5) Determine the tension in the string at the bottom of the swing using an FBD.

Experiment B (90 minutes): This activity is a variation on the previous one using

a sphere instead of a cylinder. Don’t use thin black string (it breaks); use the

braided string (white). Start with your pendulum held parallel to the ground as

shown in the figure. Recall that L=0.50 m is measured from the center of mass of

the sphere to the pivot point. Now a thin rod is used to interrupt the swing of the

pendulum. When the simple pendulum of length L is released from rest, it swings

down and around the thin rod located a distance D below the pendulum support.

Use an FBD at the top of the loop to derive Rgv =min . Hint: what must be true

about the tension at the minimum velocity?

Think: If a light rod was used instead, how would vmin differ?

How does R relate to L and D?

Where will the ball have trouble making it through the loop?

Do a conservation of energy problem (Ei + W = Ef) using the initial point and this trouble spot.

Set the bottom of the swing as y=0. Notice that both initial and final will have GPE!!!!

Derive the min value of D for which the sphere completes the circular motion (�%V i7 � where n is an integer).

Now try it. Experimentally determine the position of the rod where the pendulum just makes it all the way around

without losing tension in the string. Look carefully at the path of the swinging ball; if it spirals inward to the

interrupting rod then you will know the string was always under a little bit of tension. If the ball makes it over the

rod just once you will know it flew over as a projectile and that the ball did not actually swing in a loop.

Conclusions Exp B:

1) What is your derived result for Dth?

D

L

vmin

R

h

L

m

m

Ei=?

Ef=?

90

2) Your initial angle here was 90° from the vertical. How would a smaller angle affect Dth? Would it tend to

make Dth bigger, smaller, or would it have no change?

3) Estimate your %errors in L and D. Add these %errors to estimate the %precision.

4) Compare this to your %difference and state if your experimental result is in good agreement with the theory

of the conservation of energy.

Experiment C (40 minutes): Build the apparatus shown below using a the long metal tracks on the shelves above

the computers. Starting with the spring initially unstretched, release the system from rest. The 1 kg mass should be

large enough that it doesn’t even move (so you don’t need to worry about the energy of the 1 kg mass). The mass

m=200g will stretch the spring some maximum amount xmax. Ask your instructor for the manufacturer’s value for k

or determine it by hanging a mass and using an FBD. Use Ei + W = Ef to derive µth in terms of m, k, xmax, θ, and g.

Conclusions Exp C:

1) What is your derived result for µth?

2) Estimate your %errors in m, k, xmax, θ, and g. Add these %errors to estimate the %precision.

3) Determine the normal force on the 1 kg block when the 200 g mass is at the xmax position.

1 kg

200g

table

floor

θ

91

Experiment D (40 minutes): Build the apparatus shown at right.

Starting with the spring initially unstretched, release the system from

rest. The 1 kg mass should be large enough that it doesn’t even move

(so you don’t need to worry about the energy of the 1 kg mass). The

spring will stretch some maximum amount xmax. Use Ei + W = Ef to

derive kth in terms of m1, m2, xmax, θ, and g.

Conclusions Exp D:

1) What is your derived result for kth?

2) Estimate your %errors in m1, m2, xmax, θ, and g. Add these

%errors to estimate the %precision.

3) Compare this to the manufacturer’s rating for the spring

kspec=3.5 N/m using a %difference. Does your experiment

indicate the manufacturer is providing springs at the stated

specification? Don’t forget to consider your %precision in

making your statement.

Experiment E (40 minutes): Level an air track. Obtain an adjustable end stop and place it at about the midpoint of

the airtrack. Raise the air track to about 5-10° above the horizontal using wood blocks (look at the figure for

Newton’s Second Law Part 2). Place a pulley at on the raised end of the airtrack. Hang a mass of about 100 grams

over the pulley. Attach the other end of the string to the glider using a small spring connector. On the other side of

the glider use another small spring connector to attach a spring. Connect the other end of the spring to the adjustable

end stop using a third spring connector.

Now turn on the air track (to about the 2 o’clock position). Hold the glider at a position such that the spring is

unstretched. Release the glider from rest and the spring will stretch some maximum amount xmax.

Use Ei + W = Ef to derive kth in terms of m1, m2, xmax, θ, and g.

Conclusions Exp E:

1) What is your derived result for kth?

2) Estimate your %errors in m1, m2, xmax, θ, and g. Add these %errors to estimate the %precision.

3) Ask your instructor for the manufacturer’s value for k or determine it by hanging a mass and using an FBD.

Compare this to your experimental result for the spring using a %difference. Does your experiment

indicate the manufacturer is providing springs at the stated specification? Don’t forget to consider your

%precision in making your statement.

1 kg

100

200g

92

93

Inelastic Collisions

Apparatus: Air track gliders (2 per track), air tracks, air supplies, air supply hoses & power cords, air track

accessories kit, photogate stands, photogate heads, photogate interface cables, PASCO Science Workshop 750

Interface & Power Supplies.

Theory: Inelastic collisions on an air track will be studied in today’s experiment. According to theory �ÙÚÚÚÛ �lÚÚÚÚÛ where �ÙÚÚÚÛand �lÚÚÚÚÛ are respectively the combined initial and final momenta of all objects involved in the collision.

For a perfectly inelastic collision the objects will stick together and form a single object with a mass equivalent to

the sum of the masses of the colliding object. This can be written as

fii vmmvmvmvvv

)( 212211 +=+

where the m’s are glider masses and the v’s are glider velocities. Don’t forget that velocities can be positive or

negative (they are vectors).

For a 1D problem, the vector sign can be dropped (assuming one uses +#’s for forward momentum and -#’s for

backward momentum). The percent change in momentum can then be calculated using the following formula: %∆� �l − �`!�` × 100%

Similarly the percent change in kinetic energy can be written as %∆Ü Ül − Ü`!Ü` × 100%

where 2222

12112

1iii vmvmK += and, for inelastic collisions only ( ) 2

2121

ff vmmK += .

Check your textbook to see if energy, momentum, or both are to be conserved for this experiment. If a physical

quantity is conserved, what should be the percent change in that physical quantity? The answer to that

question is the hypothesis of today’s lab.

Set-up for inelastic collisions:

• Set up the glider with bumpers on each end.

• Remove the small glider attachment that has a piece of cork on it. Remove the cork and don’t poke your

eye out.

• Attach this needle attachment to one end of a glider. On the opposite side of the glider attach a metal fin to

keep the glider balanced.

• On the other glider, attach the attachment filled with wax. On the opposite side of the glider attach a metal

fin to keep the glider balanced.

• Set the two gliders on the air track in such a way that the needle will penetrate the wax when the two

gliders collide. The gliders should also be set up so that the metal fins on each glider make contact with the

bumpers.

• Put flags on each glider. You can measure the length of the flag using the ruler on the airtrack.

• Set up photogates to act as timers for each glider. Use the appendix on Data Studio to set up the photogates

properly.

• Be sure to enter the correct length of the flag on the constants tab…see the appendix to learn how.

• Turn on the air supply and verify it is level. The gliders should stay relatively motionless.

94

Perform the following inelastic collision experiments:

• 1D inelastic collision with two objects of equal mass, one initially at rest. Be sure to mass the gliders

with all the attachments on them or you will get inaccurate results!

• 1D inelastic collision with two objects of unequal mass, one initially at rest. First try a heavy glider hitting

a stationary light glider. Be sure to symmetrically weight the glider.

• 1D inelastic collision with two objects of unequal mass, one initially at rest. Now try a light glider hitting a

stationary heavy glider. Be sure to symmetrically weight the glider.

For each experiment record the mass of each glider and the velocity for each photogate. Be sure to mass the

gliders with all the attachments on them or you will get inaccurate results!

Once you have the initial and final momenta and kinetic energies you can calculate the percent change in

momentum and percent change in energy. The results might surprise you.

To find the theoretical change in kinetic energy you should do the following:

• Assume that your initial speed is known (given by v1) as well as your two masses.

• Do a conservation of momentum problem to figure out what vf should be.

• Use this theoretical vf to determine what Kf should be. Write this answer in terms of the masses and v1.

• Determine the %∆K using %∆Ü Ül − Ü`!Ü` × 100%

You should show that %∆Ü − F�F%=%J­

Conclusion:

1. The air tracks are not completely frictionless. Obviously both measurements are lowered by friction. Since

we don’t really care about what initial speed is used, it really only impacts the second measurement in our

system. Will friction make %∆pexp more positive or more negative? Will it make %∆Kexp more positive or

more negative?

2. During the collision of a heavy mass with a stationary light mass state which object:

o has the greater force acting on it

o has the longer collision time

o has the largest acceleration

o has the largest change in momentum

3. How do your answers to the above question change if it is a stationary heavy mass that is hit by a moving

light mass?

4. According to theory, what is %∆p for an inelastic collision? Assuming 5% precision, did each experiment

match theory?

5. According to theory, what is %∆K for each inelastic collision? Assuming 5% precision, did each

experiment match theory?

6. According to theory, which collision should lose the most energy? Even if you have large %differences,

does your data qualitatively support this aspect of the theory?

7. Energy is conserved in the universe. Our experiments in the lab always show a slight loss of energy.

Where does the “lost energy” go in these problems?

8. Suppose your collision was not perfectly inelastic but actually elastic. What are the theoretical values of

%∆p and %∆K for this elastic collision?

95

Elastic Collisions

Apparatus: Air track gliders (2 per track), air tracks, air supplies, air supply hoses & power cords, air track

accessories kit, photogate stands, photogate heads, photogate interface cables, PASCO Science Workshop 750

Interface & Power Supplies.

Theory: Elastic collisions on an air track will be studied in today’s experiment. According to theory �ÙÚÚÚÛ �lÚÚÚÚÛ where �ÙÚÚÚÛand �lÚÚÚÚÛ are respectively the combined initial and final momenta of all objects involved in the collision.

For an elastic collision the objects will bounce off each other and each travel with a different final velocity. For

elastic collisions the following equations hold true:

ffii vmvmvmvm 22112211

vvvv+=+

222

211

222

211 ffii vmvmvmvm +=+ .

Don’t forget that velocities can be positive or negative (they are vectors).

For a 1D problem, the vector sign can be dropped (assuming one uses +#’s for forward momentum and -#’s for

backward momentum). The percent change in momentum can then be calculated using the following formula: %∆� �l − �`!�` × 100%

Similarly the percent change in kinetic energy can be written as %∆Ü Ül − Ü`!Ü` × 100%

where 2222

12112

1iii vmvmK += and (for the elastic case)

2222

12112

1fff vmvmK += .

If a physical quantity is conserved, what should be the percent change in that physical quantity? The answer

to that question is the hypothesis of today’s lab.

Set-up for elastic collisions:

• Set up the glider with bumpers on each end.

• Put flags on each glider.

• Attach bumpers to either side of one of the gliders.

• Attach the small metal fins to each end of the other glider.

• Be sure that both ends of the air track have a stopper on them.

• The glider that has bumpers on both sides of it will bounce off the stoppers. On the end of the air track that

stands to be impacted by the other glider (the one with two metal fins), you will need to place the last

bumper.

• Set up photogates to act as timers for each glider. Use the appendix on Data Studio to set up the photogates

properly.

• Turn on the air supply and verify it is level. The gliders should stay relatively motionless.

Perform the following elastic collision experiments:

• 1D elastic collision with two objects of equal mass, one initially at rest. Be sure to mass the gliders with

all the attachments on them or you will get inaccurate results!

• 1D elastic collision with two objects of unequal mass, one initially at rest. First try a heavy glider hitting a

stationary light glider. Be sure to symmetrically weight the glider.

• 1D elastic collision with two objects of unequal mass, one initially at rest. Now try a light glider hitting a

stationary heavy glider. Be sure to symmetrically weight the glider.

For each experiment record the mass of each glider and the velocity for each glider (before AND after the collision).

Be sure to record the masses of the gliders with all the attachments on it (or your data could be off

significantly).

96

Conclusion: 1. The air tracks are not completely frictionless. Obviously both measurements are lowered by friction. Since

we don’t really care about what initial speed is used, it really only impacts the second measurement in our

system. Will friction make %∆pexp more positive or more negative? Will it make %∆Kexp more positive or

more negative?

2. During the collision of a heavy mass with a stationary light mass state which object:

o has the greater force acting on it

o has the longer collision time

o has the largest acceleration

o has the largest change in momentum

3. How do your answers to the above question change if it is a stationary heavy mass that is hit by a moving

light mass?

4. According to theory what should be the %∆p for an elastic collision? Assuming 5% precision, did each

experiment match theory?

5. According to theory what should be the %∆K for each elastic collision? Assuming 5% precision, did each

experiment match theory?

6. Suppose your collision was not elastic but actually perfectly inelastic. Derive a result showing that %Q ∆Ü −F�F5 + F� × 100%

The second page of the Inelastic Collision experiment gives some hints on how to do this derivation.

7. Energy is conserved in the universe. Our experiments in the lab always show a slight loss of energy.

Where does the “lost energy” go in these problems?

97

Ballistic Pendulum

Apparatus: Ballistic pendulums, ballistic pendulum accessories, projectile launchers, projectile accessories,

projectile metal spheres, meter stick, photogate heads, photogate interface cables, PASCO Science Workshop 750

Interface & Power Supplies.

Purpose: To test the principles of conservation of momentum and mechanical energy.

Experiment 1 – Getting vth 1. Take the pendulum off the support and measure the mass M of the

pendulum and the mass m of the metal ball.

2. Determine the distance Rcm from the pivot to the center of mass of

the pendulum with ball in it by finding the balance point.

3. Estimate the %error associated with finding Rcm.

4. Fire the ball into the pendulum 5 times and record the angle.

5. Use geometry to show that the center of mass rises to ∆hcm=Rcm(1-

cosθ).

6. Using conservation of energy, determine an expression for the

combined ball and pendulum speed just after impact.

7. Using conservation of momentum, determine an expression for the

speed of the ball just before impact.

8. Combine 5, 6, & 7 to get an expression for the theoretical speed of the ball. You final result should look

something like: E%V tjÝt √2? ? ?. 9. Determine vth for each angle.

10. Use the AVERAGE and STDEV functions in Excel to determine the average and standard deviation of vth. If

your standard deviation is ridiculously low, assume the precision of the experiment is given by the

%error in your Rcm. 11. Calculate the initial theoretical kinetic energy of the ball before the collision (using vth).

12. Calculate the final experimental gravitational energy of the pendulum with ball in it (using ∆hcm).

Experiment 2 -Measuring the muzzle velocity directly. 1. Connect both photogates to the end of the spring gun using the special adapter. The ball will pass though the

beams of the photogates just as it leaves the spring gun.

2. Assume the diameter of the ball is 25 mm.

3. Measure the distance between gates and enter that data on the constants page of DataStudio so the computer

will compute the velocity for you.

4. Measure the “Velocity Between Gates”.

5. Shoot the ball several times and record the average value as your experimental velocity.

Conclusion: 1. Compare experiment 1 to experiment 2 with a %difference. Do you find that conservation of momentum and

energy theories, when used appropriately, accurately determine the muzzle velocity? Compare %difference to

%precision to discuss.

2. Friction tends to slow motion of both the ball and the pendulum. Will this cause your experiment to have more

positive or more negative percent differences?

3. Compare the initial kinetic energy of the ball as it exits the cannon to the final potential energy. Is energy

conserved? Should mechanical energy be conserved in this experiment? Should energy be conserved in the

universe as a whole? Explain where the lost energy goes.

4. When one considers the ball and the pendulum as the system, are there any external forces acting on the

system? Are there any NET external forces on the system during the collision?

5. To apply conservation of momentum to this problem, one must assume that the collision time is short. Explain

why. Hint: think about the net external force after the rod starts moving upward…

Going Further: In this lab we see that the pendulum is swinging (which implies rotational motion). A better

treatment is given on page 5 of: ftp://ftp.pasco.com/Support/Documents/English/ME/ME-6830/012-05375B.pdf.

98

99

Rotation WARNING: Today’s lab requires the massing of objects that could damage our electronic balances. Do not

mass these large objects on the electronic balances; use the old fashioned one in the SE corner of M205.

Apparatus: Rotation set disks and rings, rotation set aluminum rods and point masses, hanging mass sets, pulley

cord, stopwatches, scissors, digital calipers, meter sticks, pulleys for rotation sets, rotation set turntable, large triple

beam balance (from SE corner of M205)

Today you will be comparing theoretical moments of inertia to experimentally determined ones. This will be done

by observing an object undergoing rotational motion and comparing observations to predictions using kinematics,

forces, and torques.

Linear motion equations and angular motion equations look very similar as seen in the table below:

The variables are defined in your class notes or in the text somewhere. In circular motion the following

relationships exist between the above sets of variables:

s = rθ

v rω=

ta rα=

A torque can be exerted on an object and cause it to rotate. A torque is essentially (in a sloppy, non-technical kind

of way) a force that causes rotation. To determine the size of a torque consider the diagram and equation below:

The torque τ caused by the force F is given by the equation

θτ sinrF=

where r is the distance from the pivot point to the point where the force is applied and θ is the angle between the r-

vector and the F vector as shown in the figure.

The analogy between rotational equations and translational equations holds for Newton’s 2nd Law as well. The

rotational equivalent to mass is the moment of inertia given by the letter I. The N2L eqt’n and its rotational analog

are listed below.

∑ = maF ∑ = ατ I

2

0

1

2x x

x v t a t∆ = + 2

02

1tt αωθ +=∆

)(22

0

2xavv xxx ∆+=

2 2

0 2 ( )ω ω α θ− = ∆

tavv xxx += 0 0 tω ω α= +

The x indicates where

the axis of rotation (or

pivot point) is located.

θ

F

r

100

It can be shown that a falling object (dropped from rest) of mass m attached to a spindle of radius r causes the object

to rotate in accordance with the following equations:

The first equation is the 1D kinematics equation which describes the linear fall of the mass towards the floor.

The second equation relates the linear acceleration of the falling mass to the angular acceleration of the spindle.

These two are related because they are connected by a string.

The third equation comes from doing the sum of forces on the falling

mass. Show an FBD in your notebook and derive this equation. The

fourth equation comes from the sum of torques on the spindle. Show a

torque picture and derive this result in your notebook as well. Note that

only one force causes the spindle to rotate and that force is the tension in

the string. Because the string comes off the spindle at a right angle, the

angle θ from the torque equations is 90° (which in turn makes sinθ=1).

By fooling around a bit with the third and fourth equations you should be

able to derive an expression for I in terms of m, g, ∆y, t, and r. Derive this

equation in your notebook. Your final result should end up in the form

Iexp=mr2(…-1).

In today’s lab you will measure ∆y with a meter stick and t with a stopwatch. To get r you can measure the radius of

the spindle with calipers and to get m you can use the balances in the lab. You can then use this information to

determine the moment of inertia of the turntable and spindle.

Use five measurements to get an average and standard deviation for Iexp.

I suggest using a small hanging mass m for the turntable and a larger m when you have heavy objects on the

turntable. Also, try spinning the pulley; if it is sticking try asking for some lubricant.

To further study rotational motion, additional experiments can be performed using the ring, the disk and the point

mass and rod attachments. WARNING: do not mass these large objects on the electronic balances, use the old

fashioned one.

Let half the groups do both Version A & B while the other half do Version C. VERSION A: Determine Iexp for a disk and ring. Don’t forget to subtract of the Iexp of the turntable.

VERSION B: Determine Iexp for the rod with the point masses close to the center then again with the point masses

further from the center. Don’t forget to subtract of the Iexp of the turntable (although it may be negligible).

VERSION C: Determine determine Þ for all possible positions of the point masses. Subtract off the turntable’s

moment of inertia. Plot Þ as a function of point mass position k. Show both theory and exp curves (type II graph).

It can be shown that the theoretical moments of inertias for these objects are as follows:

Repeat the experiment using the disk & the cylinder. You can obtain theoretical values for the moments of inertia

by massing the objects (with the old-fashioned balance) and measuring their dimensions with a meter stick.

You will obtain an experimental value for the moment of inertia using the method described above. Subtract from

this value the moment of inertia of the turntable found previously. Compare this result to the theoretical moment of

inertia of the disk, ring, or bar with point masses using a percent difference.

1) 2

2

1aty = 2) αra = 3) maTmg =− 4) αIrT =

2

2

1DiskDiskDisk RMI = ( )22

2

1InnerOuterThickRingRing RRMI +=

2

12

1RodRodRod LMI =

2

pntpntpnt RMI =

2r

m

101

Conclusions:

1. Estimate the precision and compare it to the %difference for the moment of inertias of your two objects.

Are the theoretical formulas for moments of inertia in good agreement with your experimental

observations?

2. Which of your objects should have a larger angular acceleration? Explain why. Was this shown to be true?

3. Technically, we ignored friction in both the pulley and in the turntable. By neglecting these would that

tend to make the percent differences more positive or more negative? Does this make sense based on the

sign of your %differences?

4. Derive an equation for the speed of the hanging mass using Ei=Ef. Verify your result makes qualitative

sense by describing what should happen to v in the following cases:

a. By increasing the height the equation shows the velocity __________(increases/decreases/stays

the same)

b. By increasing the moment of inertia the equation shows the velocity

__________(increases/decreases/stays the same)

c. By increasing the radius of the spindle the equation shows the velocity

__________(increases/decreases/stays the same)

d. By increasing the hanging mass the equation shows the velocity

__________(increases/decreases/stays the same)

102

103

Rotation Madness Apparatus: Enough materials for two copies each of options 3 & 5, one copy each of options 1, 2, & 4

OPTION 1: Create a system wherein a rod (meter stick) is free to pivot about one end of the rod.

You should be able to hold the rod parallel to the ground and release it from rest. The rod should be

able to swing down until the rod is vertical. Furthermore, you should be able to securely affix a

point mass to the rod at an arbitrary location. The variables used in this option are rod length (�),

rod mass (F), point mass (X), point mass distance from pivot (k), and the angle of the rod from

horizontal (±). Notice this is not the usual angle we use. I think it will be easier to understand your

experimental data if we use this angle. Hint: if you make the point mass a multiple of the rod

mass, say ß «�, the math will be a lot cleaner.

You should be able to drop the rod with the point mass at various locations. For each location, use a rotary motion

sensor to record �, |, and ± data. I recommend using k 10.0, 20.0, … 100.0cm. If time permits, get a few extra

data sets at k 15.0&25.0cm as the graph is usually most interesting there.

Note: if the point mass is too close to the axis of rotation, it is no longer sensible to consider it as a point mass! Doing a little math one can show

that treating a sphere as a point mass introduces less than 1% error when the sphere is positioned more than 3.2 diameters from the axis of

rotation. How can you use this information? First determine the size of your point mass with a ruler. Ensure your point mass is at least four

times this size from axis for all x values you use in the experiment.

GOALS:

1) Is angular acceleration constant for a swinging system?

2) Do energy methods accurately predict the rotational motion

a. dependence of the rotation rate upon angle

b. rotation rate at bottom of swing

c. position of point mass which produces the fastest rotation

Understand the problem theoretically for arbitrary angle

1. Determine an expression for the center of mass position (�àá) of the system at an arbitrary angle θ.

Answer in terms of �, F, X, k, and ±.

2. Determine the moment of inertia of the rod in terms of �, F, X, and k.

3. Derive the theoretical angular velocity when the rod is some arbitrary angle θ from the vertical. Answer in

terms of �, F, X, �, k, and ±. Use energy methods.

4. Derive the theoretical angular acceleration in two ways

a. Take the derivative of your theoretical expression for ω. You’ll need to use the chain rule on any

terms involving θ.

For example: ÖÖ% sin ± cos ± 4ÖâÖ%6 cos ± (|).

b. Do it again using a torque problem. Hint: don’t forget that the gravitational force is (X + F)�

and it is applied at the center of mass!

c. Is the use of the instantaneous pivot is applicable in this case? Explain.

5. Determine an expression for the rotation rate at the bottom of the swing (assuming the rod was released

from rest while parallel to the ground). Write your expression in terms of �, F, X, k, and �. Take an

appropriate derivative to determine the value of k which gives max speed at the bottom of the swing.

6. For each value of ã, use a rotary motion sensor to obtain ±, |, ä, &� data.

k

E

|

±

104

GOALS: (consider showing goal slide multiple times to help people stay on track in your talk)

1) Is angular acceleration constant for a swinging system?

2) Do energy methods accurately predict swinging motion

a. dependence of the rotation rate upon angle

b. rotation rate at bottom of swing

c. position of point mass which produces the fastest rotation

Maybe discuss the procedure briefly then immediately revisit your goals so everyone is clear on what you are trying

to do. Maybe this is a good time to ask the class to make a guess…when people guess they care more about finding

out the answer and will listen better. Note: in any procedure show a picture of your equipment with something in

picture for scale. Fortunately, your apparatus itself is a perfect way to show the scale of the device…

Then do the FBD/torque problem for arbitrary angle. Use this result to derive the angular acceleration as a function

of angle.

To discuss point 1, show plots of angular displacement versus time and angular velocity versus time.

• Think: the slope of one of these plots gives you your answer. The concavity of the other plot gives you a

double check.

• Think about the bottom of the swing…what is tangential acceleration there? What does this imply about

angular acceleration at the bottom of the swing?

• Think: when should the slope of the angular velocity plot be steepest?

• According to your coordinates, when should the signs on each plot be positive/negative?

• Think about the signs and slopes of the angular displacement plot compared to the angular velocity plot.

Now do your energy problem derivation.

To discuss point 2a, create a plot of | vs ±.

• You should be able to create a table of theoretical values using step 3 from the previous page.

• You should have experimental data from tracker.

• Show the theory as a smooth line and the experimental points as dots. Include theory equation on plot.

• Ask for ideas about getting a numerical value to describe the agreement of experiment to theory.

To discuss 2b and 2c, create a plot of | vs k.

• You should be able to create a table of theoretical values using step 3 from the previous page.

• You should have experimental data from photogate measurements.

• Show the theory as a smooth line and the experimental points as dots. Include theory equation on plot.

• Ask for ideas about getting a numerical value to describe the agreement of experiment to theory.

• Point out to the class which value of k is predicted to cause the fastest rotation at the bottom of the swing.

Before giving your talk, remind yourself of all assumptions you made in your theoretical calculations.

Quantify your % error and compare it to a % difference. Qualitative agreement? Quantitative agreement?

105

OPTION 2: Predict E vs � and a for various yo-yo’s as they fall in terms of its mass (F), center-

of-mass moment of inertia (Þàá), �, and the spindle radius (�). Start with a yo-yo from rest.

Record a video and use Tracker to collect �, �, and E data so various plots can be made. Repeat

the experiment with different yo-yo’s. I’m thinking three yo-yo’s plus two solid cylinders

(with different diameters) and/or two thin walled pipes (with different diameters).

GOALS:

1) Do energy methods accurately predict speed versus displacement for falling yo-yo?

2) Is an unraveling yo-yo accurately modeled as rolling without slipping?

3) For two yo-yo’s with equal moment of inertia, how does spindle size affect acceleration?

Understand the problem theoretically for fall distance å

1. Record the dimensions and masses of all different yo-yo’s, cylinders, and pipes. You

might also take a photo of all of them side by side for your talk. You can number them in this photo to

make it easier to discuss things at the end of your talk. Include a ruler for scale in any photo. Note: from

this info you can generate values of the moment of inertia for each object using the tables in your

workbook. These moments of inertia can then be used to predict the theoretical acceleration (or velocity)

of each yo-yo.

2. Derive the theoretical velocity after falling a distance � using an energy problem. Use Þàá for the moment

of inertia (instead of your result for part 1) to keep your work simpler.

3. Derive the theoretical translational acceleration using a torque problem. Be sure to distinguish between Þàá and Þ∥. a. Do this once using the center of mass as the pivot point.

b. Do this again using the instantaneous pivot point.

c. Do it a third time using your result from part 1 and kinematics. Note: you will not need to explain

2c in your talk but it is definitely worth doing for exam practice and general understanding of your

situation.

4. Take a video of each object falling. Record a video and use Tracker to collect �, �, and E data so various

plots can be made. Note: the cylinders and pipes will probably be quite sensitive to the way the string is

wound. Try to get the string wrapped up near the center of mass. Try a few times to get a decent video for

the pipes and cylinders then move on.

5. On the big wooden yo-yo, check the mass of the string with a balance…it may not be negligible. We will

still assume it is but this may help explain discrepancies later.

y

106

In your talk you always want your goal slide first.

Then explain the procedure. Include a picture of your various yo-yo’s including a ruler for scale.

Mention how you got your theoretical moments of inertia.

Don’t go into gory detail on the math…do list final numerical results? Then revisit and beef up the goals…

GOALS REVISITED:

1) Do energy methods accurately predict speed versus displacement for falling yo-yo?

2) Is an unraveling yo-yo accurately modeled as rolling without slipping?

a. Should the acceleration magnitude be more than, less than, or equal to � as it falls? Why?

b. Should the acceleration of a falling yo-yo be constant as it falls? Why or why not?

3) For two yo-yo’s with equal moment of inertia, how does spindle size affect acceleration? Why?

After revisiting goals, show energy derivation.

To discuss point 1, show plots velocity versus fall distance.

• Your work for part 2 on the previous page should help you tabulate theoretical data.

• Your Tracker videos should give you experimental data.

• Show the experimental data as dots and the theory as smooth lines. Include theory equation on plot.

• Might start with one case and show clearly. Then show all cases on same plot if easy to color code. May

need to be creative to get through these quickly…we can discuss options once plots are made.

Now show torque derivation using both styles (see parts 3a and 3b on previous page).

Consider showing goal slide again after derivation. The torque derivation for the acceleration should tell you the

answers to goals 2a, 2b, and 2c. Consider pointing this out the class at this time.

Use your results to create a table of predicted accelerations for each yo-yo/cylinder/pipe. Show this on a slide.

Then show a E�-plot for a single yo-yo. Use a trendline to get the acceleration. Then animate in a line showing the

theoretical acceleration on the same plot (ask if you need help).

Then show all yo-yo’s on the same plot with experimental data as dots and theory data (not trendlines) as smooth

lines of matching color.

Show a slide with a table comparing your experimental accelerations to theoretical accelerations. Include a percent

difference.

Note: This comparison directly address Goal 2a.

Note: if the slopes are constant for each E�-plot that addresses Goal 2b.

Finally, address Goal 3 by comparing two yo-yo’s with nearly identical mass but different spindle radius. Your

derivation of translational acceleration from part 3 on previous page should help answer which should accelerate

faster. Then make a E�-plot with data from those two yo-yos. Hopefully the steeper slope matches what theory

suggests!

OPTIONAL: Use your technique to predict the moment of inertia of yo-yo’s of more complicated design. In

particular, use your system to find the moment of inertia of a fourth yo-yo (black with rubber tubing on it). Use

your previous work to estimate % uncertainty on your measurement of this oddball yo-yo.

Tie it together by revisiting the goal slide one last time and summarize how you answered each goal question.

Before giving your talk, remind yourself of all assumptions you made in your theoretical calculations.

Quantify your % error and compare it to a % difference. Qualitative agreement? Quantitative agreement?

107

OPTION 3 (two versions means two groups): Start the objects from rest. You will try at

least 6 different objects. One group uses 3 solid spheres (bowling ball plus 2 other sizes)

and 3 spherical shells (basketball plus 2 other sizes). The other group uses 3 thin rings

with differing radii and 3 sold disks with differing radii.

Predict E vs k and for an arbitrary round shape (disk, ring, ball, etc) as it rolls down an incline.

Compare to data from Tracker.

Also, use E vs � data from Tracker to experimentally determine ap(® for each case. Use ap(®

to determine and experimental value of the moment of inertia Þp(® for each rolling object. Compare Þp(® to

theoretical values predicted by formulas from a textbook.

GOALS:

1) Do energy methods accurately predict speed versus displacement for rolling objects?

2) In general, what factors determine which object gets downhill fastest? Mass, radius, shape, frictional

coefficients, etc.

Understand the problem theoretically:

1. Get the mass and radius of each object. Take a photo of all together and number them. Use textbook

formulas and this data to generate theoretical moments of inertia for each object. We won’t need the gory

details in the talk but we do need a number for each Þ%V at some point. Note: for rolling objects we expect

all moments of inertia take the form Þàá ·X��. Know the value of · for each object…

2. Derive the theoretical velocity after rolling distance x using an energy problem. Use I for the moment of

inertia to keep your work simpler. Note: for rolling objects we expect all moments of inertia take the form Þàá ·X��. At the end of your derivation, use this fact to clean things up.

3. Do a torque problem and solve for Þàá in terms of the translational acceleration (a, not ä). Be sure to

distinguish between Þàá and Þ∥. OPTIONAL: use Þàá ·X�� to clean up the formula. If you do this you

get a prediction for · in terms of a and it should look a lot simpler.

a. Do this once using the center of mass as the pivot point.

b. Do this again using the instantaneous pivot point.

c. Do it a third time using your result from part 1 and kinematics. Note: you will not need to explain

2c in your talk but it is definitely worth doing for exam practice and general understanding of your

situation.

4. Take a video of each rolling object experiment. Use Tracker to get �, k, and E data for making plots.

Think: how could you use this information to also make ± vs � and | vs � plots. I’m not sure if you need

to do that, but understanding how you might do this is worth considering.

Think: look back at the picture and you will notice I said the angle was ½. Be ably to clearly explain what ± and ½ mean in this experiment. If not sure…take a guess then check with me.

5. At some point you will need a precise value of the angle. Rather than use a protractor or angle

indicator…use height and length of the ramp along with some trig to get the angle. That said, double check

this number with a protractor or phone.

6. Make video of two different races to show class (ask who wins each race before starting vids). Could use

this as a final slide to see who was listening?

a. One of two objects with same shape but differing radii.

b. One race with two objects of differing shape but same radii.

c. Ask bonus question, in which case, if either, do objects have same rotation rate at finish line?

Before giving your talk, remind yourself of all assumptions you made in your theoretical calculations.

You’ll want to emphasize this case is not on verge of slipping; that is why you had to do sum of forces in step 3a.

Emphasize objects with identical shapes have same translational speed/accel but not same rotational speed/accel.

Quantify your % error and compare it to a % difference. Qualitative agreement? Quantitative agreement?

k

½

E |

108

GOALS REVISTED:

1) Do energy methods accurately predict speed versus displacement for rolling objects?

2) In general, what factors determine which object gets downhill fastest? Mass, radius, shape, frictional

coefficients, etc.

Explain procedure.

Include picture showing each object with number. Include a ruler for scale.

Consider animating in Þàá and/or · for each object?

Consider revisiting goal slide after explaining procedure to segue into energy problem? Remind us of goal 1?

Then do energy derivation slide.

To discuss Goal 1, show plots of E vs k.

• Your work for part 2 on the previous page should help you tabulate theoretical data.

• Your Tracker videos should give you experimental data.

• Show the experimental data as dots and the theory as smooth lines. Include theory equation on plot.

• Might start with one case and show clearly. Then show all cases on same plot if easy to color code. May

need to be creative to get through these quickly…we can discuss options once plots are made.

• Emphasize objects with identical shapes have same translational speed but not same rotational speed.

Consider a revisit to Goal Slide to remind us what we’ve learned and what is left to answer?

Do torque derivations (once with CM pivot and once with instantaneous pivot). Distinguish between Þàá and Þ∥. To discuss goal 3, first show plot of E vs � for single object.

• Use Tracker data to get plot.

• Add trendline to get slope (slope is the experimental translational acceleration, ap(®).

• Animate in algebraic equation from torque derivation onto this plot. It will either say Þàá ⋯ (or · ⋯).

• Finally animate in a numerical value for Þàá (or ·).

Summarize the entire process you just did to find Þàá (or ·) for this one object.

Perhaps show the data for all six objects on one E�-plot?

Perhaps a final table summarizing theoretical values of Þàá (or ·). Include pics of each object if space allows?

Include theoretical value, experimental value, and % difference.

Revisit the goal slide one last time and make sure you hit everything.

109

OPTION 4: Obtain one of the unusual pulleys that look the figure shown at right.

For now, assume the pulley is a uniform disk with center of mass moment of inertia Þàá. Two hanging masses F5 and F� are attached at radii �5 and �� respectively.

Assume we may ignore axle friction for the pulley. Assume there is sufficient

friction between the strings and the edge of the pulley such that the strings do not

slip relative to the pulley as it turns. Finally, assume F� travels downwards.

NOTE: the strings are not wound up the same way!

NOTE: rather than tie knots, it may help to tape the strings to the pulley to make it

easy to change radii.

NOTE: it may be useful to let F� � 15g and F5 � 10or12g. You may need to

tape together custom hanging masses. If too light, try F� � 50g and F5 � 20g or F� � 200g and F5 � 100g?

Start with F� on the outer radius and F5 on the inner radius. Release the system

from rest. Record a video of either the pulley spinning or one of the masses

translating. Using this video in tracker you can get �, k, and E data (for a translating

mass) or �, ±, and | data (for rotating pulley).

Update: try using the pulley with holes in tandem with a photogate drilled in it to

act as an encoder. In data studio, do not select photogate. Instead, select super

pulley. You will likely need to adjust the super pulley constants before taking data.

Use data studio to directly acquire �, ±, and | data. This avoids having to do tons

of tracker vids. That said, if you go the data studio route also take some photos and

videos to help with your presentation.

Repeat the experiment with the same F5 and F� with F� still on the outer radius but move F5 the next radius.

Keep repeating the experiment with F� always on the outer radius but keep moving F5 the next radius.

Notice for the last trial both masses will be on the same radius.

I think you will have either 4 or 5 data sets at this point…I forget if the pulley has 4 or 5 radii.

Now retake all 4-5 data sets but use a smaller radius for F�. If possible, cut the radius by a factor of 2 or 3 to make

comparisons easier later on. In the end you should now have about 8-10 data sets of �, ±, and | data.

GOALS:

1) Do energy methods accurately predict the rotation rate of a pulley versus angular displacement?

2) Do torque methods accurately predict the rotate rate of a pulley versus time?

3) Is our pulley approximately a disk (as far as moment of inertia is concerned)?

Understand the problem theoretically:

1. Derive the theoretical rotation rate | after pulley rotates through angle ±. Answer algebraically in terms of Þ, F5, F�, �5, ��, �, and ±.

2. Derive the theoretical angular acceleration using a torque problem. Answer algebraically in terms of Þ, F5, F�, �5, ��, �, and ±.

a. Do this using the center of mass as the pivot point.

b. Do it again using your result from part 1 and kinematics.

c. Is the instantaneous pivot is applicable in this case? Be prepared to explain why or why not.

3. Measure all the radii with calipers if possible. If the largest radius is too big for calipers wrap a string

around it a few times and measure the length to determine the radius. Note: don’t forget to divide by two if

you measure the diameters! We may have jumbo calipers in the back…

4. Don’t forget to record the masses of all items (including the pulley itself) using a balance!

Side view

Front view

F5

F�

110

GOALS:

1) Do energy methods accurately predict the rotation rate of a pulley versus angular displacement?

2) Do torque methods accurately predict the rotate rate of a pulley versus time?

3) Is our pulley approximately a disk (as far as moment of inertia is concerned)?

Things to emphasize/think about before giving presentation:

• String should not slip on edge of pulley. This allows us to relate translational variables to rotational

variables.

• Axle friction is assumed negligible (in practice you’ll observe the friction is probably non-negligible).

• The strings are at different radii. As the pulley spins through angle ± each string will experience the same

angular displacement ± but different arclengths of string are wound (or unwound). This means the masses

will travel different amounts for the same angular displacement!

• Similarly, the masses will travel at different speeds as they rise and fall. They will also have different

accelerations!

• The string connects the super pulley to the big pulley at radius ��. The edges of these two pulleys have the

same translational speed. The two pulleys do not have the same angular speed. The angular speed of the

big pulley should equal the translational speed of the super pulley divided by ��. Thoroughly understand

this tricky spot.

After this rather long complicated procedure, revisit goal slide and remind us what goals are.

Now do energy derivation to determine | in terms of Þ, F5, F�, �5, ��, �, and ±.

Show plot of | versus ±:

• Get experimental data from super pulley (or Tracker vids).

• Get theoretical prediction from step 1 on previous page.

• Show theory as smooth line and experiment as points. Include theory equation on slide.

Now do torque derivation to determine ä%V in terms of Þ, F5, F�, �5, ��, �, and ±.

For one trial (one of your 10 data sets), show plot of | versus �: • Get experimental data from super pulley (or Tracker vids).

• Do a trendline to get the slope of | versus � (this is äp(®)

• Consider animating in a smooth theoretical line using theoretical prediction from step 2 on previous page.

Think: you should know the value of ä%V and |%V(�) |` + ä%V�. • Summarize what was predicted by theory and what was found in experiment. Compare with a percent

difference.

Now show | versus � data for the first half of the data sets on a single plot. Include only the experimental data and

theoretical data (no trendlines). Coordinate the colors to make it easy to interpret the data sets. Comment on any

unusual discrepancies.

Now show | versus � data for the first half of the data sets on a single plot. Include only the experimental data and

theoretical data (no trendlines). Coordinate the colors to make it easy to interpret the data sets. Comment on any

unusual discrepancies. Ask the class why one (or more) of the slopes is negative)? Make sure you know why…

Note: you may have one data set that has no acceleration. Explain why ignoring axle friction causes major problems

when analyzing this data set.

111

Consider making a summary table showing the predicted/theoretical values of angular acceleration, the experimental

angular accelerations, and the % difference. Consider taking the absolute value of each % difference then averaging

them.

It is typically wise to avoid data tables if you can present the information graphically. Perhaps a better thing to do

would be to use each data set to calculate the moment of inertia of the pulley. Then get the average of those

numbers (since that should be the same for each trial). You can determine an average and standard deviation to

figure out an estimate on the error. Then compare this experimentally obtained value of the pulley’s moment of

inertia to a theoretical value assuming the pulley is a disk (Þ%V 5�X��). You should then be able to find a percent

difference to compare to a percent error.

Discuss both qualitative agreement & quantitative agreement?

Revisit the goal slide one more time telling us how you answered those questions.

You might end by mentioning if you believe it was reasonable to treat the pulley as a solid disk.

Try a web search for images with the keywords “change drill press speed”. You should see a practical application of

this type of system. That might be a fun way to end your talk as well.

112

OPTION 5 (two versions means two groups):

GOALS:

1) Do energy methods accurately model motion in system with translational and rotational motion?

2) Test a system for determining moment of inertia of arbitrary objects.

Connect a small hanging mass (50 grams?) to the spindle of the

turntable. Release the hanging mass from rest. Acquire data for

position and velocity versus time for the small hanging mass. Tip:

rather than use tracker for this step, consider using a smart pulley

and data studio. This will hopefully give you �, �, and E data

directly with no need for video capture.

Experimentally determine the magnitude of the acceleration of the

hanging mass. Use that acceleration to determine an experimental

value for the moment of inertia of the turntable.

Repeat the experiment with various objects on top of the turn

table. Note: for the rest of the day you will probably want to use

the same large hanging mass…probably 500 g or 1kg. For each

different object on the turntable, acquire data for position and

velocity versus time.

Version A: This version uses the disk, ring, and rod with point

masses close to axis and rod with point masses far from axis. Also

cut a rectangular wooden board to fix on the apparatus to use as a

rectangular plate. If possible, also flip the board on its edge. Could

you safely flip the ring on its edge as well? Discuss with instructor

before proceeding.

Version B: Use the rod and point masses with all different positions

of the point masses.

If you are doing things correctly, you should have �, �, and E data for the hanging mass for the turntable with

nothing on it as well as the turntable with each object on it.

The acceleration in each E�-plot should gives ap(®. Each ap(® can be used to determine Þp(® for each object. Don’t

forget to subtract off the moment of inertia of the turntable from the other objects!!!

Understand the problem theoretically:

1. Determine the equation for Þ%V of each object using the tables from text. You will need the masses, length,

and radii of your various shapes. For the point masses you will have several different theoretical values

since you will have several different values of k. Use your formulas to also compute numerical values of Þ%V for each object you will place on the turntable.

2. Derive a theoretical velocity E of the hanging mass after falling a distance � using an energy problem. Use Þ for the moment of inertia (instead of your result for part 1) to keep your work simpler.

3. Do forces on the hanging mass and torques on the turntable. Solve for moment of inertia (Þp(®).

a. Do this once using the center of mass as the pivot point.

b. Do it again using your result from step 2 and kinematics.

c. Does it make sense to do this problem using the instantaneous pivot instead?

2k

�

Rod with point masses, one object you will place on turntable

�

2� F

E

|

TIP: adjust system such that mass F barely impacts ground

when string is fully unwound from spindle. This avoids the

turntable spinning out tons of string to get caught in the axle.

113

GOALS:

1) Do energy methods accurately model motion in system with translational and rotational motion?

2) Test a system for determining moment of inertia of arbitrary objects.

Briefly explain the procedure.

VERSION A: Include a picture showing all the various objects you placed on the turntable. Include a meter-stick

for scale. Animate in numerical values of each objects moment of

inertia (we are calling these values Þ%V).

VERSION B: Include a picture of turntable with rod and point masses

on it. Also show a sketch similar to the one shown at right.

Explain why Þ%V 58F¥=Ö�� + 2 F®.t.k�!.

Show a plot of Þ%V vs k as a smooth line with no points. Include

numerical values for F¥=Ö, F®.t., and � somewhere on this slide.

Consider a revisit to the goal slide. Remind us of the purpose of your talk.

Now do energy problem to derive speed of hanging mass as function of distance fallen.

Mention if we plug in numerical values of Þ%V into the equation for E we can predict the shape of a plot of E vs �.

Show plot of E vs �:

• Theory comes from steps 1 & 2 on previous page.

• Experiment comes from Data Studio or Tracker data

• Experiment is dots, theory is smoothed line. Include theoretical equation on slide.

Revisit goal slide…was goal 1 met? Can you estimate an average % difference?

Now do your forces and torques derivation to derive a direct determination of Þp(®.

Emphasize us that � is the spindle radius…not the radius of the object on the turntable.

Emphasize you remembered to subtract off moment of turntable before comparing results.

Show a E�-plot for case only:

• Data points come from Data Studio or Tracker.

• Use a trendline to get the slope (this is ap(®).

• Animate in your equation that relates the acceleration to moment of inertia Þp(® F��(… ). • Animate in a numerical result for Þp(® based off of this one graph.

If time permits, show a single E�-plot with all data color coded and a legend. Color code the trendlines if possible

but you can leave off the Þp(® equations and calculations for this plot.

VERSION A: create a table showing all shapes. Consider include tiny pics of each object if you can squeeze it in to

make it painfully obvious which shape goes with which entry in the table. Include Þ%V, Þp(®, and % difference for

each shape. You can then estimate % errors.

Finally, revisit your goal slide and make sure you covered everything.

Before you talk, think about all the different assumptions you made. Is there axle friction? Is the friction between

the edge of the pulley and the string? Why does friction between the string and the edge of the pulley do no work?

Why did you include the rotational energy of the turntable but not the smart pulley? Is air resistance more of a

factor on certain shapes? Is it completely negligible? To help you think about drag, first consider how fast things

moved (translational speed) for your fastest trial in miles per hour?

VERSION B on next page…

2k

�

114

VERSION B: Add in one final graph showing Þ vs k

• The theory values (smooth line) should come from Þ%V 58F¥=Ö�� + 2 F®.t.k�!

• Include theory equation on chart.

• Animate in experimental points (ask me how).

• Remind that class that each experimental point was determined by first getting ap(® from Data Studio (or

Tracker) and subsequently using Þp(® F��(… ). Animate in this equation as well.

• Animate in

• Discuss why data shouldn’t match perfectly for smallest value of k. State if the theory equation used above

is an underestimate or overestimate for Þ%V for small k. Explain why.

Before you talk, think about all the different assumptions you made. Is there axle friction? Is the friction between

the edge of the pulley and the string? Why does friction between the string and the edge of the pulley do no work?

Why did you include the rotational energy of the turntable but not the smart pulley? Is air resistance more of a

factor on certain shapes? Is it completely negligible? To help you think about drag, first consider how fast things

moved (translational speed) for your fastest trial in miles per hour?

115

OPTION 6: Program a computer simulation of any of the above experiments. Include sliders for typical parameters

(e.g. in OPTION 1 make sliders for selecting the values for m, M, x, and L). Ensure that even an idiot like your

instructor can pick some values and then hit play to watch the simulation.

OPTION 7: Propose something else requiring roughly equal effort AND possible with equipment available. We

have a lot of weird stuff (and tons of tape) so we can probably work up something for most ideas.

116

Prepare a PowerPoint Presentation.

• State why is the audience supposed to care/listen, support claims with visual evidence, minimize

unnecessary details/words or other distracting information

• Post your video to the web somehow (e.g. post it to YouTube) so it can be played during your presentation

if there are any questions that come up. Edit out the unimportant stuff or know the appropriate times to use

in the video so we don’t need to watch a lot of worthless material.

• Each slide has one key point supported by pictures, an eqt’n, or a graph (not bulleted lists of words)

• Keep fonts big (20 points for axis labels, everything else bigger than that)

• Include a title slide that gives your names, a candid picture of you performing your experiment, a short title

of your experiment.

• Include a goal slide.

• Include a derivation slide (or two). Try to minimize clutter. Show the important equations you started with

and the also the final result but skip all the gritty details in between. You should have all those details

written out on a piece of paper while you are presenting but not on the slide (just in case I ask a question).

At the end of your derivation, consider re-emphasizing your goal since people have probably forgotten.

• Include a procedure slide. Briefly summarize the steps required to get your graphs or data. Don’t say

every little detail...if I really want to know something detailed I’ll ask a question. If you have a big picture

or sketch of your apparatus (or before and after picture) you can simply explain by pointing at the picture

and telling us how it worked.

• If your video is really short, this might be a good time to show it to the class. You could escape out of

PowerPoint, have your video ready to go in another window, play it, and then go back to your PowerPoint.

Note: sometimes trying to embed the video in your PowerPoint can cause frustrations and delays. Please

do not frustrate and delay everyone. Practice showing your video on at least two different computers

before coming to class.

• Show any data or graphs. Mention how they relate to goal. Point out any unusual trials of your

experiment. Discussion the experimental precision of your experiment relative to your % differences. Use

20 pt or larger font size so we can actually read your axis labels, etc.

• Have a final slide summarizing how you met (or didn’t meet) your goal. Were the theoretical predictions in

good agreement with the experimental results? Don’t just say yes or no, discuss the % precision and %

difference.

• Practice the presentation. Each group member should speak an equal amount. The total presentation time

should be 8-10 minutes. Anyone exceeding the 10 minute time limit will be stopped and lose points.

• Save the file on each member’s flash drive and, as a back-up, email it to yourself. Think, what if somebody

gets sick? Make sure all team members have all the info.

• Consider reviewing http://writing.engr.psu.edu/speaking.html or google other sites to improve your talk.

Be prepared to give your presentation next week in lab. You will also be required to give feedback to your peers on

their presentation.

117

What is good science?

I want to make your lab experience useful not only in this class but also after you transfer and eventually get a job. I

asked myself, what is good science? What are the most important things I want you to take with you from lab?

• Form a good question (falsifiable claim)

• Design an experiment (tests claim, can be repeated by others, wisely restricts parameter space)

• Collect adequate data (data covers broad swath of restricted parameter space, multiple trials per case)

• Appropriate error analysis (honestly interpret data, even if errors are large or claim is shown to be false)

• Proper citations (be generous and honest about contributions from outside sources)

This is, of course, some variation on the classic scientific method.

We now have several different well-accepted theories we can use to design falsifiable questions:

• Kinematics (constant acceleration, separation of variables, etc)

• Newton’s laws (∑YÛ FaÛ & ∑èÛ ÞäÛ) • Work-Energy ( `̂ +ép(%&i.Õ. l̂)

Furthermore, in lab we typically measured the following parameters:

• Determine elapsed time with a stopwatch

• Determine velocity with a photogate or encoder (e.g. a smart pulley)

• Determine acceleration using one of the above tools plus distance

• Plot position and velocity versus time with Tracker

• Determine acceleration form the slope of a vt-plot made using Tracker

It makes sense, then to use our theories to predict elapsed time, final speed, acceleration, etc. These predictions can

then be shown to be false (or not false) with our experimental equipment. By testing not just a single case but

multiple heights/angles/masses we can see if the result is valid over a broad range of conditions. We can then do

proper error analysis to determine percent error (% precision) of our experiment. By comparing the % error to the %

difference we can state if our results are in good quantitative/qualitative agreement with the widely accepted

theories. Eventually you will learn how to quantify your confidence in such statements. That is good science.

118

Option1:

1) �Õt tê&jÝ(tjÝ (1 − cos ±) where ± is from the vertical

2) Þ 58F�� + Xk�

3) Energy equation gives | 34tëj�Ý(ì 6� cos ± where I is moment of inertia of the rod & point mass combined

4) ä 4tëj�Ý(ì 6� sin ±…you should be able to get this result either way

5) At the bottom the angle is ± 0°. The velocity is given by E �| where, for the end of the rod, � �. One

finds E �3tëj�Ý(ì � �3 tëj�Ý().të&jÝ(& �. Notice that in theory you can predict a value of v for given values of x,

L, m, and M.

12) You can use your v vs x plots to compare vexp. Put error bars on the points so we can quickly see if the

experimental results and in quantitative agreement with the theoretical line.

If not in good quantitative agreement, do the experimental points suggest good qualitative agreement? Is the shape

of vexp vs x similar to the shape of vth vs x? For ath and aexp, just make a little table and include % difference & %

error for each of the two cases.

Feeling über hard-charging? Try to make a theoretical plot of ω vs t. I think this will require a numerical technique

such as Euler’s Method. This is very interesting…and probably a bit challenging. Essentially, you start with a

known initial value. Then use | Δ±Δ� 9:F� + 2XkÞ ; � cos ±

This gives Δ± Δ�9:F� + 2XkÞ ; � cos ±

In theory, for a small enough step size, one can then predict the next value of theta ±5 ±Ê + Δ±. This corresponds

to the time �5 0 + Δ�. By repeating this process you can eventually get a column of data for both t and ± to

compare to your experimental values.

119

Option2:

1) Þ%V ÞÕ$­ + 2Þ¬2,J¥p 5�FÕ$­�� + 58F¬2�� where R is the radius of the cylinder and s is the side length of the

square.

2) E%V 3 �LV5jì/tí&

3) ä%V Lí 4 55jì/tí&6

4) � 3�VL (1 + Þ/F��)

9) From your plot of |(�) you should be able to use a trendline to determine αexp. You can use the LINEST

command in Excel to get an error associated with αexp. You can compare this to your theoretical value derived in

part 3) with a % difference. Put error bars on the plot of v vs h so it is easy to see if the experimental points are in

quantitative agreement with the smooth theoretical line.

If the experiment is not good agreement, one can still look for qualitative agreement. Also note if the experiment is

in qualitative agreement by looking for the following two things: 1) is the slope of |vs. � roughly constant & 2) is

the magnitude of the slope in the ballpark. Does the v vs. h plot take the shape of the square root function?

120

Option3:

1) E%V 3�Lë nïðñ5jì/tí& where ½ is the angle of incline as opposed to the usual ± for reasons that will become obvious

later

2) ä%V L nïðñí 4 55jì/tí&6

3) � 3 �ëL nïðñ (1 + Þ/F��)

9) From your plot of |(�) you should be able to use a trendline to determine αexp. You can use the LINEST

command in Excel to get an error associated with αexp. You can compare this to your theoretical value derived in

part 3) with a % difference. Put error bars on the plot of v vs L so it is easy to see if the experimental points are in

quantitative agreement with the smooth theoretical line.

If the experiment is not good agreement, one can still look for qualitative agreement. Also note if the experiment is

in qualitative agreement by looking for the following two things: 1) is the slope of |vs. � roughly constant & 2) is

the magnitude of the slope in the ballpark. Does the v vs. L plot take the shape of the square root function?

121

Option 4:

1) This is trickier than I first suspected. We know the two objects will rotate the same angle but the amount of

height traveled by the blocks differs! We know that block 1 will travel distance ℎ5 �5± while block 2 will travel

distance ℎ ��± A ratio gives ℎ5 ¥)¥& ℎ. Similarly, if m1 is travelling with speed v, m1 is travelling with speed E5 ¥)¥& E. The energy equation becomes F��ℎ F5�ℎ �5�� + 12F�E� + 12F5 :�5�� E;� + 12 Þ|�

The final result for speed is

E%V ò2�ℎ F� −F5 �5��F� + F5 4�5��6� + Þ���

2) The two masses do not have the same acceleration! As before, if m2 has acceleration a, then m1 has acceleration a5 ¥)¥& a. Ultimately one finds ä � ��F� − �5F5F���� +F5�5� + Þ 3) � 3�V t&¥&&jt)¥)&jì!¥&L(¥&t&g¥)t))

4) Þp(® Lz (��F� − �5F5) − F���� − F5�5�

11) From your plot of |(�) you should be able to use a trendline to determine αexp. You can use the LINEST

command in Excel to get an error associated with αexp. For each video you can determine a value of Iexp. Assume

the % error is the same as the % error associated with αexp. Average these Iexp’s values to give a result for the

moment of inertia of the pulley.

While not strictly a disk, you may compare your experimental results to the theoretical value for a disk given by Þó`¬Ø 5�F¥�� and give a % difference. Note that the error in the average will be the average % error divided by √M where N is the number of cases you tried. State if it is appropriate to treat the pulley as a disk by noting if the

%difference < %error?

Put error bars on the plot of v vs h so it is easy to see if the experimental points are in quantitative agreement with

the smooth theoretical line.

If the experiment is not good agreement, one can still look for qualitative agreement. Also note if the experiment is

in qualitative agreement by looking for the following two things: 1) is the slope of |vs. � roughly constant & 2) is

the magnitude of the slope in the ballpark. Does the v vs. h plot take the shape of the square root function?

Feeling frisky? You could track both masses and plot velocity versus time for both masses on the same plot. You

could compare the slopes of the two lines and see if the relationship a5 ¥)¥& a� is in quantitative agreement with the

values of your slopes!

122

Option 5:

1) For a rod with two point masses one finds Þ%=%J­ Þ¥=Ö + 2Þ®.t. 55�F¥=Ö�� + 2 F®tk�!

A thick ring is one that has an inner radius noticeably different from the outer radius. For a thick ring the moment of

inertia is given by ÞrV`ÕØí`iL 5�F¥(�=,%p¥� + �`iip¥� ). A solid disk can be found with the same formula noting that

the inner radius is 0!

2) E 3 �LV(5jì/t¥&) where m is the hanging mass and r is the spindle radius.

3) ä L¥ 4 55jì/t¥&6

4) � 3�VL (1 + Þ/F��)

Note: re-arranging the equation 3) and substituting ä J¥ gives Þp(® F�� 4�a − 16

10) Use the slope of each v vs t plot to get a value for aexp. The error for each aexp can be obtained using the LINEST

command in Excel. You can determine Ith using the equations from part 1). Watch out! Don’t forget to subtract the

moment of inertia of the turntable from the Iexp’s obtained for all the other graphs! Compare using a percent

difference. Version B: plot I versus x showing the theory values as a smooth line and the experimental values as

dots (with error bars).

123

Angular Momentum and Rotational Kinetic Energy

Apparatus: Small rotation set disks & rings (2 disks and 1 ring per rotary group), rotary motion sensor, small rods,

small bases, PASCO Science Workshop 750 Interface & Power Supplies

Theory:

This experiment is essentially a rotational version of the inelastic collisions lab. Briefly review that lab.

Recall that angular momentum is represented by the eqt’n

ωIL =

where L is the angular momentum of an object, I is the moment of inertia for that object and ω is the angular

velocity of that object. In this case, saying that angular momentum is conserved means

fi LL = or ffii II ωω =

where the Ii and If are the initial and final moments of inertia and ωi and ωf are the initial and final angular velocities.

Rotational kinetic energy is defined as

2

2

1ωIRKE = .

PROCEDURE

In this experiment you will change an object’s moment of inertia by adding a second object to it. You will

determine the initial and final moment of inertia (see the text). In this lab you will use a disk and a thick ring.

Be sure that the disk is screwed on to the rotary motion sensor. Start the disk spinning. Hold the ring a cm or two

above the spinning disk. Drop the ring onto the spinning disk. Notice that it is difficult drop the ring such that it

lands (and stays) centered above the disk. Practice dropping the ring onto the disk a few times before going further.

You might try looking down on the spinning disk from above to see if that helps.

After setting up the experiment you will need to open

the Data Studio program on a lab computer. If you

have not already done so verify the Pasco interface

box is “ON”. Verify you have the yellow cable from

the Pasco “Rotary Motion Sensor” in digital input 1 on

the interface box. The black cable goes into digital

input 2.

You can set up the Rotary Motion Sensor to read

angular velocity (just like setting up a photogate).

You can unclick the Angular Position button and click

on the Angular Velocity button. DONT FORGET

TO CHANGE THE UNITS to rad/s! Then drag the

angular velocity (on top left side of figure) down to the

graph (bottom left side of figure).

Spin the disk and hit the start button. Verify the computer is taking data. You should see a graph of angular speed

versus time being plotted by the computer. It will look a little like the graph below. Think: if your graph is upside

down the values are all negative. How could you fix the negative sign? How does the negative sign relate to the

spinning? Try using the “xy tool”, the 6th button from the left in the graph tool bar (see circled in figure below).

When you click this button the dotted lines will appear. You can use these dotted lines to quickly determine the xy

124

coordinates of any point. Simply bring the crosshairs over the point and the xy tool should lock onto the point and

read out the coordinates to you. Notice that my graph left out the units…you are expected to figure them out.

On your graph in Data Studio you will notice the angular velocity suddenly drop! Does this agree with conservation

of angular momentum? Do you think the energy is conserved? Notice in the graph above that the point selected

must be just before the collision. You can quickly get the experimental angular velocities of just before the

collision (ωi) and just after (ωf).

Repeat this experiment once for the ring and two more times with a 2nd disk instead of a ring.

You will need to calculate the moments of inertia of both the annular cylinder and the ring. Measure the mass of the

disk and the ring. Measure the radius of the disk as well as the inner AND outer radii of the ring (it is a thick ring,

not a thin ring). These formulas are available in the text or from the previous lab.

Calculate the initial and final angular momentums and energies using your experimental values of ωi and ωf and

your calculated moments of inertia. Is L conserved? Is RKE? Should they be?

Experiment 2: Repeat the same procedure as above except this time drop a second disk onto the first disk. Again

note the values of ωi and ωf and calculate the initial and final angular momentums and energies.

To determine the theoretical change in kinetic energy, do the following:

• Assume that your initial angular speed (ωi) is known. Also, assume your moments of inertia are known.

• Do a conservation of angular momentum problem to figure out what ωf should be.

• Use this theoretical ωf to determine what RKEf should be. Answer in terms I1, I2, and ω1. Note: in this

case I1 is the moment of inertia of the first object (the initially spinning disk) and I2 is the moment of inertia

of the second object that is dropped onto the initially spinning disk.

• Using RKEi (determined from ω1) figure out what the %∆RKE should be and compare it to theory. You

should find that %∆�Ü^ − Þ�Þ5 + Þ�

125

Conclusions:

1. Was angular momentum conserved in each experiment?

2. Was rotational kinetic energy conserved in each experiment?

3. Which collision caused the greatest change in angular speed; does it make sense?

4. Are there any trends relating moment of inertia and RKE change? Which collision should cause the

greatest change in RKE?

5. What happens to the “lost” RKE?

6. When dropping the disk onto the disk it is very easy to keep the two disks centered. When dropping the

ring onto the disk it is almost never centered after the collision. By being off-center, how is the moment of

inertia of the disk affected (increased, decreased, not affected)?

7. How should your answer to the previous question affect your values for ωf?

8. Will it the collision lose more or less energy when the ring is off-center? Will this tend to make your

%differences more positive or more negative? Is that what you found?

126

127

Buoyant Force

Apparatus: Aluminum cylinders, pulley cord, scissors, small rods, right angle clamps, small bases, force sensors,

lab jacks, 600 mL beakers, water, mystery fluid (about 5 liters of fluid, preferably with a density more than 20%

different from water), digital calipers, hanging mass sets, PASCO Science Workshop 750 Interface & Power

Supplies.

Purpose:

The purpose of this lab is to demonstrate Archimedes’ Principle by submerging a small mass of known volume and

comparing experimentally measured forces with theoretically calculated forces.

Theory:

According to Archimedes’ Principle, the buoyancy force Fb of a body is equal to the weight of displaced fluid wf.

This is written in equation form as

VgwF ffb ρ==

where ρf is the density of the fluid, V is the volume of the submerged object and g is the magnitude of the

acceleration due to gravity.

The volume V can be rewritten in terms of cross-sectional area A and height h as

AhV =

for objects with a constant cross-sectional area. By substitution we then have

gAhF fb )(ρ= .

Thus, if Fb is compared to h for a known A the density of a fluid may be determined.

In today’s experiment it will be difficult to measure the buoyant force directly.

Typically this is experiment is done by plotting T versus h and determining ρf from

the slope of a linear fit.

Procedure:

• Setup the PASCO interface and computer as demonstrated by your lab

instructor. Connect a force sensor. Double click on the appropriate input (the one to

which you connected the force sensor). In the pop-up window find the “FORCE

SENSOR” and click on it.

• An icon will appear in the upper left corner that says Force. Drag this icon

down to DIGITS. Hit the start button on the very top row of buttons.

• Set up a base and stand. Use a right angle clamp to set up a horizontal rod. The

rod should be high enough above the ground to mount a beaker on a lab jack AND

still have room for a hanging force sensor.

• Mount the force sensor on the horizontal rod with the hook end down. It should

be hanging vertically.

• Hang several known masses on the force sensor. Verify that the force sensor is

taking accurate measurements. If the numbers are way off your force sensor may not

be perfectly vertical. Try adjusting the angle that it dangles. If this doesn’t improve

things, read the calibration guide at the end of the lab.

• Select an experimental mass and determine its volume. For example, if your

selected mass is in the shape of a cylinder, then measure the diameter and height, then

calculate the volume as equal to the area of the base (πr2) times the height. Use the

calipers to get accurate measurements.

• Hang the metal cylinder from the force sensor hook with a string.

• Fill the beaker with enough water so that the mass will completely submerge without touching the bottom

of the beaker.

• Mark the actual cylinder with 1.0 cm increments as precisely as possible before submerging the cylinder.

WATCH OUT! If you look closely, rasing the lab jack by 1.0 cm actually raises the water level by a slightly

different amount (because the cylinder displaces fluid as it is submerged). If the cylinder diameter is non-

negligible compared to the beaker diameter you’ll run into 5-10% errors!

• Now you are ready for data. Partially submerge the mass 1.0 cm beneath the surface of the fluid by raising the

beaker from underneath. Wait 5-10 seconds with the mass submerged until the force reading stabilizes.

force sensor

measures

tension = T

h

128

• Record the values of the force and h in your spreadsheet. Raise the beaker by another cm or so and record a

second data point. Remember too that you want to record the depth of the water h and not the distance the lab

jack moves. Continue until you have 8 data points.

• Use a linear trendline (show eqt’n and R2 value on graph).

• Draw a free body diagram. Derive a result that relates T to h. In particular, try to solve for T=???.

• Notice that on your graph h is like the x coordinate while T is like the y coordinate. Identify all the garbage in

front of the x coordinate as the slope. Write down an eqt’n on your graph that says “slope=???”.

• Rearrange that equation algebraically to solve it for the density of the fluid (ρfl). I expect to see this

rearrangement on your graph in the form of “ρfl =???”. This will be your experimental value of ρfl.

• Determine a percent difference for the experiment using water. NOTE: use this %difference as your

estimate of precision for today’s lab. This is a little different than our usual method.

• Repeat the data gathering process using another liquid with an unknown density different than that of water.

Use your measurement data to calculate the density of the unknown liquid. This will be your theoretical value

of the density.

• Determine the experimental density by measuring the mass of the liquid on the laboratory scale, then dividing

by the volume. Don’t forget to subtract out the mass of the beaker itself.

• Obtain a percent difference between the experimental and theoretical values of density for the mystery fluid.

Conclusions:

1. Suppose the aluminum cylinder was not solid but was actually hollow. Would you expect a positive or negative

percent difference on your value of ρfl? Think: would your tension be increased or decreased? How would that

affect the slope? How would that affect the value of ρfl?

2. Suppose this experiment was performed using a cylinder with identical dimensions but made from solid steel

instead of solid aluminum. You are told that steel is roughly three times denser than aluminum. How would the

graph differ? Consider both the intercept and the slope. Sketch what the graph of T vs h should look like.

Calibration Procedures:

Once you have your force sensor running, click on the Calibrate Sensors button above the picture of the interface

box in the Experiment Setup window.

Hang a 50 g mass on the sensor. This will be calibration point 1, standard value of 0.49 N.

Hang a 100 g mass on the sensor. This will be calibration point 2, standard value of 0.98 N.

Click Done.

129

Contour Plots

One way to describe a contour plot is to think of a topographic map. For each point on the map, an elevation is

specified. We could say the height (ℎ) is a function of latitude (�) and longitude (k). In other words ℎ is a function

of k and �. In math class we would say ℎ ℎ(k, �).

I tried to think of a simple variation where some parameter in physics depends on two different

terms. The simplest case I could think of was an Atwood’s machine shown at right. An Atwood’s

machine is simply a pulley with two masses on it. The black arrow indicates F5 will accelerate

upwards and F� will accelerate downwards with magnitude a. The rate at which the masses

accelerate will depend on the size of each mass and is given by a F� − F5F� + F5 �

where � is the magnitude of the acceleration due to gravity…� is NOT gravity (gravity is a force).

I considered the four cases below and punched in the numbers to get a feel for this equation.

CASE F5(kg) F�(kg) a(m/s2)

1 0 2.0 9.8

2 1.0 0 -9.8

3 1.0 1.0 0

4 1.0 2.0 3.3

Then I thought I might make a plot for some of these cases. In Excel I first considered F5 1.0kg and allowed F�

to vary from 0 to 10 kg. In jargon we say I fixed F5 at 1.0 kg and swept F� from 0 to 10.0 kg. I used Excel

formulas to compute the values. I did a second case fixing F5 at 2.0 kg while sweeping F� from 0 to 10.0 kg.

The formula for the top data set in cell B5 was

=$B$2*(A5-$A$2)/(A5+$A$2)

• Notice that CASE 2 (F5 1.0kg

&F� 0kg) is represented by the point

where the line crosses the �-axis.

• Notice that CASE 3 (F5 1.0kg

&F� 1.0kg) is represented by the

point where the line crosses the k-axis.

• Notice that CASE 4 (F5 1.0kg

&F� 2.0kg) is 3.3 m/s2

The formula for the bottom data set in cell B21

=$B$18*(A21-$A$18)/(A21+$A$18)

F5

F�

130

We still haven’t gotten to a contour plot…what is the point of all this?

I wanted to point out that for each different value of F5 we get a different graph of a versus F�. Now I want to

build a 2D data set. Rather than calculate all 100 values by hand I used an Excel formula to turn the crank for me!

The formula I used in cell C3 was

=9.8*(C$2-$B3)/(C$2+$B3) • The bold numbers show the

value for F� across the top

(row 2) and the values of F5

(column B)

• The values for acceleration are

NOT in bold in the bulk of the

chart.

• Notice first cell gives an error.

If both masses are zero out

formula shouldn’t apply;

seems reasonable.

MANUALLY SET CELL C3

TO ZERO.

• Notice row 4 gives us the first

data set on the previous page

where F5 is fixed at 1.0 kg

and F� is swept from 0 to

10.0 kg

• Notice row 5 gives us the

second data set on the

previous page where F5 is fixed at 2.0 kg and F� is swept from 0 to 10.0 kg

• Notice whenever the masses are equal (main diagonal of the matrix) the acceleration is zero.

• Notice whenever F5 > F� (lower left portion of the matrix), the acceleration is negative.

• Notice whenever F� > F5 (upper right portion of the matrix), the acceleration is positive.

• In general the acceleration has large magnitude when one of the masses is zero (or near zero) and the other

is large.

• In general the acceleration has small magnitude when both of the masses are large.

Holy cow! Look how much more we can learn from this more complicated data set than by simply looking at one or

two cases. Even more exciting is to plot this as (you guessed it) a contour plot. In Excel a contour plot is called a

“surface plot”. If you need help, try searching the help file for “insert surface”. My attempt at instructions is

below.

1. Highlight all the numbers. Use only the values for acceleration. Do not include the masses in column B or

row 2.

2. Click on the INSERT tab near the top left corner of the screen.

3. Near the middle-top of screen click on a chart type…any type for now will do.

4. Once the chart pops up, right click on it and select “Change Chart Type”.

5. Look around for the surface plot option, click on it, and hit ok.

6. If you right click and hit “Format plot area” you can tinker with the 3D rotation options to spin the graph

around and pick your favorite view point!

Check out what I choose to do on the next page…

131

• Notice the top plot shows a 2D

surface that curves in 3D space.

• On the left side of the plot we see the

scale for the acceleration ranging

from -10 to 10 along the ô-axis.

• I removed the axis labels for the k-

axis and �-axis. I let F5increase

going into page while F�increases

going to the right.

• At the bottom of the plot we see a

legend that helps us interpret which

color goes with which range of

values.

• Notice whenever F5 > F� (lower left

portion of the graph), the acceleration

is negative.

• Notice whenever F� > F5 (upper

right portion of the graph), the

acceleration is positive.

• In general the acceleration has large

magnitude when one of the masses is

zero (or near zero) and the other is

large.

• In general the acceleration has small

magnitude when both of the masses

are large. This is represented by

points in the back corner (back

middle) of graph.

• The second plot is a lot easier to see. I

removed the axis labels for the k-axis

and �-axis. I let F5increase going

upwards while F�increases going to

the right.

• Notice the legend below the graph

shows when the acceleration is

positive and negative as well as large

or small.

• Something doesn’t seem quite right

with the main diagonal…

-10.0

-5.0

0.0

5.0

10.0

Chart Title

-10.0--5.0 -5.0-0.0 0.0-5.0 5.0-10.0

Chart Title

-10.0--5.0 -5.0-0.0 0.0-5.0 5.0-10.0

132

For Contour Plots I prefer the the formatting options in MATLAB to Excel. To do this we must Export the data as a

CSV file, upload the file to Matlab, then use code in Matlab to create a better contour plot.

1. Copy the data, acceleration numbers only, and paste them into a new file in Excel.

2. Click “Save As” and save the file as a comma delimited file (CSV file).

3. Once the file is saved, you will need the file path. Simply

find the icon in your computer, right click on it, and select

“Properties”. From that screen you can determine the

filepath. As an example, My filepath is

E:\AHC\Fall15\161\2DAtwoodsData.csv

Notice I got the first part from “Location:” and the rest

from the top two lines of the pop-up window.

4. Now open up Matlab and type the following lines of code:

Filename=‘E:\AHC\Fall15\161\2DAtwoodsData.csv’ M=csvread(Filename); contourf(M); title(‘a versus m1 and m2’); xlabel(‘m2 (kg)’); ylabel(‘m1(kg)’);

5. You should have a nice contour plot pop up in a second window. If you look closely

you should see the tiny rectangular rainbow icon near the middle-top of the pop-up

window. Click on it to get the legend showing which values of a are positive or

negative. My plot ends up looking like the one shown below.

Notice the axis labels are still messed up!!!

They range from 1 to 11 kg instead of 0 to 10 kg. To

fix this, create two vectors in Matlab. Then ask the

contour plot to use these vectors as your axis labels.

Use the following code: m1=[0:1:10] m2=[0:1:10] contourf(m1, m2, M);

Notice the axis numbers are better but you need to

redo the lines of code putting on the words. Rather

than retype, hit the up arrow several times until you

see the correct line of code. Alternatively, you could

look at the bottom right side of your Matlab window

for the “Command History” window. From there you

can simply scroll around and find the line of code you

need and double click on it!

Look at the final graph on the next page.

133

I think you’ll agree there is an amazing amount of information on this graph. Also, I think you will agree that, while

it took a little bit of work, the final product is much more easily understood than in the default Excel surface plot.

• The color bar clearly shows the values in an ordered fashion (compared to the random color choices of

Excel). More blue means more negative; more red means more positive.

• We see the clear main diagonal of zero acceleration when F5 F�.

• Above the main diagonal (F5 > F�) all accelerations are negative.

• Below the main diagonal (F5 < F�) all accelerations are positive.

• When F5 and F� are both small we see the acceleration can change dramatically for even a small variation

in either F5 or F�.

134

Going further: you might consider learning how to use Matlab to do this entire calculation. You could create the

vectors F5 and F� using the following code: m1=[0:1:10] m2=[0:1:10]

Form there you could use the meshgrid function to create a matrix for the acceleration a. Think about the matrix

element a`õ (ith row, jth column of matrix a). You would want to use the ith element of F� and jth element of F5 to

compute a`õ . It should only be a few lines of code. Perhaps you could figure it out or take a course on Matlab…

Feeling motivated? Try this code. m1=[0:1:10]; m2=[0:1:10]; [M2, M1]=meshgrid(m2, m1); A=9.8*[(M2-M1)./(M2+M1)]; contourf(m2, m1, A);

Notice the following:

• Five lines of code takes less than a minute to write and produces an amazing plot that is easy to

comprehend while filled with tons of information

• Capital M2 is not the same as lower case m2 in Matlab (in Excel they are the same)

• The period is necessary when doing the division. Matlab assumes all operations are matrix operations

unless otherwise specified. What we want to do is divide the elements of the matrix (M2-M1) by the

elements of the matrix (M2-M1).

• Notice the defect that occurs when F5 F� 0. To fix the defect, it is reasonable to manually set that

particular value of a to zero by typing A(1, 1)=0;

• You may have noticed the colorbar showing accelerations ranges from -10 to 8 (not -10 to 10). I believe

this can be adjusted in Matlab but I didn’t have the time to fix that in these notes.

• If you change the code from m1=[0:1:10]; to m1=[0:0.1:10]; the values of F5 will go by 0.1 kg increments

instead of 1 kg increments.

• When you type the semi-colon after your commands it prevents Matlab from trying to print the entire

massive matrix on your screen. If you leave the semi colon off it Matlab will list out the entire matrix on

screen in full detail.

135

If you want more practice:

A ball is kicked horizontally from the top of a building of height ℎ with initial speed E. Assuming

air resistance is negligible, the horizontal distance traveled (� =“the range”) of the ball is

� = E92ℎ�

Make a contour plot showing the range of the projectile as both E and ℎ are varied.

• Think: when you set E 0 does the ranger result makes sense? Why does it make sense?

• Think: when you set ℎ = 0 does the ranger result makes sense? Why does it make sense?

• We chose to ignore air resistance. In real life, at some point, this assumption fails. Consider what it feels

like putting your hand outside the window of a moving car. At what speed, in mph, does it feel like the

force is strong. Obviously this is extremely subjective, but it might give you an idea of where your works

is completely useless. Recall 1mn � 2.2mph.

Consider the picture at right. Newton’s second law tells us Y cos ± Fa

Perhaps an engineer wants to use a large range of masses and angles. In each case the engineer

wants the block to have an acceleration of 2� � 20 mn&. Rearranging the equation to solve for force

we find Y = 2F�cos ±

Create a contour plot showing the required applied force as a function of both angle and mass. Here are some things

you will have to consider:

• If you use Excel or Matlab, do you need to use radians or degrees?

• Typically people think in degrees so you want the final data on the chart in degrees not radians. Is there a

function in Excel or Matlab that will convert degrees to radians for you? How would you incorporate that

function into your equation.

• Before pumping out all the numbers, consider the limitations of your equation. Under what circumstances

will your equation break down? For instance consider what should happen to the equation when you use

large angles or small masses.

• Select a range of values for F and ± that will make the chart useful (say angles between 0°and80° and

masses between 0.5 kg and 5.0 kg).

• Choose an increment size such that you have a decent amount of points. Since the computer is doing all the

work, you could go by increments of 1° and 0.1 kg.

θ F

E

136

What about a ball launched from the roof of a building at an angle? Suppose you

always throw the ball with speed E. Allow the height of the building x and the launch

angle to vary ±. Assume the ground below the building is level.

• First verify the range equation for such a flight is

k = E cos ± ÷E� sin� ± 2�x E sin ±� = E�2� sin 2± ø1 91 2�xE� sin� ±ù

• Notice the re-arrangement at far right looks slightly easier to code.

• Notice the re-arrangement at far right will have an artificial singularity (denominator goes to zero).

• Since I am not a programmer, I would probably stick to the uglier yet singularity-free version.

• Set up your code so you first choose the velocity.

• You could use a fixed range of x values (say 0 to 20 m going in 0.25 m increments).

• You could instead allow x to scale relative to E. Why the latter? For bigger E’s, larger heights become

more interesting. Even though they don’t have the same units, you might set the number for x equal to 4E

(number only, not units). You could set the increment size by dividing the largest x by 100 or something

similar.

• The angle probably ranges 0° to 90° in 1-degree increments. Actually…why not try D90° to 90°! • Think about the special cases that might arise and use these to check your code for bugs. Perhaps consider

the following:

o When shot straight up (or down) the range should be zero for any x or E combination.

o If you drop the rock (by setting E = 0) the range should again be zero for any x or E combination.

o If you use level ground (when x = 0 on your plots), the range should max out at 45°. For larger

heights the range should max out at smaller and smaller angles.

o For slow speeds, increases the height should change the angle for max range more rapidily than

high speeds

o While a bit tricky to check in practice, consider what should happen if you use ± = 0°. When shot

horizontally, you results should get the same results as your earlier, simpler code. The only

problem? Earlier we varied x and E, not x and ±. Still, I think you should be able to compare one

strip of this contour plot to one strip of that contour plot.

o Feeling gangbusters? In theory I suppose you could check all angles between D90° to 270°! Angles between 90° to 270° correspond to shooting the ball to the left. You could verify your

code is symmetric as it should be.

o The more ways you have to check your code the better. Think: you spent all this time writing

something, you want to show your boss, save the company money, and look good to get a raise.

Why not make sure the damn thing works before you look like an idiot, cost the company tons of

money, and get fired? No one will care how hard you worked getting the thing to almost

work…they will only remember it didn’t work. Is it fair? No. Is it reality? Yes.

±

E

k

�

137

What about a totally different topic? For instance, static electricity. We will learn much

later about a concept called electric potential (voltage). For now, it suffices to say the

electric potential near a single point charge is given by <5 = ·'5�5

where '5 is the size of the charge (in units of Coulombs = C), �5 is the distance from the

center of the charge (in units of m), and · is the Coulomb constant (· � 9 � 10� ý∙m&à& �.

In the picture at right the point charge is located at the point �kÊ, �Ê�. I want to calculate

the electric potential (voltage) at the other point �k, ��. • Notice the distance can be determine using �5 = ÷�k D k5�� �� D �5��.

• The voltage blows up for small values of �5 (close to or at the point charge).

• Consider trying to figure out how to use an “IF” statement in your calculation. Basically, you use this type

of statement to tell the computer “IF the voltage calculation gives an error or is greater than some number

(say 100) use the number 100 instead of the calculated value.” Perhaps a better way would be to tell the

computer “IF the distance to the point charge is too small, say �5 < 0.001m, set the voltage for such points

to 100”. You can learn more about how to handle this later.

Once you can handle a single point charge, try adding a second point charge. The voltage at any one point will be

the sum of the voltages of each charge (<%=%J­ = <5 <�). This is called the superposition principle.

Much later we will learn the electric potential can be used to determine the electric field. Form the electric field we

could determine the electric force. The electric force could be used to determine an acceleration and thus the motion

of any charged particle near this assemblage of point charges. Unfortunately, the acceleration would not be

constant.

We could, however, consider very small time scales. For each small time scale problem the charges won’t move

much. For each small time scale problem the acceleration is approximately constant. The procedure for handling

problems this way is discussed next.

�

�k5, �5�

�k, ��

138

139

Stepping it up a Level

Until now, all of our equations have been exact. That is to say we have used no approximations to compute each

data point. Unfortunately, doing only this type of coding severely limits our abilities to only problems we can solve

exactly. In real life, we might have air resistance being non-negligible. How might we handle that situation?

One powerful tool is to use time steps. The general procedure is this:

• Determine an equation that applies at each instant in time.

• Determine your initial state (e.g., initial height, speed, launch angle, etc).

• Apply your equation over a tiny time interval ∆� starting with the first time instant.

• Your equation applied to the initial state (� = 0) determines the state at time (� = 0 ∆� = ∆�) • Repeat the process. Equation applied to new condition (� = ∆�) determines state at next new time � = 2∆�. • Think about how many times you want to repeat the process. Keep in mind if you repeat this process N

times you will model all times between � = 0 → M∆�. • Check: if your graph seems to be doing weird things, perhaps your time step was too large (making the

approximations crappy). Maybe you had a typo equation? Maybe the equation you tried to apply did not

accurately model the system?

Examples begin on the next page…

140

Example

Let’s do something we already understand pretty well so it will be easy to check our ability to use the

time-step technique. One simple problem that comes to mind is drop the rock. Notice I have chosen

down as the positive direction in my coordinate system at right to get rid of minus signs. I am also

setting the initial position as the origin to reduce clutter in the equations. To keep the math easy, I

will assume � = 10 mn&. At any random point in time the position and velocity as functions of time

are � = �` E`� 12��� = �` E`� 5�� E = E` �� = E` 10� We can rewrite these equations in terms of time steps. We say the �� 1�%V position will depend on

the �%V velocity using �ij5 = �i Ei�∆�� 5�∆��� Eij5 = Ei 10�∆�� In this case our initial conditions are �Ê = 0 and EÊ = 0. Let’s start the problem using a time step size of ∆� = 1s. We will use M = 10 so we plot the first 10 seconds of motion. Doing the procedure one step at a time we find �5 = �Ê EÊ�∆�� 5�∆��� �5 = 0 0�1� 5�1�� = 5 E5 = EÊ 10�∆�� E5 = 0 10�1� = 10

Ok…now we use the values of �5 and E5 to determine �� and E� another ∆� = 1s later. �� = �5 E5�∆�� 5�∆��� �� = 5 10�1� 5�1�� = 20 E� = 10 10�∆�� E� = 10 10�1� = 20

Now use �� and E� to determine �8 and E8 another ∆� = 1s later. Show �� = 20 20�1� 5�1�� = 45 E� = 10 10�∆�� E� = 10 10�1� = 30

Now it would make sense to use a computer to calculate all the info. The table and plot are shown below. While the

data you see below is nothing new, the way it is calculated is new.

The function I used to compute cell B6 was

=B5+C5*$D$2+0.5*$C$2*$D$2^2

The function I used to compute cell C6 was

=C5+$C$2*$D$2

�

141

Drop the Rock Including Air Resistance

Now we want to include air resistance. An FBD at a random instant in time gives the force equation F� D qE� = Fa

For a baseball dropped near sea level I estimated q = 0.0013 ¦§m . The mass of a baseball is

approximately F = 0.15kg. So we can compare to the previous problem, again assume � = 10 mn&. This time we see that the acceleration will change. The equation for the acceleration is a = � D qFE� a = 10 D 0.00867E�

We want to use the �%V value of E to estimate �%V value of a. This is an only an estimate because we

during the time step the speed increases and the acceleration will decrease. If our time step is small

enough this estimate should still produce an accurate model. In terms of time-steps we should say ai = 10 D 0.00867Ei�

We will still model the motion as constant acceleration. As long as the times steps are small enough this model

should be approximately correct. This means we will still use �ij5 = �i Ei�∆�� 12 ai�∆��� Eij5 = Ei ai�∆�� Let’s go through the first few calculations by hand to get a feel for the problem. Let’s start the problem using a time

step size of ∆� = 1s for M = 10 steps. Note: this step size will probably prove too large but that we can check that

later. Finally, again assume our initial conditions are �Ê = 0 and EÊ = 0.

Compute �

(Compute Acceleration For Initial Stage)

aÊ = 10 D 0.00867EÊ� aÊ = 10 D 0.00867�0�� = 10

Use å�, ��, & Ð�to determine åÑ and �Ñ

�5 = �Ê EÊ�∆�� 12 aÊ�∆��� �5 = 0 0�1� 5�1�� = 5 E5 = EÊ aÊ�∆�� E5 = 0 10�1� = 10

Ok…now let’s do the next stage.

Compute ÐÑ

(Compute Acceleration For Next Stage)

a5 = 10 D 0.00867E5� a5 = 10 D 0.00867�10�� � 9.91

Use åÑ, �Ñ, & ÐÑto determine å« and �«

�� = �5 E5�∆�� 12 a5�∆��� �� = 5 10�1� 4.96�1�� = 19.96 E� = E5 a5�∆�� E� = 10 9.91�1� = 19.91

Ok…now let’s do the next stage.

Compute Ы

(Compute Acceleration For Next Stage)

a� = 10 D 0.00867E5� a� = 10 D 0.00867�19.91�� � 6.56

Use å«, �«, & Ыto determine å� and ��

�8 = �� E��∆�� 12 a��∆��� �8 = 19.96 19.91�1� 3.28�1�� = 42.79 E8 = E� 6.56�∆�� E8 = 19.96 6.56�1� = 26.52

On the next page we will let the computer do the work, discuss the problems in our model and time-step-size.

�

142

Below I am showing a picture of the Excel sheet I made. The Excel formulas I used are shown above the screen

capture image.

In cell D5 I determined a using

=10-$F$2*C5^2 In cell B6 I determined � using

=B5+C5*$G$2+0.5*D5*$G$2^2

In cell C6 I determined E using

=C5+D5*$G$2

Things to notice:

• As speed increases we expect drag force increases. Eventually it should balance the weight force. We

expect acceleration to decrease over time (see bottom plot).

• At some point we expect the ball will reach terminal velocity (Er). By definition, at Er the acceleration is

zero and drag force equals weights force. According to our theoretical equation

a = � D qFE�

0 = � D qF Er�

Er = 3F�q � 34ms

Notice our numerical model accurately predicts this!

• I was worried about the size of this time step being too large. As it turns out, it seems a decent choice.

How did I figure this out? I redid the problem using a smaller time step (I chose ∆� = 0.1s) and noticed

the plots were nearly identical! While this type of check works for this problem, in general one should

think carefully about the size of the time step. Take a numerical ODEs class or something similar.

• Going further: one powerful tool is to learn how to use a variable step-size. For times when acceleration is

changing rapidly (at the beginning of the problem) one would choose a small time step size. For regions

where the acceleration is changing slowly (close to terminal velocity) one could get away with a larger time

step size. This technique can make larger codes run more efficiently and cut down computation time. Not

a big deal here but a huge deal in large codes.

143

Air Resistance for a 2D Projectile

To determine drag force I used the equation ��a� = qE� q = 12²¤y

where ¤ is the density of air (approx. 1.2 ¦§m.), y is the cross-sectional area of the ball (approx. >��), and ² is a

coefficient between 0 and 1 (I chose ² = 0.5). To learn how the coefficients are computed, read about Reynolds

number.

If you want to extend this problem to 2D motion it gets trickier. Suppose we launch from initial height x with initial

speed EÊ at launch angle ±. Since drag force always opposes the direction of motion we can actually rewrite the

drag force as �ÚÚÛ = DqEEÛ = DqEE(�̂D qEE$�̂ Notice this equation works on the way up or down. On the way down E$ < 0 causing an upwards component in the

drag force. Notice we will have to do kinematics for both the k- and �-directions using a$ = � D qFEE$

a( = D qFEE(

E = 3E(� E$�

� = �` E`$� 12 a$�� E$ = E`$ a$� k = k` E`(� 12 a(�� E( = E`( a(� �Ê = x kÊ = 0 EÊ( = EÊ cos ± EÊ$ = EÊ sin ±

You could check your work by comparing your code to the projectile motion PhET created by the University of

Colorado. Maybe you can appreciate even more all the amazing work they do.

Another way to check your work is to compare your plot to the ideal case with no air resistance (which has an exact

solution). Your plots should be identical if you set ² = 0. As you increase the coefficient, the plot should deviate

from the ideal path by more and more. With drag engaged in your model, you should find it actually takes slightly

less time to rise to max height than it does to fall back to earth!

If you crank up the drag force by artificially setting ² = 10or1000 you would expect the object to hit terminal

velocity instantly. You can imagine the projectile will have massive drag and fall very slowly to earth (just like a

cotton ball or coffee filter falls to earth more slowly than a brick).

144

Extra problems

The magnitude of the electrical force between two point charges is given by Y�=i5 = ·'5'���5�

where '5 & '� are the sizes of the two charges (in units of Coulombs = C), ��5 is the center-to-center separation of

the two charges (in units of m), and · is the Coulomb constant (· � 9 � 10� ý∙m&à& �. You likely have heard already

that opposite charges attract and like charges repel.

Suppose two point charges are constrained to move along the k-axis. Assume gravitational forces are negligible.

The first charge is placed at the origin and will be released from rest. The second charge will be fixed in place at k� = D0.10m. Assume both charges are positive with '5 = '� = 1.0μC. Assume the mass of the charge at the

origin is 10g = 0.010kg. We expect the charge at the origin, upon being released from rest, will move to the right.

As it gets farther away the acceleration will decrease. The equations of motion will be a5 = Y�=i5F ��5 = b k k = k` E`(� 12 a(�� E( = E`( a(� kÊ = 0 EÊ = 0 Remember, the acceleration is not constant. However, if we choose a small enough time step, the acceleration is

approximately constant.

Try plotting the motion (with k�-, E�-, and a�-plots) for the charge that is fee to move.

Think about how things change if the second charge is negative instead of positive. Think about how you would

choose to handle the imminent collision. Would you let the moving charge pass right through the second? Would

you have it bounce backwards? In either case, what, if anything would you need to do to the code to handle the

collision? Do you see how an oscillation will occur?

What if you added in a third charge (also fixed)?

What if, in the three charge case, you allowed motion in both k- and �-directions?

What if you wanted to include gravitational force, or drag force, or both?

What if, all the charges were free to move? Then you would be required to get the net force on each charge at each

instant in time, compute each acceleration (in both k- and �-directions), then (assuming your time step is small

enough) compute the next position of each charge using constant acceleration kinematics, then repeat ad nauseum.

You would need to concern yourself with the issue of collisions. What would constitute a collision? What is the

actual size of your charges? Do they bounce off each other or pass through each other?

Check out the PhET for Electric Field Hockey. In this PhET you set out a number of fixed charges and release one

additional charge from rest. No air resistance or gravitational forces are included. This one is amazing!!!