Laboratory Manual

LABORATORY MANUAL

Phys1101, Spring 2011

University of Minnesota-Morris

3

Table of Contents

Pre-Lab Assignment for Experiment 1: ..................................................................................................... 5

Laboratory Experiment #1: Measurement ................................................................................................. 7

Pre-Lab Assignment for Experiment 2: Basic rules of Uncertainty propagation ..................................... 15

Laboratory Experiment #2: Motion with Constant Acceleration .............................................................. 17

Pre-Lab Assignment for Experiment 3: ................................................................................................... 21

Laboratory Experiment #3: Motion in two dimensions............................................................................. 23


Laboratory Experiment #4: The Addition of Vectors ................................................................................ 29


Laboratory Experiment #5: Newton's Second Law ................................................................................. 35

Pre-Lab Assignment for Experiment 6: Dealing with Random Errors ..................................................... 39

Laboratory Experiment #6: Centripetal Force ......................................................................................... 41


Laboratory Experiment #7: Collisions in One Dimension ........................................................................ 47

Pre-Lab Assignment for Experiment 8: Logarithm Rule .......................................................................... 51

Laboratory Experiment #8: Ballistic Pendulum ........................................................................................ 53


Laboratory Experiment #9: Static Equilibrium ......................................................................................... 59

Pre-Lab Assignment for Experiment 10: ................................................................................................. 63

Laboratory Experiment #10: Hooke's Law and Simple Harmonic Motion ............................................... 65

Pre-Lab Assignment for Experiment 11: Estimation ................................................................................ 69

Laboratory Experiment # 11: Standing Waves in a String ...................................................................... 71

Pre-Lab Assignment for Experiment 12: ................................................................................................. 77

Laboratory Experiment # 12: Calorimetry - Specific Heat ....................................................................... 79

Appendix A: Forms of Lab Reports ......................................................................................................... 81

Appendix B: Sample Lab Report ............................................................................................................. 82

Appendix C: Significant Figures .............................................................................................................. 86

Appendix D: Uncertainty ......................................................................................................................... 90

Appendix E: Propagation of Uncertainties .............................................................................................. 95

Appendix F: Finding the Slope and Slope of Linear Graphs ............................................................... 103

Appendix G: Excel Tips for Physics Labs .............................................................................................. 104

Laboratory Experiment 1: Measurement

5

Name: _________________________________________________ Score: _______of 15

Pre-Lab Assignment for Experiment 1:

This assignment needs to be completed before you come to lab. Turn in at the beginning of lab.

A. Study the use of significant figures. You can find information in Appendix C of the Lab manual or in section 1.5, page 8, of the text book.

B. For the following numbers, determine the number of significant figures, and write them in scientific notation.(4)

Physical quantity Number of SFs Scientific notation

520 m

420. miles

2000 years

C. Compute the net value and write it after the equal sign using the proper number of significant

figures.(4)

6.44 7.0

57.6 8

(3.74 - 2.83) 59.16

D. Study the role of uncertainty in Appendix D or section 1.5, pp 8ff textbook. Acquire and understanding of the following: (a) Precision error, (b) accuracy error, (c) random error, (d) absolute error, (e) relative error.

E. You use your wristwatch (showing time as in 3:25:12 PM) to measure the time it takes to walk from your door to the door of the physics lab, and find: 12 minutes 20.33 seconds. For this time measurement, analyze the uncertainty:

Type of Uncertainty Why does the measurement have this type of error?

Size of this error (number and unit, if applicable)

Precision error (limited by scale; equal to of smallest scale unit)

Accuracy error (limited by measuring method and calibration; small)

Random error (non-reproducible variations

Total absolute uncertainty

Total relative uncertainty


6


7

Laboratory Experiment #1: Measurement

Objective:

To learn proper measuring techniques. To see and understand the uncertainty inherent in all physical measurements. To become familiar with the use of significant figures and simple uncertainty calculations.

Introduction:

Meter Stick: All physical measurements are of limited precision. The instrument, the technique, and the quantity being measured all contribute to the uncertainty of a measurement. The meter stick, for example, has a least count of 1 mm. Then the uncertainty due to scale limitations is 1/2 mm. For an experimenter with a sharp eye, further discernment to the nearest 1/2 mm might be possible, reducing the uncertainty to 1/4 mm. Precision beyond this is, however, not very reliable with the common meter stick because of the coarseness of the black lines that mark off the millimeters. Another example is the Vernier caliper--an instrument designed to measure lengths more accurately than the meter stick. Its use is explained below. The least count of a standard Vernier caliper is 1/10 mm which leads to an uncertainty of 1/20 mm. But a more precise Vernier caliper, such as is used in this lab, can have a least count of 1/50 mm. An even more precise instrument is the micrometer caliper (also explained below). It has a least count of 1/100 mm. An experienced user might be able to discern tenths of the least count. The proper use of the meter stick might seem to need little explanation--one simply holds it against the object being measured and computes the difference between the readings at the extremes of the object. Yet, even during such a simple operation, a couple of errors are commonly made. Both involve the manner in which the meter stick is held. The parallax (line-of-sight) error can occur when the actual measuring scale is not up against the object being measured. When such is the case, the reading will depend upon the line-of-sight of the experimenter. (See Figure 1) The resulting reading will be of questionable value. This error is considered a blunder and should be avoided.

Figure 1.1: The parallax plunder

By holding the measuring scale directly against the object, a good reading can be taken irrespective of the line of sight of the experimenter. (See Figure 2) A "zero-error" is also common because the end of the meter stick is often assumed to correspond to a reading of "zero". For real meter sticks, such is usually not the case. The discrepancy might be due to the manufacturing process or to the wearing-down of the meter stick with use. In Figure 3, a technique that avoids the zero-error is illustrated. (Notice that a couple of units are worn off the left end of the stick.) The end of the meter stick is not used. Rather some convenient reference is chosen such as the 10 unit mark. The length is then the difference between the extremes.

5 6 7

Reads 7.4 Reads 7.6


8

10 20 30

Reading = 31 units

Figure 1.2: Better positioning of a meter stick

10 20 30

Measured length = 29 - 10 units = 19 units

Figure 1.3: Avoiding the zero error.

Vernier Caliper: A typical Vernier caliper is pictured in Figure 4. It consists primarily of two parts: the main frame and the movable frame. Often the frame will include both English and metric scales and the number of lines on the Vernier can be more than the number shown here. The outside caliper jaws are used to measure the outside dimensions of an object while the inside caliper jaws are used to measure inside dimensions (such as the inner diameter of a tube). The depth gauge can measure the depth of holes.

inside caliper jaws

outside caliper jaws

movable frame

main frame

depth gauge

Figure 1.4: A Vernier caliper

To read the Vernier caliper, one first looks to see where, on the main scale, the zero line of the movable scale (the "index line") lies. In general it will not line up exactly with any line on the main scale but will exceed some line by some fraction of the least count. (See Figure 5.) The purpose of the movable or "Vernier" scale is to


9

indicate that fraction to the nearest tenth. One counts the lines of the Vernier scale from the left (starting with zero) until one comes to a line that is best aligned with a main-scale line. The number of that Vernier line indicates the number of tenths by which the reading exceeds the number just to the left of the index line.

Example

Main Scale

0

5

10

Vernier Scale

30 40

Figure 1.5: A basic Vernier scale.

The index line indicates 26 mm + some fraction. Since the 7th line of the Vernier scale is aligned with a main-scale line, that fraction is .7 mm and the reading is 26.7 mm. Since the alignment of the seventh line was clear, we assign an uncertainty of one-half the least count, namely .05 mm. The measurement value is then 26.7 .05 mm.

Example

Main Scale

0

5

10

Vernier Scale

60 70

Figure 1.6

The index line indicates 60. + a fraction. Since the 4th line of the Vernier scale is aligned, the fraction is .4 mm. Thus the reading is 60.4 mm. If the alignment was clear, we assign an uncertainty of one-half the least count, namely .05 mm. Then the recorded value should be 60.4 .05 mm.


10

Example

Main Scale

0

5

10

Vernier Scale

90 100

Figure 1.7

The proper reading here is 91.9 .05 mm. The Vernier scale works because its lines are only 9/10 as far apart as the main scale lines. Figure 8 shows the relationship between the two scales when the zeroes are aligned.

Main Scale

0

5

10

Vernier Scale

10

5

0 1/10 3/10 5/10 7/10 9/10

2/10 4/10 6/10 8/10

Figure 1.8: The fundamental principle of the Vernier.

Observe that when the Vernier's zero line is aligned, all the other Vernier lines are not aligned. (The "ten" line is not counted.) Now, if the Vernier scale was moved just 1/10 of a unit to the right, then line number "one" of the Vernier scale would be aligned. If the scale was instead moved 6/10, then line "six" would be aligned. By studying Figure 8, we can understand how the Vernier scale indicates fractional readings. A systematic zero-error can also occur while using the Vernier caliper. When the calipers are closed all the way, the reading ought to be zero. If it is not, it is simply necessary to adjust all readings accordingly. For example, if a Vernier caliper reads .3 mm when closed, then all readings will be .3 mm too large. Just subtract .3 mm from each reading. The stainless steel (Scherr-Tumico) Vernier calipers that will be available in lab have a Vernier scale with fifty lines rather than the standard ten lines discussed above. In this case, by determining which of the lines on the Vernier is best aligned with a line on the frame scale, one can measure with a least count of .02 mm. Further estimating is quite difficult. In fact, it will sometimes be difficult to decide which single line is best aligned. The following example illustrates this difficulty though artificially.

Example: Notice that the Vernier is now the upper scale and the units are cm.


11

Vernier scale

0 5 102 3 4 6 7 8 91

main scale2 3 4 5 6 7

Figure 1.9: An example with a fifty-line Vernier scale. The digital nature of this computer rendition causes the erroneous appearance

of five lines that all seem to be perfectly aligned.

The zero line of the Vernier indicates that the value of the measurement is 2.1 cm plus some additional fraction. The additional fraction is determined by looking at which line of the Vernier is best aligned with a line of the main scale. Since five lines all look nicely aligned (58, 60, 62, 64, and 66), we will take the reading that lies in the center of the five, namely 62. Then the reading is 2.162 cm. A reasonable estimate of uncertainly might be .002 since it was not very clear which line was best aligned. Written value: 2.162 .002 cm. Micrometer Caliper: A typical micrometer caliper is shown in Figure 10.

Figure 1.10: The micrometer caliper.

The movable jaw advances or recedes when the thimble is rotated due to screw threads inside the sleeve. (Important: when closing the jaws, do not turn the thimble directly but rather turn the slip knob. This will prevent damage to the instrument and guarantee a consistent exertion of pressure on the jaws.) Because one revolution of the screw advances the jaw exactly .5 mm, the fifty equally spaced markings around the circumference of the thimble each correspond to 1/50th of .5 mm or .01 mm. Furthermore, as the thimble turns it reveals more of the sleeve scale. Between these two scales, very accurate readings are possible. An experimenter with a sharp eye can even make estimates of readings that lie between lines on the thimble scale.


12

Example

thimblesleeve

0 5

25

30

35

Figure 1.11

From the location of the thimble edge on the sleeve scale, one reads 7.5 mm + a fraction. From the location of the index line (the horizontal line on the sleeve) on the thimble scale one reads 30 hundredths. Thus: reading = 7.80 mm. Actually, because the index line was so well lined up with the 30, one might dare to say the reading is 7.800 .0005 mm.

Example

thimblesleeve

15 20

20

25

Figure 1.12

From the sleeve scale, the reading is 23 mm + fraction. From the thimble scale, the fraction is .23 plus some little bit. Estimating that little bit to be 8/10 of the distance between "23" and "24" on the thimble scale, one gets a thimble reading of .238. Then the total reading is 23.238 mm. Since the last digit, the 8, was an estimate, its uncertainty might 2. Then the reading should be recorded as 23.238 .002 mm.

Example

thimblesleeve

5 10 35

30

Figure 1.13

From the sleeve one reads 13.5 mm plus a fraction. From the thimble: fraction = .322 .002. Then the reading is 13.822 .002 mm.

The possibility of a zero-error also occurs with the micrometer caliper. To check the calibration, simply close the jaws completely, using the slip knob, and take the reading. If the reading is non-zero, all readings with that particular instrument will have to be adjusted accordingly.

Equipment:

Meter stick, Scherr-Tumico Vernier caliper, micrometer caliper, 150g (m=.01g) digital mass balance, sphere, small cylinder, block (iron).


13

Procedure:

Measure the dimensions of the sphere using the meter stick, the Vernier caliper and the micrometer caliper. When measuring, watch for uncertainty inherent in the quantity being measured as well as for uncertainty of the measuring devices. (Irregularities in the dimensions of the object being measured may exceed the uncertainty associated with the measuring device. Check for this by measuring the dimension at different locations and noting variations of readings.) Make sure the measurements made with the three devices agree with each other (within your assigned uncertainties). Also measure the dimensions of the cylinder and the block using the three measuring devices, keeping in mind the tips above. Measure the mass of each object. Compute the volume and density for each, complete with uncertainty. (Express the volume and density uncertainties as percentages.)

Uncertainty of volumes:

V W L H

V W L H 3

V D

V D 2

V D H

V D H

d m V

d m V

Comments:

Density, d, is defined as mass, M, divided by volume, V. m

dV

Express your densities in units of g/cm3. For your information, the textbook values of a few densities are:

aluminum--2.70 .01 g/cm3

steel--7.9 .05 g/cm3

brass--8.4 .05 g/cm3

copper--8.93 .01 g/cm3

gold--19.28 .05 g/cm3.

Question:

See Spreadsheet. 1. Make three statements about the composition of the three objects, using the micrometer caliper

measurement. State whether your result agrees with the standard densities within uncertainty, and give possible sources for the discrepancy.

2. Give reasons for your choice for the value of uncertainty for the diameter of the cylinder as measured

with a Vernier caliper. Submission: Completed spreadsheet by e-mail

Experiment 2: Motion at constant acceleration

14


15

Name: ________________________________________________ score:______of 10

Pre-Lab Assignment for Experiment 2: Basic rules of Uncertainty propagation

A. Study the rules for uncertainty propagation in Appendix E of the Lab manual. Note, in particular,

a. The addition/subtraction rule: 2Y A B C D then 2Y A B C D

b. The multiplication/division rule: A B

YC D

then Y A B C D

Y A B C D

c. The power rule: nY A A A A then

Y A A A An

Y A A A A

B. In Lab Experiment #2 the acceleration due to gravity g is calculated using

g

2DL

t2h

In which D, L and h are length measurements, and t is a time measurement. One measurement gave the following values:

50.0 0.1L cm , 2.7 0.1h cm , 90.6 0.1D cm , and 1.864 0.047t s .

a) Calculate the value for g from this measurement.

b) Find the % error for each (L, h, D, and t).

/ %L L

/ %h h

/ %D D

/ %t t

c) Using the Multiplication/Division and Powers Rule, write an equation for the relative uncertainty of g,

containing four terms (one for each of the measured variables).

d) What is the result for g g from this measurement?


16


17

Laboratory Experiment #2: Motion with Constant Acceleration

Objective: To determine, experimentally, the local acceleration due to gravity. To observe and analyze the motion of an object having constant acceleration. Familiarize with the tools of graphing by hand. Part A: objects in free fall Part B: object on frictionless inclined track

Introduction:

All objects in the vicinity of the earth's surface are drawn toward the center of the earth. In the absence of any restraints, such objects will fall with velocities that increase uniformly with time. The rate of

change of these velocities (their acceleration) is the same for all objects, namely about 9.8 m/s2. Free Fall: When an object is falling with this constant acceleration, its velocity will be uniformly increasing in the downward (negative y) direction. Unfortunately this all occurs so rapidly or over such short distances that one cannot visually discern the rate of change of the velocity. One might notice that the velocity starts out small and increases but without making some measurements one cannot say how much it was at a given time. For a quantitative analysis, some permanent record of the motion is needed such as might be provided by a strobe-illuminated time-exposure photograph or by a spark-timer output tape. Such records would show the location of the falling object at equally-spaced intervals of time. From this kind of data the average velocity during each time interval can be computed:

vavg = y / t

t. (Since the motion is one-dimensional,

vector notation can be omitted.) It turns out that one can also determine the instantaneous velocity at certain times from the above type of data provided the acceleration is constant. In fact, the average velocity calculated above for a given time interval is equal to the instantaneous velocity at a time halfway through that time interval. To

prove this, consider an interval t to t + t. Then

vavg = [v(t) + v(t + t)] / 2.

But

v(t + t) = v(t) + a t

so that vavg = [v(t) + v(t) + a t]/2

= v(t) + a t / 2 = v(t + t / 2)

Thus the average velocity during the interval t to t + t is equal to the instantaneous velocity at the mid-time t + t /2. Once the instantaneous velocity is known at a number of different times, a graph can be made showing v(t). The data points should form a straight line whose slope equals the acceleration of gravity, g. Thus g can be experimentally determined.


18

Inclined Track: A second method for measuring g is to time the fall of an object through a measured

distance, y. From y = gt2

/2, one could then obtain the acceleration. One difficulty with this method is that falls over human-scale distances occur in short time intervals that are challenging to measure accurately. To address this difficulty, it is possible to make the "fall" last longer. This can be accomplished by letting an object slide down an incline. Then, in the absence of friction, the component of g parallel to the incline will cause acceleration.

LD

g

g sin

h

Figure 2.1: An inclined plane.

time of "fall" down a certain length, D, of frictionless incline, the acceleration can be obtained from

D = a t2/2. Substituting for the acceleration, we obtain D = (g h / L) t2/2. Solving for g yields

g2DL

t2h

(1)

Equipment:

One free-fall apparatus with spark timer and D.C. power supply, transparent plastic ruler, meter stick, air track with air supply, glider with flag, electronic timer with two photogates (t=1%), and vernier caliper; graph paper.

Procedure:

Part A: The free-fall apparatus is designed to make a permanent record of a falling object's motion. The object falls between two vertical wires. A spark generator sends high voltage jolts through the falling object (via the two wires) at equally spaced time intervals. The resulting sparks make spots on a waxed strip of paper so that the location of the object is recorded. An analysis of those spots will yield g.

Step 1: Obtain data by setting the spark timer to 30 sparks per second (t = 1/30 second .1%) and release

the object. Remove the paper tape for analysis. (An alternative is to set the spark timer to 60 sparks per second and use every other spot.)

Step 2: Consider ten consecutive, clear spots on the tape. Number them from top to bottom. Measure the

distances between the spots, y, and record these in the suggested tabular form. Measure as precisely as possible using the transparent rulers! Be sure to record uncertainties.


19

Step 3: Compute the average velocity during each time interval vavg = y / t.

Step 4: Assuming t=0 corresponds to spot #1, compute the times, tM that correspond to the midpoint of each interval. (tM is the time halfway between the times corresponding to the endpoints of an interval. See the table below that illustrates this.)

Spot t (s)

Interval

y (cm)

tM (s)

1 0 0

1 - 2 0.5 (t t)

2 (t t)

2 -3 1.5 (t t)

3 2 (t t)

3 - 4 2.5 (t t)

4 3 (t t)

4 - 5 3.5 (t t)

5 4 (t t)

5 - 6 4.5 (t t)

6 5 (t t)

6 - 7 5.5 (t t)

7 6 (t t)

7 - 8 6.5 (t t)

8 7 (t t)

8 - 9

9

9 - 10

10

Step 5: Manually, graph vavg as a function of tM. Use a ruler and a sharp pencil (no pen). Include error bars, using the uncertainties in t and y for each point. Draw the "best" straight line through the points to get g. Choose two points (NOT any of your data points), far apart, on that line. Read their positions

1 1 1 1,v v t t and 2 2 2 2,v v t t on the axes of your graph and label the points. Estimate

the uncertainties for v and t in each point by holding your ruler in the positions of the steepest and flattest line you may fit to your data points (within uncertainty) and observe by how much your values

of v and t would deviate. The slope 2 1

2 1

v vslope a

t t is the acceleration component. Record it on the

graph. Find the he uncertainty of the acceleration (using add/subtraction rule and multiplication/division

rule) by 1 2 1 2

2 1 2 1

v v t tslope a

slope a v v t t, see discussion in appendix F.

Part B:

The air track is a device on which a glider can move with almost no friction. If the air track is tilted slightly from the horizontal, then a glider released from rest at the top of the incline will accelerate down with acceleration a = g sin.

Step 1: Carefully level the air track and adjust the air supply so that the glider can move freely. With too little

airflow, the glider may drag on the air track. With too much airflow, the glider may wobble and/or experience friction from individual air jets. Never slide a glider along the track with the air supply off -- you may scratch the track and/or the glider! If the air track is slightly bowed, level it as best you can, then document where it has peaks and valleys. Try to perform this experiment to minimize the affect of any bumps.


20

Step 2: Raise one end of the air track by placing an object of known height, h, under one of the feet of the air track. Since the feet are not at the ends of the air track, notice that in your analysis L is not the total length of the air track!

Step 3: Place the two photogates above the air track so that the glider can pass completely through both of

them. Set the photogate timer resolution to 1 ms. (t inherent is 1%.) Allow enough space below the second photogate to catch the glider with your hands before it smashes into the end stop, if necessary. Use the light emitting diodes (LEDs) on the sides of the photogates and the scale on the air track itself to determine how far the glider travels between tripping the two photogates. You will have to decide which of the three timer modes to use. Remember not to move either photogate once you have made this measurement.

Step 4: Practice releasing the glider smoothly from rest as close as possible to the upper photogate. Use the

LED on the photogate and the scale on the air track to help you do this reproducibly. To characterize the variability in this measurement, time the glider for 6 good runs.

Remember, Equation 1 assumes that the glider starts from rest at t = 0. With this timing setup, this

cannot be precisely true, but it will be a good approximation if you do the experiment carefully.

Step 5: Compute a value of g g from the average of the measured times. Step 6: Repeat the experiment for a different inclination angle. Then do it again for a third time. Remeasure L

and D each time.

Comment

The accepted value for the acceleration of gravity in Morris, MN is 980.57 0.05 cm/s2.

Question:

In Part 1, if the graph of vavg as a function of tM is extrapolated back to vavg = 0, what is the corresponding value of t? What is the physical meaning of this time? Submission:

1. One hand-drawn graph per student 2. Electronic lab spreadsheet

Experiment 3: Motion in two dimensions

21

Name: ________________________________________________ Score: ___________of 10


For a ball thrown from the right across the room, the following motion diagram was observed:

A. Sketch the x-t graph and y-t graph for this motion.

B. Sketch the vx-t and the vy-t graphs. Place axes as needed.

C. Sketch the ax-t and the ay-t graphs for this motion. Place axes as needed.

D. Using the Addition/Subtraction rule, and the Multiplication/Division rule, find the equation for the relative

uncertainty for 1 1

2

n nx xvt

, in which 1nx , 1nx and t are measured quantities.

x

y

x

t

y

t

vx

t

vy

t

ax

t

ay

t


22


23

Laboratory Experiment #3: Motion in two dimensions

Objective:

To study position, velocity and acceleration for situations in which motion occurs in two dimensions simultaneously.

Introduction:

When the motion of objects is confined to a plane, we refer to it as two-dimensional (2-D) motion. Itis instructive to examine the horizontal and the vertical components of such motion independently. When this is done, many familiar examples of 2-D motion are revealed to consist of either constant velocity or constant acceleration motion horizontally, coupled with either constant velocity or constant acceleration motion vertically. Motion in two dimensions may be studied by analysis of video clips in which the motion occurs in a plane perpendicular to the line of sight of the camera. A characteristic feature on the moving object is marked in each frame of the video so that the X-position and Y-position of that feature may be tracked as time goes by. By knowing the time separation of the frames in the video and the scale of the motion across the screen, one may deduce position, velocity, and acceleration versus time (and more!). The time information and distance scaling have already been documented in the video clips that we will examine. Choose two of the following examples:

A. A toy wind-up car rolling across the floor

B. A ball thrown between two students.

C. The bob of a pendulum is moving back and forth.

The movies will be analyzed frame by frame, positions in x and y will be measured.

Equipment:

Large graph paper, toy car, ball, tape, pendulum, a digital camera, a tripod, meterstick, connector cable to computer, jump drive to transfer the movies to all work stations

Procedure:

Choose two of the options. The procedure is exactly the same for each of the two types of motion. Follow these steps:

1. Cover the reference background with large-scale graph paper. That would be the floor for the car, or the wall for projectile motion and glider. Make a decision as to x, y directions and mark every tenth box in each direction. Record the size of 20 units in x and y, as well as their uncertainties.

2. Bring the camera into a position that allows to view a large enough area to follow your object in its motion. Use the movie setting at 30 frames per second. Set the maximum frame size possible for better esolution.

3. Film the motion of your object. Make sure that the object moves through a large portion of the view. For the projectile, try to throw the ball close and parallel to the marked wall, so that the influence of perspective in the movie will be minimized, when you determine the position of the ball later-on.

4. Transfer the movies to your workstation.


24

5. Open them in a program called ImageBrowser. The icon is located on the left-hand panel on your screen. Double-click on the selected movie and open it in full screen. The play button () will run the movie, which can be stopped with (). On the right, you find the buttons to move by single frames (| and | ). You can leave the full-frame view with a double-click on the right side of the screen.

6. Choose a point on your object, which is visible throughout the movie. Step through the movie frame by frame (as long as your object is visible), and record the x- and y- position in the data table provided. Also, assign an uncertainty to your x and y values. Since the movie is using the screen, you have to record on paper first.

7. Transfer your data to the Lab3 spreadsheet.

Evaluation:

8. Plot the trajectory of the object in a x-y graph in a separate sheet in your spreadsheet. Label the sheet, the axes and the graph. Include error bars in both directions. Help with graphing in Excel can be found in Appendix F.

9. Plot each, the x-t graph and the y-t graph in the spreadsheet. Include error bars in both directions, and label the graphs properly. Ask for assistance if you are not sure how to do this.

10. For each, the x- and y-direction, find the velocity versus time. This can be best achieved in the following

way: at time tn, the instantaneous velocity of the object can be approximated as 1 1

2

n nn

x xv

t. Use Excel

features wisely to do this if not sure ask for help. Also, determine the uncertainty for each velocity value. Note that you will not have a velocity value for your first and your last data point.

11. Analyze the acceleration for the y component in the projectile motion In a hand-drawn graph, plot the y velocity versus time. Use a ruler and a sharp pencil (no pen) for everything. Label axes and graph properly. Include error bars based on your data uncertainty on each data point..

12. Draw a best-fit line. Choose two points (NOT any of your data points), far apart, on that line. Read their

positions 1 1 1 1,v v t t and 2 2 2 2,v v t t on the axes of your graph and label the points. Estimate

the uncertainties for v and t in each point by holding your ruler in the positions of the steepest and flattest line you may fit to your data points (within uncertainty) and observe by how much your values of v and t would deviate.

13. The slope 2 1

2 1

v vslope a

t t is the y component of the acceleration. Record it on the graph.

14. Find the he uncertainty of the acceleration (using add/subtraction rule and multiplication/division rule) by

1 2 1 2

2 1 2 1

v v t tslope a

slope a v v t t, see discussion in appendix F.

15. Now, that you have the acceleration components, go back into the spread sheet, and draw a vector with the direction of the acceleration into the x-y graph for the data set you chose into three different data points.

Submission:

1. Electronic spreadsheet for each group. 2. One hand-drawn graph for each student.


25

Motion A:___________________________________________

Grid dimension (20 units Uncertainty (20 units)

X direction (cm)

Y direction (cm)

Frame rate (fps) 30

Frame-by-frame positions (beginning when motion is visible):

Frame X position in grid units x Y position in grid units y

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25


26

Motion B:_____________________________________________________

Grid dimension (20 units Uncertainty (20 units)

X direction (cm)

Y direction (cm)

Frame rate (fps) 30

Frame-by-frame positions (beginning when motion is visible):

Frame X position in grid units x Y position in grid units y

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Experiment 4: Addition of vectors

27

Name: ________________________________________________ Score: ___________of 10

Pre-Lab Assignment for Experiment 4: basic uncertainty approximation

A. Study Appendix E again. You will need the basic uncertainty approximation equation for the absolute error

f f ff a t y

a t y

if f is a function of the measured variables a, t and y. A partial derivativef

a treats all variables except

the a (in this case) as constants, but is otherwise executed as any normal derivative. For the work in B through E, use the space below and the backside of the paper.

B. For the equation cosxF F show that cos sinxF F F where is expressed

in radians. Find Fx Fx when F = 2.92 .05 N and = 10o .5

o.

C. For the equation sinyF F , show that sin cosxF F F where is expressed

in radians. Find Fy Fy when F = 2.92 .05 N and = 10o .5

o.

D. For the equation 2 2

x yF F F , show that x x y yF F F F

FF

. Find F F when Fx = 1.02

.02 N and Fy = 1.95 .03 N.

E. For the equation arctany

x

F

F, show that

2

x y y xF F F F

F (in radians). Find when

Fx = 1.02 .02 N and Fy = 1.95 .03 N. Remember that will come out in radians. Be sure to report both and in the same units.

1

1 The following partial derivatives may be useful:

2 2arctan

a b

a b a b and

2 2arctan

a a

b b a b


28


29

Laboratory Experiment #4: The Addition of Vectors

Objective:

To observe and investigate the manner in which vector quantities add.

Introduction:

In order to clearly describe many physical quantities, both a magnitude and a direction must be specified. Such quantities are called vector quantities. Some examples are position, velocity, acceleration, and force. The addition (or subtraction) of vector quantities is more complicated than the addition of ordinary numbers (scalars) because of the effect of the directions. For example, a walk of 4 miles to the East followed by a walk of 3 miles to the North results in a net displacement of only 5 miles:

3 miles

4 miles

5 miles

Figure 4.1

Two different methods exist for computing the sum of vectors--the graphical method and the analytical (component) method. In the graphical method, the vectors to be added are drawn, with some scale factor, on graph paper. The first vector is drawn outward form the origin at the angle of the vector with a length corresponding to its magnitude. The second vector begins where the first ended and continues in its direction for a distance corresponding to its magnitude. Any further vectors are added consecutively in like fashion. (The orientation of the vectors is very important.) The sum or "resultant" vector is the vector from the origin to the end of the last vector added. Figure 4.2 shows the resultant, R, of three vectors A, B, C.

A

B

C

R

Origin

Figure 4.2

The "equilibrant" vector, E, is that vector which, when added to a sum of other vectors, yields a total of zero. For example:

E + (A + B + C ) = E + R = 0

Then

E = - R . So the equilibrant is simply the opposite of the resultant. In the analytical method, the vectors to be added are first resolved into components along some set of axes (e.g. x and y). All the x-components are then added to yield the x-component of the resultant and the y-components are added to give the y-component of the resultant.


30

Equipment:

Force table, each with four mounted pulleys and weight hangers, 600g or 1200g digital balances (m=.1g), and assorted masses.

Procedure:

The force table is a small, round, horizontal table around whose circumference all 360 are indicated. A small ring at the center of the table is tied to several weight hangers on which various masses can be placed. The force of gravity on those masses therefore causes forces on the central ring via the strings. (The vertical force of gravity is turned into a horizontal force by the (frictionless?) pulleys.) See Fig. 4.3.

Figure 4.3: The force table.

The force of gravity is F = mg where m is the mass expressed in kilograms (kg) and g = 9.8057 .0005 m/s

2.

Then the unit of force is the Newton (N). For example, a mass of 100. grams will experience a gravitational force (.100 kg)(9.8057 m/s

2) = .981 N.

When all of the forces on the central ring are balanced (i.e. their sum is zero), then the ring will not be pulled preferentially in any particular direction. Then any one of the forces acting on the ring is the equilibrant of all the others. Trial #1 Hang 300.g at 10 and 240.g at 120. (Be sure to include the mass of the weight hanger in the 300.g total. Some of the weight hangers are 50g each and some are 8g each.) Find the experimental equilibrant on the force table by hanging masses on a third string. Vary the amount of mass as well as the angle until balance is achieved. To characterize the variability of this measurement, determine the mass range and angle range over which balance is maintained. Dont rely on the values stamped into the masses--put them on the digital balance to get the actual mass value and use that value in the experimental, analytical, and graphical methods. The balancing vector is the equilibrant. Record the experimental equilibrant in your results table. Also determine the equilibrant graphically by making a scale drawing on graph paper. Put only one drawing per page and make the drawing large so as to fill the page. Indicate the scale on each drawing (for example: 1N = 5 cm.) Also determine the equilibrant analytically. Use the experimental uncertainties of the specified masses and angles to calculate the uncertainty of the analytical resultant. Trial #2 Hang 100.g at 10, 200.g at 300, and 250.g at 200. Find the equilibrant in the three usual ways.


31

Properly determined uncertainties will be expected on the experimental and analytical results but you are excused from any error analysis of the graphical additions. It's not that it's excessively difficult, but it is quite tedious and time-consuming.

Comments

Please do not overtighten the pulley mounts! For best results, keep the strings aligned with the radial direction:

Right Wrong

Figure 4.4: Proper string alignment on the force table ring.

Some new Excel functions you will need are: ABS(...) = absolute value of argument in parentheses ATAN(...)=arctangent of argument in parentheses PI()=3.14159. . . (dont put anything inside the parentheses) Submission:

1. completed spreadsheet by e-mail 2. Two hand-drawn graphical solutions from each student

Experiment 5: Newtons Second Law

32


33

Name: ____________________________________ Score: __________of 10


A. Refer to appendix E again: The logarithm rule allows to calculate the relative error of a function as

follows: ln ln lnf f f f

x y zf x y z

. You will need this rule below.

B. For an arrangement as used in Experiment 5, the theoretical acceleration of the glider is found as

mA g

M m. Note that the mass m appears twice in this equation, prohibiting the use of the

multiplication/division rule. Use the logarithm rule to show that the corresponding error equation is

A g M m M

A g m M m M m.

C. If m = 0.065 .0005 kg, M = 0.1245 .0005 kg, and g = 980.57 .0005 cm/s2 find acceleration and

uncertainty A A/A (as a percent).

D. Write the equation for the relative uncertainty of acceleration 2

2Da

t, if both, D and t are measured

quantities.


34


35

Laboratory Experiment #5: Newton's Second Law

Objective: To explore how the one-dimensional acceleration of an object depends upon the net force applied to the object and its mass.

Introduction:

The net force on a glider moving on a leveled air track is essentially zero. The airflow provides the normal force to counter the weight of the glider, so there is no net vertical force. The glider may move along the track with very little friction because it is not in direct contact with the track, so there is no net horizontal force either. According to Newton's 1st Law, a glider at rest on a level air track should remain at rest, and a glider moving along a level air track should move with a constant velocity (until it hits an end stop). In fact, these expectations may be used to help level an air track, since a glider will accelerate down any slope. If an air track is not perfectly straight, the slight acceleration of the glider at different points along the track can help identify even miniscule peaks and valleys. For the subsequent discussion, it is assumed that the air track is perfectly level. A net force may be applied to a glider on an air track by running a string from one end of the glider, over a pulley at one end of the air track, to a block which drops vertically as shown in Figure 5.1. The magnitude of the acceleration of the glider will match that of the dropping block, as long as the string connecting them does not stretch. Assuming the mass of the string may be neglected and the pulley is nearly ideal (i.e. frictionless and massless), this one-dimensional acceleration should be a constant.

m

M

a

a

stringpulley

glider

air track

block

Figure 5.1 The acceleration of the glider and the dropping block may be determined using two photogate timers as follows. Place the photogates as far apart as possible so that the same feature of the glider, such as the leading edge of the flag, trips first one gate and then the other. Set the timer mode to record the time between these two events. For one-dimensional constant-acceleration motion, the experimental acceleration (call it a) is related to the elapsed time t and the distance traveled d the glider travels by

D = (1/2) a t2+v0

If the glider starts from rest v0 = 0 so this simplifies to

D = (1/2) a t2 (1)

This may be rearranged to deduce the acceleration a from measurements of D and t. Remember that Equation 1 only applies when the glider is timed starting from rest! This may be accomplished experimentally by placing the glider so that it does not quite block the first photogate (the indicator light on the side of the photogate will be helpful here) then smoothly releasing the glider so that it


36

does not wobble as it picks up speed. In this way, the glider will be essentially at rest when the timing begins so Equation 1 may be used. The vertical forces on the glider remain balanced, so the net force on the glider is simply the horizontal tension applied by the string. Notice that the tension in the string is not the same as the weight of the dropping block! If that were the case, the net force on the dropping block would be zero and it should move at a constant velocity, which contradicts the observation that the dropping block in fact accelerates downward. Predict the acceleration of the glider (call this theoretical value A), for comparison with the experimental acceleration a, as follows. Draw a free-body diagram of the dropping block then apply Newton's 2nd Law

F = ma to the block to determine how the weight of the block w = mg (directed downward) and the tension in the string T (directed upward) are related to the acceleration of the block A (directed downward). Considering the direction of A, which must be larger, T or w? Now rearrange your expression to deduce T in terms of m and A. Next, draw a free-body diagram of the glider and apply Newton's 2nd Law separately in the horizontal and the vertical directions. Be sure to use a different variable for the glider mass (perhaps M) so as not to confuse it with the mass of the block m. Even though the glider accelerates sideways and the block accelerates downwards, remember that the magnitude of their accelerations A is the same because an inextensible string connects them. Thus you may use the same variable A for both the block and the glider. By combining the force equations for the block and the glider, eliminate the variable T and solve for A. This will allow you to predict the acceleration A simply from the masses m and M (and constants like g). Finally, compare the predicted acceleration A to the experimental acceleration a, determined earlier from measurements of D and t. These accelerations should match (within their uncertainties, of course) if Newton's 2nd Law is correct and if the assumptions made about the motion are reasonable. Notice that the experimental acceleration a will have an associated uncertainty which stems from the uncertainties in D and t, and the predicted acceleration A will have an associated uncertainty which stems from the uncertainties in m and M.

Equipment:

Air track, glider with flag, electronic timer with two photogates (t=1%), light string or monofilament fishing line, pulley, collection of masses for dropping block, and a 1200g (m=.1g) digital balance.

Procedure:

Step 1: Derive an expression for the acceleration A and its uncertainty A and show it to your instructor before going on to Step 2. Include the derivation (with proper force diagrams) in your report.

Step 2: Carefully level the air track and adjust the air supply so that the glider can move freely. With too little

airflow, the glider may drag on the air track. With too much airflow, the glider may wobble and/or experience friction from individual air jets. Never slide a glider along the track with air supply off -- you may scratch the track and/or the glider! If the air track is slightly bowed, level it as best you can, then document where it has peaks and valleys. Try to perform this experiment to minimize the affect of any curvature.

Step 3: Attach the string between the glider and the dropping mass, as shown in Figure 5.1. Place the

photogates at the two ends of the accelerated motion (see Comments below). Set the photogate to 1 ms resolution. Deduce the range of masses over which you can safely perform the experiment, which means catching the glider after it passes the second photogate but before it smashes into the end stop. (Let x1 be the position of the first gate and x2 the position of the second.) Practice releasing the glider smoothly from rest as close as possible to the first photo gate. Use the LED on the side of the photo gate and the scale on the side of the air track to help you do this reproducibly. These two


37

devices will also allow you to determine the travel distance, D, accurately. Remember not to move the photo gates once you have measured this distance! Don't forget to determine the mass of the glider M as well.

Step 4: Select five different mass values m for the dropping block, starting at 20g and adding 10g increments.

For each value of m, measure t 5 times to help characterize the variability in the timing. Use the average t in your calculation to determine the experimental acceleration, a. (For the uncertainty on the average time, take the average of the individual time uncertainties plus the standard deviation of the time values.)

Step 5: For each value of m, determine the experimental acceleration a + a as well as the predicted

acceleration A + A. Present your 5 pairs of results in tabular form.

Comments:

Be sure that the glider is still accelerating when it reaches the second photo gate. In other words, the glider must trip the second photo gate before the dropping block strikes the floor (at which point the string goes slack and the glider proceeds with constant velocity).

You may use the accepted value of g = 9.8057 + 0.0005 m/s2 in your analysis, which applies in Morris, MN.

Questions:

(a) One of the assumptions in the analysis was that there was no friction in this case. What effect would friction have had on the results (the relationship between your experimental and theoretical values) you found? Is there evidence of any friction in your calculated results? Where might friction arise in this apparatus? List several possible sources.

(b) One of the assumptions in the analysis was that the glider was at rest when the timing began. If you

were careful, this assumption should be quite good, though it cannot be perfectly accurate. If you were not so careful and the glider had some modest velocity when it tripped the first photo gate, what would be the affect on the relationship between your theoretical and experimental acceleration? How can this possible error be distinguished from the possibility of friction in the apparatus?

(c) The larger the mass of the dropping block, for a given glider mass, the larger the acceleration of the

glider/block system. What is the limiting value of the magnitude of the acceleration that can be achieved in this way? For example, what acceleration would you expect if you used your textbook as the dropping block? How about tying your lab partner to the end of the (unbreakable) string and dropping him or her? If your initial response was that unlimited accelerations could be achieved, think again! No calculations are necessary.

Submission: Completed electronic lab report by e-mail (spreadsheet)

Experiment 6: Centripetal Force

38


39

Name: ____________________________________ Score: __________of 10

Pre-Lab Assignment for Experiment 6: Dealing with Random Errors

A. In experiment 6, the centripetal force is calculated from the mass m, the radius r and the period T as 2

2

4c

mrF

T. Write the error equation for the relative error for Fc.

B. Study this: In a certain measurement run, the period T has been measured 3 times using a photo gate,

see table. The systematic uncertainty of each time value is 1% of the time value, due to systematic variations in the quality of the photo gates. We will need the average period with its respective uncertainty in this experiment. However, the uncertainty of the period is not only determined by the systematic error of the photo gate, but also by random factors during the execution of each single experiment. Hence, the uncertainty of the average is larger than the individual uncertainties of the measurement values. In such a measurement, the uncertainty of the average time is calculated as

2

1, 1,

1 1i i i

i N i N

T T T T TN N

The first term is the average of the uncertainties of the individual measurements; the second term is the standard deviation of the measurement values.

C. Apply this step by step by filling in the white fields in the table below in order to find average time and its

uncertainty.

Time Ti (s) Uncertainty iT (s)

(1% of Ti)

2

iT T

Measurement 1

0.695

Measurement 2

0.689

Measurement 3

0.692

Average 3

1

1

3i

i

T T

1,3

1

3i i

i

T T 2

1,

1i

i N

T TN

Standard deviation

2

1,

1i

i N

T TN

Uncertainty of average

iT T

In the spreadsheet, such tedious calculations are much simpler. Average and standard deviation can be programmed as

i iT AVERAGE T STDEV T


40


41

Laboratory Experiment #6: Centripetal Force

Objective: To study how the centripetal force required to make an object undergo uniform circular motion varies with the mass and speed of the object as well as with the radius of its circular path.

Introduction:

When an object moves under the influence of a net force that acts along the direction of motion, its speed either increases or decreases. If the net force always acts at right angles to the direction of motion, it changes the direction of motion of the object while leaving its speed unchanged. In both cases the objects velocity vector is changing so the object is being accelerated. An object moving in a circular path at a constant speed is said to be undergoing uniform circular motion. As the object moves around the circular path, its direction changes continuously. Thus a net force whose direction is continuously changing must be acting on it. The direction of this net force must be radial since the velocity is tangential to the circular path and the net force must be perpendicular to the velocity if the speed is to remain constant. A more careful analysis shows that the net force Fc must be directed toward the center of

the circle and must have a magnitude

Fcmv2

r

where m is the mass of the object, v is its speed, and r is the radius of the circle. A net force that always points toward the center of a circle is called a centripetal force. If the object takes time T to complete one revolution, its speed v is

v2 r

T

so

The experiment makes use of a centripetal-force apparatus in which a metal bob hanging is rotated in a horizontal circle. A sketch of the apparatus with the bob rotating is shown in Figure 6.1(a).

(a) (b) Figure 6.1: The centripetal force apparatus.

The metal bob of mass m hangs from a rod attached to a rotating axle. The bob is connected to the axle by a horizontal spring. As the axle is spun by hand, the metal bob moves faster and faster, stretching the spring as more and more horizontal force is needed to keep the bob moving in a circular path. The radius of the circular path can be determined by moving a pointer horizontally until its tip is directly below the tip of the metal bob. In actuality, the pointer is positioned first at some desired radius then the metal bob is spun till it passes directly over the tip of the pointer. The person spinning the axle must watch the alignment between

Fcm 2 r / T

2

r

4 2mr

T2 . (Eq.1)

pointer

m axle

cross arm

bob


42

the metal bob and the pointer and continuously adjust the spinning rate, to compensate for slowing due to the small amount of friction in the axle bearing. A photogate timer is used to time the rotational period T of the metal bob while is it being spun at the desired radius. The radius r is determined by measuring the distance from the pointer to the axle, then adding half the diameter of the axle. The radius may be adjusted by moving the pointer in or out. The metal bob can be taken off the apparatus to measure its mass m. Additional slotted masses can be piled on top of the metal bob to adjust m. The horizontal force exerted by the spring can also be determined when the metal bob is not rotating. A second mass M is suspended from a string that runs over a pulley then horizontally to the side of the metal bob opposite the spring (shown in Figure 6.1 (b)). The horizontal force exerted by the spring Fs must be

sufficient to support the weight of the hanging mass, so Fs Mg. (Eq. 2)

The hanging mass M can be increased till the spring is extended so that the metal bob lines up with the pointer. At this point the force exerted by the spring can be determined using Equation 2. When the metal bob is rotating the only horizontal force acting on it is that of the spring, so in this case the centripetal force required for circular motion is due to the spring. Thus Fc Fs in this situation, so

Mg4 2mr

T2 . (Eq. 3)

Notice that increasing M stretches the spring, thereby increasing r, but is completely independent of changes in m. Similarly, m can be increased without affecting r (or M). Thus when M (and hence r) are fixed, the spring force is a constant. Experimentally this means that changes in m must result in corresponding

changes in T to keep the right hand side of Equation 3 (more specifically, the ratio m /T2) a constant. On the

other hand, m /T2 is not fixed when r changes. This can be seen because r is not directly proportional to M.

As M approaches zero, r approaches the relaxed length of the spring, not zero. If the metal bob mass m is held constant and the hanging mass M is varied, the radius r and the period T will both change in such a way

to ensure that the ratio r / T2

is directly proportional to M.

Equipment:

Centripetal force apparatus, weight hanger, weights including 500g sizes, ruler or meter stick, photogate timer (t=1%), 10 kg capacity (m=1g) digital balance, and a piece of 3x5 card.

Procedure:

Place the centripetal force apparatus on the lab table so that the hanging mass M will clear the table edge when it hangs down from the pulley. Adjust the apparatus' legs as necessary to ensure that the cross arm rotates in a level horizontal plane. Step 1: Set the pointer location at its minimum distance from the axle. Disconnect the spring from the metal

bob and move the cross arm as necessary so that the freely hanging bob is lined up with the pointer. Attach a strip of 3x5 cardstock horizontally to the top of the mass, extending out away from the spring (see Figure 6.1(a)). Place the photo gate timer so the strip will trip the timer when rotating at this radius. Select the appropriate timing mode to measure the period of rotation T. Set the photo gate to 1 ms resolution.

Step 2: Remove the metal bob from the apparatus and measure its mass. Also measure the radius r of the

circular path of the metal bob by measuring the distance from the center of the pointer to the center of the axle.


43

Step 3: Reattach the bob to the apparatus. Attach the mass hanger and string to the metal bob using the hook provided. Add mass to the hanger until the metal bob lines up with the pointer. Record the value of M.

Step 4: Changing mass: Remove the mass hanger, string, and hook from the bob. Twist the axle to start

the metal bob moving. Increase the rotational rate until the tip of the metal bob passes directly over the pointer on every pass. Practice keeping the metal bob going at this rate for many revolutions. Do not allow the rotating mass to strike the photo gate timer! Take 10 good measurements of the period, clearing the photogate timer between measurements. (It helps to have the memory function ON so that the timer does not restart on its own the next time the bob comes around.) The measurements will not be during consecutive rotations because you need to write down each period before taking the next.

Step 5: Change m by adding a 100-gram slotted mass securely to the top of the metal bob using the

knurled nut on top of the bob to hold the added mass in place. (Do you need to measure M again?) Measure the period 10 more times for this mass and radius combination.

Step 6: Changing Radius: Leaving the amount of the mass on the bob the same, move the pointer to a new

location approximately 3/4 of the way out from the axle and determine the value of M needed to stretch the spring that far. Be sure to adjust the cross arm when you move the pointer so that the metal bob is lined up with the pointer when hanging freely. Measure the value of the radius corresponding to this new pointer location. Measure the period 10 more times for this mass and radius combination.

Step 7: Calculate the average period T for each of the three data sets. Think about what uncertainty T to

associate with these average Ts. Beware! The variability in the periods is likely to exceed the scale-limit of the photogate timer. Calculate the standard deviation of each set of periods to quantify the reproducibility-uncertainty in this experiment.

Step 8: Use Equation 1 to determine the centripetal force, complete with uncertainty, for each data set. Step 9: Use Equation 2 to determine the force exerted by the spring, complete with uncertainty, for each

data set.

Questions:

Do your data support the proposal that the centripetal force is independent of mass when the radius is held fixed? Explain.

Submission:

Electronic spreadsheet.

Experiment 7: Collisions in one Dimension

44


45

Name: ____________________________________ Score: __________of 10


A. Write down the error equation for the momentum L

p mt

and the kinetic energy

21

2

LKE m

t.

B. For one measurement of an inelastic collision, an incident glider is colliding with a resting glider. The momenta of all glicers before and after the values for the momentum of two gliders before and after the collision are:

initial mom pi (kg m/s) pi (kg m/s) final mom pf (kg m/s) pf (kg m/s)

incident glider

0.0873 0.002 0.0316 0.0007

stationary glider

0.000 0.002 0.0466 0.0004

Total momentum

pf pi

Find the total momentum of both gliders before and after the collision, together with their uncertainties. Find the difference in momentum before and after, together with the uncertainty of that value. You will need the Addition/Subtraction rule for uncertainty propagation.

C. Why is there an uncertainty associated with the initial speed of the resting glider? Explain.

D. The expected value for the difference in momentum before and after the collision is 0. Was this experiment successful? Explain.


46


47

Laboratory Experiment #7: Collisions in One Dimension

Objectives:

To observe, qualitatively and quantitatively, the behavior of particles when they collide and to see whether momentum and kinetic energy are conserved in such collisions.

Introduction:

When two or more particles collide, there is a transferring of momentum between them. It is an observed fact that the total momentum of the particles is the same after the collision as before. Momentum (p) is defined as the mass of the object multiplied by its velocity:

p = mv.

The conservation of momentum is then written

p f pi where the summation symbol indicates that the momenta of all particles are to be added. The subscripts "f" and "i" denote final and initial values, respectively. For a collision involving only two particles,

p1f + p2f = p1i + p2i.

If, furthermore, the motion is restricted to one dimension, this becomes

p1f + p2f = p1i + p2i

or

m1v1f + m2v2f = m1v1i + m2v2i.

This conservation of momentum will hold true as long as there is no external force acting to change the velocities (such as gravity). Kinetic energy, on the other hand, is not always conserved in a collision. When it is conserved, the collision is called "elastic". Then

K f K i

or

m12

v1f2 m2

2v2 f

2 m12

v1i2 m2

2v2i

2.

When the kinetic energy is not conserved, the collision is "inelastic". In an inelastic collision, the kinetic energy will decrease as a result of the collision (unless an explosion occurs during the collision). The lost kinetic energy is converted into heat, sound, or some other form of energy. The air track is a very convenient device for studying collisions in one dimension. If the track is properly leveled, the downward force of gravity on the glider is completely negated by the normal force that the track exerts on the glider. Then gravity will not influence the velocities of the gliders and momentum will be conserved. Elastic collisions will result if two gliders are sent toward one another so that their contact is via the band springs that are mounted on one end of each glider. (During a collision, the springs will temporarily store some energy in potential form but, because they are conservative in nature, this energy will be returned to


48

kinetic form as the gliders separate. Thus kinetic energy is conserved and the collision is elastic.) On the other hand , if the two gliders somehow stick together after the collision, it will be inelastic. Such a collision can be achieved by using the needle and socket ends of the two gliders--in a collision, the socket will grip the needle, holding the two gliders together.

Equipment:

Air track, two gliders with flags, additional glider-masses, appropriate bumpers for elastic and inelastic collisions, two electronic timers with photo gates (t=1%), Vernier caliper, and a 1200g (m=.1g) digital balance.

Procedure:

Carefully level the air track using a glider with no attachments on it (no flag, no bumpers). If the track is not perfectly straight, try to characterize where it has dips and/or peaks so that you can do the experiment in such as way as to minimize the effect of such imperfections. Load the gliders symmetrically and adjust the airflow as necessary, to make sure that they never scrape on the air track. Step 1: Qualitatively study elastic collisions, in which the gliders bounce apart after the collision. Record your

observations on your data sheet. Limit your studies to collisions where a moving glider (call it glider A) strikes a stationary glider (call it glider B). Look for trends in the collision outcome as the mass of glider A is adjusted with respect to that of glider B. (It should be sufficient to check (1) a light glider strikes a light glider, (2) a light glider strikes a heavy glider, (3) a heavy glider strikes a light glider, and (4) a heavy glider strikes a heavy glider. To change a glider from "light" to "heavy", add two 50-g masses to it, one on each side.) Your report should include a paragraph summarizing your observations. Be sure to answer (at least) the following questions.

(1) Under what conditions (if any) does glider A continue forward/bounce backward/not move after

the collision? (2) Under what conditions (if any) does glider B move forward/move backward/not move after the

collision? (3) Under what conditions (if any) is the speed of glider B after the collision greater than/less

than/equal to the speed of glider A before the collision? Step 2: Qualitatively study totally inelastic collisions, in which the gliders stick together after the collision.

Record your observations on your data sheet. Limit your studies to collisions where one glider is initially at rest and label the gliders as above. Again, let the relative masses of the gliders be the primary adjustable parameter. Be sure to answer (at least) the following questions in your summary-of-observations paragraph.

(1) Under what conditions (if any) do the two gliders (now locked together) move forward/move

backward/not move after the collision? (2) Under what conditions (if any) is the speed of the gliders after the collision greater than/equal

to/less than half the speed of the incident glider? Step 3: Now quantitatively study elastic collisions and inelastic collisions when a light incident glider strikes a

heavy stationary glider. Attach the elastic collision attachment on one end and the inelastic attachment to the other end of each glider. Measure the length of the glider flags using the Vernier caliper. Set the photo gate resolutions to 1 ms.

Perform two good trials of each type of collision. For each trial, determine the initial and final

velocities of both gliders by measuring the time required for each glider to traverse a pre-determined distance (i.e. by measuring the time it takes for a glider flag to pass completely through a photogate). No more than three different velocities will ever need to be determined, since the heavy glider always begins at rest. Each trial is independent because the initial push given to the incident glider is


49

different every time. Thus it is not possible to compare different trials as a check of the accuracy of the measurements. Good technique is vital! Estimate your uncertainties thoughtfully! Read the comments below.

In each trial for each type of collision, compute the momentum, p

mL

t, of each glider, both before

and after the collision. Remember that p's are vectors, so sometimes they can be negative. Also

compute the change of the total momentum: p pf pi . Next, compute the kinetic energy,

Km

2

L

t

2

, of each glider before and after the collision, as well as the change in total kinetic

energy: K K f Ki . Remember that K's should never be negative. Dont forget to compute

the uncertainties of each of these quantities. Remember that some (but not all!) of the final differences are expected to equal zero (within uncertainty).

Comments on the timing systems:

If you are using two independent photogate timer systems: In some cases, you will need to take two measurements with one of the timer systems. You may

either do this by quickly reading and resetting the timer in the middle of the trial or by making use of the memory function of the timer system. If you use the latter approach, the time in the memory is actually the sum of the two times. To retrieve the time of the first event from memory, first note the time of the second event on the display, then press the toggle switch to see the value in memory. To get the time of the first event, subtract the time of the second event from the displayed value.

If you are using one timer system with two photo gates connected to it: Locate the photo gates so that only one photo gate is making a measurement at any one time. The

timer unit can only store 2 timing values simultaneously (using the memory function). Thus it will be necessary to reset the timer in the middle of any run requiring 3 timing measurements (after noting the initial time value(s), of course). Many practice trials may be necessary to arrange the photo gates so that the gliders pass through them in the correct order.

When two events occur in succession without the timer being reset, the time in the memory is actually the sum of the two times. To retrieve the time of the first event from memory, first note the time of the second event on the display, then press the toggle switch to see the value in memory. To get the time of the first event, subtract the time of the second event from the displayed value.

Submission: Electronic spreadsheet

Experiment 8: Ballistic Pendulum

50


51

Name: ____________________________________ Score: __________of 10

Pre-Lab Assignment for Experiment 8: Logarithm Rule

A. Using the spring-loaded mechanism of the ballistic pendulum in order to propel a metal ball in a horizontal projectile motion off the table, the initial velocity, v, of the projectile can be found from measurements of the vertical fall distance, y, and the horizontal range, x, using

v xg

2y.

Use the logarithm rule to find the error equation for the speed v. B. If x = 2.81 .03 m, g = 9.806 .0005 m/s

2, and y = 1.02 .01 m, find velocity v and its relative error

as v v/v.

C. For the ballistic pendulum, the initial speed of the projectile is calculated as vom M

m2gh .

Use the logarithm rule to derive the equation for the relative error vo

vo=

M

m M+

M m

m(m M)+

1

2

h

h+

1

2

g

g


52


53

Laboratory Experiment #8: Ballistic Pendulum

Objective:

To determine the velocity of a projectile using a ballistic pendulum and the laws of conservation of energy and momentum.

Introduction:

In this experiment a metal ball of known mass m is shot horizontally using a spring gun. When the ball is caught by the hollow bob of a pendulum, the pendulum and the ball swing together along the arc of a vertical circle. The velocity of projection can be determined from the height of the swing and the laws of conservation of energy and momentum.

The apparatus, shown in Figure 8.1, consists of a light pendulum rod that has a hollowed-out bob at its lower end and is suspended at its upper end from a horizontal suspension rod. It swings freely in a vertical plane about the suspension rod axis. The hollow interior of the bob has a rubber o-ring which allows the ball to enter but prevents it from coming out once inside; it thereby is able to catch the ball. The metal ball (with a hole along its diameter) is placed on the horizontal rod of the spring gun. By pushing on the ball with the palm of your hand, the spring can be compressed and locked. A squeeze of the trigger shoots the ball horizontally so that it hits the freely hanging pendulum bob where it is captured. The pendulum with the embedded ball swings forward and upward until it stops at the highest point of its swing. A ratcheting lever attached to the bottom of the bob engages the curved rack and holds the pendulum at that position . (The rack has a series of horizontal ridges that are spaced about 1 mm apart. By engaging the notch nearest to the highest position of the bob, the lever prevents the pendulum from swinging back.) An index attached to the pendulum bob marks the center of gravity of the loaded pendulum.

Figure 8.1: Ballistic Pendulum

Method 1:

To determine the initial velocity of the ball, measure the height h through which the center of mass of the loaded pendulum rises as the ball collides with the bob at speed vo. The combined pendulum and ball move with a speed v just after the collision. Let m be the mass of the ball and M be the mass of the pendulum. Then using the principle of momentum conservation

m vo = (m + M ) v . (1)


54

The speed v can be related to the height using the principle of conservation of energy. At the start of the

swing the energy is completely kinetic ((m+M)v2/2) and at the end it is completely potential ((m+M)gh). Hence we have

Ei = Ef

(m+M) v2 / 2 = (m+M)gh Then

v 2gh (2)

Combining equations (1) and (2) one finds

vom M

m2gh (3)

The corresponding error equation is given by

vo

vo=

M

m M+

M m

m(m M)+

1

2

h

h+

1

2

g

g (4)

Method 2:

The velocity vo can also be determined from the range of the ball. With the pendulum removed, the ball is

fired so that it hits the floor. The distance x along the horizontal from the gun to the point on the floor as well as the vertical distance y directly below the gun to the floor are measured. See Figure 8.2. If t is the time of flight of the ball as it travels from the gun to the floor ,

x = vot

and

ygt2

2.

By eliminating t from these two equations one obtains the trajectory:

vo xg

2y (5)

A knowledge of x and y thus enables us to determine vo.

Equipment:

Ballistic pendulum apparatus, 150 g (m=.01g) digital mass balance, meter stick, ruler, paper and a thin piece of wood or cork.


55

Procedure:

Check to see that the pendulum hangs freely and can oscillate without friction. Load the gun and then release the trigger to ensure that the rubber 0-ring operates effectively as a catch and that the lever engages with the notched rack. Method 1: Step 1: Measure the mass of the ball, m. The mass of the pendulum, M, is noted on the apparatus. Step 2: Make sure the pendulum hangs freely. Determine the height h1 of the center of mass from the base

of the apparatus using a ruler. Step 3: Place the metal ball on the rod of the spring gun and by pushing on it, compress and lock the

spring. Pull the trigger to release the ball. Step 4: Note the notch number on the rack that the lever engaged. Step 5: Lift the lever out of the notch and move the pendulum back to the vertical position. Remove the ball

carefully from the pendulum bob and place it back on the spring gun. With the pendulum stationary in the vertical position, fire the gun again and note the notch number.

Step 6: Repeat the above process for a total of ten trials. (Repeat any unsuccessful trials.) Step 7: Determine the average value of the notch numbers. Round it upward to the next integer.

(Understand why.) Move the pendulum up the rack until the lever is in the notch corresponding to that rounded average value. With the pendulum in this position determine the height h2 of the center of mass from the base of the apparatus. The difference h2 - h1 equals the height, h. Estimate h from the highest and lowest notch values and your own judgment. A second contribution to h arises from the spread in notch values. Calculate this using the standard deviation. The total h is the sum of these two contributions.

Step 8: Determine vo from equation (3) and the error vo from equation (4). Method 2: Step 1: Prepare to shoot the ball onto the floor. Place the pendulum high on the rack so that it is out of the

path of the ball. Align the base of the apparatus with a corner of the table in a way that can be easily and precisely reproduced before each firing.

Step 2: Locate the place on the floor where the ball lands by shooting the gun a couple of times. Tape a

sheet of paper (with a thin wood or cork board underneath) to the floor at this location. See Figure 2. One member of the group could stand behind the landing zone with a coat to stop the ball.

Step 3: Shoot the ball ten times. Number the marks on the paper from 1 to 10. Check the location of the

base before each shot. Step 4: Determine the values of the horizontal distances from the point where the ball leaves the gun to the

marks on the paper. Call them x1 x1, x2 x2, x3 x3, etc. Find the average range x

x .

Step 5: Determine the height above the floor of the point where the ball leaves the gun. In estimating

uncertainty, consider the levelness of the floor. Step 6: Use equation ( 5 ) to obtain vo vo. (Hand in the target piece of paper with your report.)


56

Figure 8.2: Set-up for Method 2 .

Comments:

Compare the values of vo as obtained in method 1 and method 2. Discuss whether they agree within their

uncertainties. What is the percentage difference between them? What sources of error were not taken into account by the uncertainties?

The acceleration of gravity in Morris, MN is 9.8057 .0005 m/s2.

Question:

Explain why the average notch number in Step 7 of Method 1 should be rounded up to the next integer. Submission:

electronic spreadsheet

Experiment 9: Static Equilibrium

57

Name: ____________________________________ Score: __________of 10


The torque balance for the calculation of an unknown mass leads us to the expression 1

2 1

2

pp

pp

x xM m

x x,

in which all variables are measured quantities. Show that the correct error equation for M2 is:

M 2

M 2

m1

m1

x1

xpp x1

x2

x2 xpp

x2 x1

xpp x1 x2 xpp

xpp

You can either use the basic equation for uncertainty propagation or the logarithm rule. In both cases, much care is needed to place the absolute-value brackets into the correct place.


58


59

Laboratory Experiment #9: Static Equilibrium

Objective:

To study the conditions of static equilibrium.

Introduction:

Newton's third law states:

Fnet = F = ma (1)

where F is an external force acting on the system. Note that this vector equation can be rewritten as three separate scalar equations, one along each of the x, y and z directions:

(Fnet )x = Fx = m a x (Fnet)y = Fy = m a y (2)

(Fnet )z = Fz = m a z

For a system of particles, equation (1) implies that if the center of mass of the system is to remain stationary (i.e. v = a = 0), then

F 0

or

Fx Fy Fz 0 .

We could imagine a situation in which the center of mass of a system of particles is stationary while many or all of the particles comprising the body are not (as in the case of a rigid body rotating about its center of mass. There must be a different "force-like"

quantity acting here. This quantity which causes a rigid body to change its rotational motion is called a torque, . A torque with re

Documents

Laboratory Manual