SPH4C COLLEGE PHYSICS - Google Search

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SPH4CCOLLEGE PHYSICS

REVIEW: MATH SKILLS

L Scientific Notation

(P.547)

Scientific Notation

In science we frequently encounter numbers which are difficult to write inthe traditional way - velocity of light, mass of an electron, distance to thenearest star. Scientific notation, or standard notation, is a technique,using powers of ten, for concisely writing unusually large or small numbers.

August 15, 2013 14CR - Scientific Notation

Expression Common decimal notation Scientific notation

124.5 millionkilometres

124 500 000 km 1.245 x 108 km

154 thousand picometres

154 000 pm 1.54 x 105 pm

602 sextillionmolecules

602 000 000 000 000 000000 000 molecules

6.02 x 1023

molecules

Scientific Notation

SCIENTIFIC NOTATION

� uses powers of ten to write large/small numbers


Expression Common decimal notation Scientific notation

124.5 millionkilometres

124 500 000 km 1.245 x 108 km

154 thousand picometres

154 000 pm 1.54 x 105 pm

602 sextillionmolecules

602 000 000 000 000 000000 000 molecules

6.02 x 1023

molecules

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Scientific Notation

In scientific notation, the number is expressed by:

1. writing the correct number of significant digits with one non-zero digitto the left of the decimal point, and then

2. multiplying the number by the appropriate power (+ or -) of ten (10).


Scientific Notation

For example, 2 394 0.067

= 2.394 x 1000 = 6.7 x 0.01

= 2.394 x 10 3 = 6.7 x 10 -2

NOTE!

Scientific notation also enables us to show the correct number of significantdigits. As such, it may be necessary to use scientific notation in order tofollow the rules for certainty (discussed later).


Scientific Notation

PRACTICE

1. Express each of the following in scientific notation.

(a) 6 807

(b) 0.000 053

(c) 39 879 280 000

(d) 0.000 000 813

(e) 0.070 40

(f) 400 000 000 000

(g) 0.80

(h) 68


6.807 x 103

5.3 x 10-5

3.987928 x 1010

8.13 x 10-7

7.040 x 10-2

4 x 1011

8.0 x 10-1

6.8 x 101

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Scientific Notation

PRACTICE

2. Express each of the following in common notation.

(a) 7 × 101

(b) 5.2 × 103

(c) 8.3 × 109

(d) 10.1 × 10-2

(e) 6.386 8 × 103

(f) 4.086 × 10-3

(g) 6.3 × 102

(h) 35.0 × 10-3


70

5 200

8 300 000 000

0.101

6 386.8

0.004 086

630

0.035 0

Scientific Notation With Calculators

On many calculators, scientific notation in entered using a special key,labelled EXP or EE. This key includes “x 10” from the scientific notation;you need to enter only the exponent. For example, to enter

7.5 x 10 4 press 7.5 EXP 4

3.6 x 10 -3 press 3.6 EXP +/- 3

NOTE!

Depending on the type of calculator you have, the “+/-” signs may need tobe entered after the relevant number.


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REVIEW: MATH SKILLS

L International System of Units (SI)

(P.572)

SI

Over hundreds of years, physicists (and otherscientists) have developed traditional ways (orrules) of expressing their measurements. If wecan’t trust the measurements, we can put nofaith in reports of scientific research. As such,the International System of Units (SI) is usedfor scientific work throughout the world –everyone accepts and uses the same rules, andunderstands that there are limitations to therules.

August 15, 2013 14CR - SI

SI

SI RULES

• In the SI system all physical quantities canbe expressed as some combination offundamental units, called base units. (i.e.,mol, m, kg, EC, s, ...). For example:

1 N = 1 kg@m/s 2 7 unit for force

1 J = 1 kg@m 2/s 2 7 unit for energy


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SI

SI RULES

• The SI convention includes both quantityand unit symbols. Note that these aresymbols (e.g., 60 km/h) and are notabbreviations (e.g., 40 mi./hr.).


SI

SI RULES

• When converting units the method mostcommonly used is multiplying by conversionfactors (equalities), which are memorized orreferenced (e.g., 1 m = 100 cm, 1 h = 60min = 3600 s).


SI

SI RULES

• It is also important to pay close attention tothe units, which are converted bymultiplying by a conversion factor (e.g., 1m/s = 3.6 km/h).


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SI

USEFUL CONVERSION FACTORS!

HHHH HHHH HHHH HHHH HHHH HHHH HHHH

G 1000 M 1000 k 1000 base 100 c 10 m 1000 : 1000 0

)))) )))) )))) )))) )))) )))) ))))

HHHH HHHH HHHH HHHH HHHH

m/s 3.6 km/h yr 365 d 24 hr 60 min 60 sec)))) )))) )))) )))) ))))


unit

SI

PRACTICE

1. Use the chart to convert each of thefollowing measurements to theirbase unit.

(a) 5.7 GW

(b) 72 cm

(c) 6 µC

(d) 0.50 MJ

(e) 6.8 mL

(f) 548 ηm

(g) 0.75 kg


Power Prefix Symbol

109 giga G

106 mega M

103 kilo k

100 ----- -----

10-2 centi c

10-3 milli m

10-6 micro :

10-9 nano 0

NOTE!

This is only a partial list - refer to P.661 for a complete list.

5.7 x 109 W

72 x 10-2 m

6 x 10-6 C

0.50 x 106 J

6.8 x 10-3 L

548 x 10-9 m

0.75 x 103 g

SI

PRACTICE

2. An athlete completed a 5-kmrace in 19.5 min. Convert thistime into hours.

19.5 min x

= 0.325 hours

3. A train is travelling at 95 km/h.Convert 95 km/h into metresper second (m/s).

95 km/h x x

= 26.4 m/s

August 15, 2013 4CR - SI 8

1 hour

60 min

1000 m

1 km

1 hour

3600 s

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REVIEW: MATH SKILLS

L Uncertainty in Measurements

(P. 546)

Uncertainty in Measurements

There are two types of quantities used inscience: exact values and measurements. Exactvalues include defined quantities (1 m = 100cm) and counted values (5 beakers or 10 trials).Measurements, however, are not exactbecause there is always some uncertainty orerror associated with every measurement. Assuch, there is an international agreement aboutthe correct way to record measurements.

August 15, 2013 14CR - Uncertainty in Measurements

Significant Digits

The certainty of any measurement is communicated by the number ofsignificant digits in the measurement. In a measured or calculated value,significant digits are the digits that are known reliably, or for certain,and include the last digit that is estimated or uncertain. As such, there area set of rules that can be used to determine whether or not a digit issignificant (refer to P.650 of your text).

SIGNIFICANT DIGITS

� digits that are certain plus one estimated digit

� indicates the certainty of a measurement

� rules L P.650


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Significant Digits

WHEN DIGITS ARE SIGNIFICANT ✔✔✔✔

1. All non-zero digits (i.e., 1-9) are significant.

For example: 259.69 has five significant digits

61.2 has three significant digits


Significant Digits


2. Any zeros between two non-zero digits are significant.

For example: 606 has three significant digits

6006 has four significant digits


Significant Digits


3. Any zeros to the right of both the decimal point and a non-zero digitare significant.

For example: 7.100 has four significant digits

7.10 has three significant digits


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Significant Digits


4. All digits (zero or non-zero) used in scientific notation are significant.

For example: 3.4 x 10 3 has two significant digits

3.400 x 10 3 has four significant digits


Significant Digits


5. All counted and defined values have an infinite number of significantdigits.

For example: 16 students has 4 significant digits

B = 3.1415... has 4 significant digits


Significant Digits

WHEN DIGITS ARE NOT SIGNIFICANT✘

1. If a decimal point is present, zeros to the left of other digits (i.e.,leading zeros) are not significant – they are placeholders.

For example: 0.22 has two significant digits

0.000 22 has two significant digits


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Significant Digits

WHEN DIGITS ARE NOT SIGNIFICANT✘

2. If a decimal point is not present, zeros to the right of the last non-zerodigit (i.e., trailing zeros) are not significant – they are placeholders.

For example: 98 000 000 has two significant digits

25 000 has two significant digits

NOTE!

In most cases, the values you will be working with in this course will havetwo or three significant digits.


Significant Digits

PRACTICE

1. How many significant digits are there in each of the followingmeasured quantities?

(a) 353 g

(b) 9.663 L

(c) 76 600 000 g

(d) 30.405 ml

(e) 0.3 MW

(f) 0.000 067 s

(g) 10.00 m

(h) 47.2 m

(i) 2.7 x 105 s

(j) 3.400 x 10-2 m


3

4

3

5

1

2

4

3

2

4

Significant Digits

PRACTICE

2. Express the following measured quantities in scientific notation withthe correct number of significant digits.

(a) 865.7 cm

(b) 35 000 s

(c) 0.05 kg

(d) 40.070 nm

(e) 0.000 060 ns


(4) 8.657 x 102 cm

(2) 3.5 x 104 s

(1) 5 x 10-2 kg

(5) 4.0070 x 101 nm

(2) 6.0 x 10-5 ns

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Precision

Measurements depend on the precision of the measuring instruments used,that is, the amount of information that the instruments can provide. Forexample, 2.861 cm is more precise than 2.86 cm because the three decimalplaces in 2.861 makes it precise to the nearest one-thousandth of acentimetre, while the two decimal places in 2.86 makes it precise only tothe nearest one-hundredth of a centimetre. Precision is indicated by thenumber of decimal places in a measured or calculated value.

PRECISION

� indicated by the number of decimal places in the number


Precision

RULES FOR PRECISION

1. All measured quantities are expressed as precisely as possible. Alldigits shown are significant with any error or uncertainty in the lastdigit.

For example, in the measurement 87.64 cm the uncertainty is with thedigit 4.


Precision

RULES FOR PRECISION

2. The precision of a measuring instrument depends on its degree offineness and the size of the unit being used.

For example, a ruler calibrated in millimetres (ruler #2) is more precisethan a ruler calibrated in centimetres (ruler #1) because the rulercalibrated in millimetres has more graduations.

#1

#2


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Precision

RULES FOR PRECISION

3. Any measurement that falls between the smallest divisions on themeasuring instrument is an estimate. We should always try to readany instrument by estimating tenths of the smallest division.

For example, with ruler #1 we would estimate to the nearest tenth of acentimetre (i.e. 3.2 cm); with ruler #2 we would estimate to thenearest tenth of a millimeter (i.e. 3.24 cm).

#1

#2


Precision

RULES FOR PRECISION

4. The estimated digit is always shown when recording the measurement.

For example, the 7 in the measurement 6.7 cm would be the estimateddigit.


Precision

RULES FOR PRECISION

5. Should the object fall right on a division mark, the estimated digitwould be 0.

For example, if we use a ruler calibrated in centimetres to measure alength that falls exactly on the 5 cm mark, the correct reading is 5.0cm, not 5 cm.


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Precision

PRACTICE

3. Use the two centimetre rulers to measure and record the length of thepen graphic.

(a) Child’s ruler

~ 4.9 cm

(b) Ordinary ruler

~ 4.92 cm


Precision

PRACTICE

4. An object is being measured with a ruler calibrated in millimetres. Oneend of the object is at the zero mark of the ruler. The other end linesup exactly with the 5.2 cm mark. What reading should be recorded forthe length of the object? Why?

5.20 cm should be recorded since the object falls right on division.Since the ruler is calibrated in millimetres, we need to estimate to thenearest tenth of the smallest division


Precision

PRACTICE

5. Which of the following values of a measured quantity is most precise?

(a) 4.81 mm, 0.81 mm, 48.1 mm, 0.081 mm

(b) 2.54 cm, 12.64 cm, 126 cm, 0.5400 cm, 0.304 cm

(a) 0.081 mm – has 3 decimal places

(b) 0.5400 cm – has 4 decimal places


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Uncertainty in Measurements

PRACTICE

6. Copy and complete the following table.


Measurement Precision# of Sig.Dig. Measurement

now needed rounded in sci.not.

a 63.479 km (example) 3 5 3 63.5 6.35 × 101

b 46 597.2 cm 2

c 0.5803 L 1

d 325 kg 2

e 0.067 80 mm 3

f 485.000 kW 43

5

0

4

1 6

4

3

4

6

4.7 x 104

6 x 10-1

3.2 x 102

6.78 x 10-2

4.850 x 102

47 000

0.6

320

0.0678

485.0

Accuracy & Significant Figures

Accuracy relates to how close a measurement agrees with the accepted

value. This is an indication of the quality of the measuring instrument and

the technique of the user. The difference between an observed value (or the

average of the observed values) and the accepted value is called the error.

The degree of accuracy of any instrument depends on two things: the

precision of the measuring instrument and the skill and care of the user.

The precision of a measuring instrument depends on the size of the unit

being used. It depends on the place value of the last digit obtained from a

measurement or calculation. For example, 2.861 is more precise than

581.86.

Certainty is determined by how many certain digits are obtained by the

measuring instrument.

Any measurement that falls between the smallest division on the measuring

instrument is an estimate and is therefore uncertain.

Example: Determine the length of each line with the appropriate number of

significant figures.

a)

b)

Significant figures record all digits that are certain, plus one uncertain digit.

To determine the number of significant figures in a measurement, count the

number of digits. All digits in a given measurement are significant except

for the leading zeros.

Example: State the number of significant figures for each of the following

measurements.

a) 32.58g b) 6.07cm c) 0.0025mL

d) 0.180L e) 4.148 x 103g f) 2.00 x 10-1m

CERTAINTY RULE FOR MULTIPLYING AND DIVIDING:

***The answer has the same number of significant figures as the

measurement with the fewest significant figures***

Example: State the answer with the appropriate number of significant

figures.

a) 9.3

8.21 b) )06.27)(80.9(

PRECISION RULE FOR ADDING AND SUBTRACTING:

***The answer has the same number of decimal places as the measured

value with the fewest decimal places***


figures.

a) 104.2km + 11km + 0.67km =

b) 5.5m + 0.597m + 0.1262m =

EXACT NUMBERS:

When you directly count the number of something, this is an exact value.

Objects that have set values, such as 100cm/m or 60s/min, are defined

values. Exact and defined values are said to contain an infinite number of

significant figures and therefore don’t affect the rules of certainty or

precision.


figures.

a) 2)2.16)(25(.2

1

b) 3)3.2(3

4

CONVERTING UNITS

The method most commonly used is multiplying by conversion factors,

which are either memorized or referenced.

Example: Perform the following conversions, stating your final answer with

the appropriate number of significant figures

a) 28.6 minutes to hours.

b) 18 minutes to seconds

c) 16 km/h to m/s.

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REVIEW: MATH SKILLS

L Calculations Using Measurements

(P.546-547)

Rounding

If measurements are approximate, thecalculations based on them must also beapproximate. Scientists agree that calculatedanswers should be rounded so they do not givea misleading idea of how precise the originalmeasurements were. Use these rules whenmaking calculations and rounding answers tocalculations.

August 15, 2013 14CR - Calculations Using Measurements

Rounding

RULES FOR ROUNDING

1. When the first digit to be dropped is 4 or less, the last digit retainedshould not be changed.

For example: 3.141 326 rounded to 4 digits is 3.141


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Rounding

RULES FOR ROUNDING

2. When the first digit to be dropped is greater than 5, or if it is a 5followed by at least one digit other than zero, the last digit retained isincreased by 1 unit.

For example: 2.221 372 rounded to five digits is 2.2214

4.168 501 rounded to four digits is 4.169


Rounding

RULES FOR ROUNDING

3. When the first digit discarded is five or a five followed by only zeros,the last digit retained is increased by 1 if it is odd, but not changed if itis even.

For example: 2.35 rounded to two digits is 2.4

2.45 rounded to two digits is 2.4

-6.35 rounded to two digits is -6.4

NOTE!

This is sometimes called the even-odd rule.


Adding & Subtracting

RULES FOR ADDING & SUBTRACTING

When adding and/or subtracting, the answer has the same number ofdecimal places as the measurement with the fewest decimal places.

For example: 6.6 cm + 18.74 cm + 0.766 cm

= 26.106 cm

= 26.1 cm

NOTE!

The answer must be rounded to 26.1 cm because the first measurement(6.6 cm) limits the precision to a tenth of a centimetre.


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Multiplying & Dividing

RULES FOR MULTIPLYING & DIVIDING

When multiplying and/or dividing, the answer has the same number ofsignificant digits as the measurement with the fewest number of significantdigits.

For example: 77.8 km/h x 0.8967 h

= 69.76326 km

= 69.8 km

NOTE!

The certainty of the answer is limited to three significant digits, so theanswer is rounded up to 69.8 km. The same applies to scientific notation.For example,

(5.5 x 10 4) )))) (5.675 x 10 -2) = 9.7 x 10 5


Multistep Calculations

RULES FOR MULTISTEP CALCULATIONS

For multistep calculations, round-off errors occur if you use the rounded-offanswer from an earlier calculation in a subsequent calculation. Thus, leaveall digits in your calculator until you have finished all your calculations andthen round the final answer.

For example: 5.21 x 0.45 )))) 0.00600

= 2.3445 )))) 0.00600 or = 2.3 )))) 0.00600

= 390.75 = 383.333333

= 390 U = 380 Y

NOTE!

The certainty of the answer is limited to two significant digits, so theanswer is rounded accordingly. In the second example though, roundingoccurred during the calculation which introduced a round-off error.


Calculations – Summary

ADDING & SUBTRACTING

� fewest decimal places

MULTIPLYING & DIVIDING

� fewest number of significant digits

MULTISTEP CALCULATIONS

� leave all digits in the calculator until finished and then round

NOTE!

If a combination of addition, subtraction, multiplication and division areinvolved, follow the rules for multiplying and dividing.


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Calculations Using Measurements

PRACTICE

1. Perform the following operations. Round your answers accordingly.

(a) 67.8 + 968 + 3.87

(b) 463.66 + 29.2 + 0.17

(c) 68.7 - 23.95

(d) (2.6)(42.2)

(e) (65)(0.041)(325)

(f) (0.0060)(26)(55.1)

(g) 650 ) 4.0

(h) 3.52

(i) (1.62 × 10-3)(7.3 × 10-1)

(j) (5.019 × 10-4)÷(3.1 × 10-7)


1039.67

493.03

44.75

109.72

866.125

8.5956

162.5

12.25

0.0011826

1619.0322...

= 1040

= 493.0

= 44.8

= 110

= 870

= 8.6

= 160

= 12

= 0.0012

= 1600


PRACTICE

2. Solve each of the following. Round your answers accordingly.

(a) Find the perimeter of a rectangular carpet that has a width andlength of 3.56 m and 4.5 m.

(a) 16.1 m



PRACTICE


(b) Find the area of a rectangle whose sides are 4.5 m and 7.5 m.

(b) 34 m2


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PRACTICE


(c) A triangle has a base of 5.75 cm and a height of 12.45 cm.Calculate the area of the triangle. (Recall A = ½bh)

(c) 35.8 cm2



PRACTICE


(d) On the planet Zot distances are measured in zaps and zings. If 3.9zings equal 7.5 zaps, how many zings are equal to 93.5 zaps?

(d) 49 zings



PRACTICE


(e) The Earth has a mass of 5.98 × 1024 kg while Jupiter has a mass of1.90 × 1027 kg. How many times larger is the mass of Jupiter thanthe mass of the Earth?

(e) 318 times


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REVIEW: MATH SKILLS

L Error in Measurements

(P.548)

Error in Measurements

Many people believe that all measurements arereliable (consistent over many trials),precise (to as many decimal places aspossible), and accurate (representing theactual value). But there are many things thatcan go wrong when measuring. For example:

August 15, 2013 14CR - Error in Measurements


• There may be limitations that make theinstrument or its use unreliable(inconsistent).

• The investigator may make a mistake orfail to follow the correct techniques whenreading the measurement to the availableprecision (number of decimal places).

• The instrument may be faulty orinaccurate; a similar instrument may givedifferent readings.


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PRACTICE

1. What three things can you do during an experiment to help eliminateerrors?

1. To be sure that you have measured correctly, you should repeatyour measurements at least three times.

2. If your measurements appear to be reliable, calculate the mean anduse that value.

3. To be more precise about the accuracy, repeat the measurementswith a different instrument.



PRACTICE

2. There are two types of measurement error. What are they?

random error and systematic error


Random Error

Random error results when an estimate is made to obtain the last digitfor any measurement. The size of the random error is determined by theprecision of the measuring instrument. For example, when measuringlength with a measuring tape, it is necessary to estimate between themarks on the measuring tape. If these marks are 1 cm apart, the randomerror will be greater and the precision will be less than if the marks are 1mm apart. Such errors can be reduced by taking the average of severalreadings.

RANDOM ERROR

� results when the last digit is estimated

� reduced by taking the average of several readings


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Systematic Error

Systematic error is associated with an inherent problem with themeasuring system, such as the presence of an interfering substance,incorrect calibration, or room conditions. For example, if a balance is notzeroed at the beginning, all measurements will have a systematic error;using a slightly worn metre stick will also introduce error. Such errors arereduced by adding or subtracting the known error or calibrating theinstrument.

SYSTEMATIC ERROR

� due to a problem with the measuring device

� reduced by adding/subtracting the error or calibrating the device


Accuracy & Precision

In everyday usage, "accuracy" and "precision" are used interchangeably todescribe how close a measurement is to a true value, but in science it isimportant to make a distinction between them. Accuracy refers to howclose a value is to its accepted value. Precision is the place value of thelast measureable digit.

ACCURACY

� how close a value is to its accepted value

PRECISION

� place value of last measureable digit


Accuracy & Precision

For example, the position of the darts in each of the figures are analogousto measured or calculated results in a laboratory setting. The results in (a)are precise and accurate, in (b) they are precise but not accurate, and in(c) they are neither precise nor accurate.


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Percentage Error

No matter how precise a measurement is, it still may not be accurate. Thepercentage error is the absolute value of the difference betweenexperimental and accepted values expressed as a percentage of theaccepted value.

NOTE!

The bars (||) in the equation above represent “absolute value”. Thismeans that, mathematically, if a = 3 and b = -3 then |a| = |b| = 3.


100 x valueaccepted

valueaccepted valuealexperiment error %

−=

Percentage Difference

Sometimes if two values of the same quantity are measured, it is useful tocompare the precision of these values by calculating the percentagedifference between them.

NOTE!

“Magnitude” is a term frequently used by physicists. The magnitude of aquantity is the same as its absolute value.


100 x

2

2t measuremen 1t measuremen

2t measuremen 1t measuremen difference %

+−

=


PRACTICE

3. At a certain location the acceleration due to gravity is 9.82 m/s2[down].Calculate the percentage error of the following experimental values of“g” at that location.

(a) 8.94 m/s2[down]

(b) 9.95 m/s2[down]

(a) 8.96

(b) 1.32


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PRACTICE

4. Calculate the percentage difference between the two experimentalvalues (8.94 m/s2 and 9.95 m/s2) used in question #3.

10.7


U Check Your Learning

WIKI (REVIEW)

O.... 4CR - WS1 (Math Skills)

O.... 4CR - QUIZ1 (Math Skills - Part 1)


8/15/2013

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REVIEW: MATH SKILLS

L Measuring & Estimating

(P.549)

Making Precise Measurements

In order to make precise measurements you need to use a device that hasa double scale. A double scale consists of a main scale that is an ordinarymetric scale with centimetres and millimetres and a sliding or vernierscale.

vernier scale L

main scale L

August 15, 2013 14CR - Measuring & Estimating


If you look at the diagram carefully you will see there are 10 graduationson the vernier scale that occupy the same space as 9 graduations on themain scale. Therefore, only one graduation on the vernier can line up witha graduation on the main scale.

vernier scale L

main scale L


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A double scale can be placed on various types of instruments. Onecommon instrument is the vernier caliper. It is used to measure theoutside diameter of a cylinder, the inside diameter of a hollow cylinder, orthe depth of a hole.



Another instrument with a double scale is the outside micrometercaliper.


Using a Vernier Caliper

Vernier calipers are precision measuring instruments used to makeaccurate measurements. The bar and movable jaw are graduated on bothsides, one side for taking outside measurements and the other side forinside measurements. Vernier calipers are available in metric and in inchgraduations, and some types have both scales. Digital models are alsoavailable.


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NOTE!

When using high-precision instruments, such as the vernier caliper oroutside micrometer caliper, it is necessary to check the zero setting beforetaking a reading. If, for example, the instrument is supposed to read 0.000cm but instead reads 0.002 cm, the error must be taken into considerationwith each reading.



HOW TO READ A METRIC VERNIER CALIPER

1. Find the first line (the ZERO line) on the vernier (sliding) scale. Lookon the main (stationary) scale and record the number you just passed(or are currently on) as #.# cm.

5.0 cm




2. Find the FIRST pair of lines that match up perfectly. Read the linenumber off the vernier (sliding) scale and add this to themeasurement.

5.08 cm


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3. Determine the error – half of the smallest measurement possible. Inour case the smallest measurement possible is 1 mm so the error is 0.5mm or 0.05 cm. Add this to your measurement.

5.08 """" 0.05 cm



PRACTICE

1. What is the reading (including the error) of the following metric verniercalipers?

(a)

0.69 """" 0.05 cm

(b)

3.18 """" 0.05 cm


0 4 0

0 1 2

2 6 8

0 4 0

3 4

2 6 8


PRACTICE


(c)

0.87 """" 0.05 cm

(d)

2.63 """" 0.05 cm


0 4 0

0 1 2

2 6 8

0 4 0

2 3 4

2 6 8

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PRACTICE


(e)

1.13 """" 0.05 cm

(f)

7.05 """" 0.05 cm


0 4 0

1 2

2 6 8

0 4 02 6 8

6 7 8

8/15/2013

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REVIEW: MATH SKILLS

L Trigonometry

(P.550-551)

Trigonometry

The first application of trigonometry was to solve right-angle triangles.Trigonometry derives from the fact that for similar triangles, the ratio ofcorresponding sides will be equal. For a given angle " in a right triangle,there are three important ratios: sine, cosine, and tangent. These arecalled the primary trigonometric ratios and they can be used to findthe measures of unknown sides and angles in right triangles.

August 15, 2013 14CR - Trigonometry

Trigonometry

NOTE!

DEG, RAD, and GRAD are different units/modes used for measuring angles.For this course make sure that your calculator is always in DEG mode.(Hint: if you are not getting the correct answers for a trigonometryproblem start by checking this!)


8/15/2013

2

Trigonometry

PRACTICE

1. Determine the value of each ratio rounded to four decimal places.

(a) sin 35°

(b) cos 60°

(c) tan 45°

(d) cos 75°

(e) sin 18°


0.5736

0.5000

1.0000

0.2588

0.3090

Trigonometry

RECALL!

If the value of a trigonometric ratio is known, its corresponding angle canbe found using the inverse of that ratio. For example, if cos 2 = 0.50 then2 = cos -1 (0.50).

• for sin -1 use:

• for cos -1 use:

• for tan -1 use:


2nd sin

2nd cos

2nd tan

Trigonometry

PRACTICE

2. Determine the size of ∠A rounded to the nearest degree.

(a) sin A = 0.5299

(b) cos A = 0.4226

(c) tan A = 4.3315

(d) cos A = 0.5000

(e) sin A = 0.2419


32E

65E

77E

60E

14E

8/15/2013

3

Trigonometry

PRACTICE

3. Solve for x rounded to one decimal place.

(a) sin 35° = x/8

(b) cos 70° = x/15

(c) tan 20° = 3/x

(d) sin 85° = 6/x

(e) cos 25° = 5/x


4.6

5.1

8.2

6.0

5.5

Trigonometry

PRACTICE

4. Use two different methods to find the value of the unknown(s) in eachtriangle. Round your answers to one decimal place.

X = 12.2 mm

Y = 54.1 cm

2 = 64.3 E


10 mm

X

35E

7 mm 260 cm

Y

26 cm

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