Department of Physicsand Astronomy

Physics 1050Lab Manual

Spring 2012

Contents

Lab Reports and Marks i

Sample Lab Outline iv

Sample Lab Report vii

1 Statistics 1

2 Free Fall 10

3 Projectile Motion 14

4 Vectors, Forces, and Equilibrium 18

5 Mechanics of the Human Arm 26

6 Angular Acceleration of a Disk 32

7 Principles of Mechanics 38

8 The Physics of Bicycles 42

9 The Speed of Sound 50

10 Nuclear Half-life 57

A Reading Vernier Scales 62

B Error Analysis 64

Lab Reports and Marks

This section includes an outline for your lab reports and a sample lab report based on anexperiment called "Conservation of Energy on an Inclined Plane" (also included).

Since your marks for the lab are derived from your written reports it is important to under-stand what is expected as a report. Read the sample experimental outline "Conservationof Energy on an Inclined Plane" first, then the lab report outline, and conclude by readingthe sample report.

Lab Report Outline:

Lab reports must have the following format:

Title Page: Includes the title of experiment, the date the experiment was performed, yourname, your partner’s name(s), and lab section.

Results: List data in tables whenever possible. Each table must have a title and carryunits for each column. Show one sample calculation for each step in the analysis,carrying the units through the calculations. Consult the sample lab report to see howsample calculations should be done. Any other measurements associated with thedata (parameters, constants, etc.) should be clearly labelled within this section.

Analysis: Complete all steps and questions in the analysis section of the experimentaloutline. The analysis section contains primarily equations and numerical work thatmust be carried out in order to obtain usable results. When asked to plot a graphbe sure to label each axis including units. Each graph must have a descriptive titleplaced above the graph (note: y vs x is not a descriptive title).

Discussion: A few written paragraphs outlining the important facets and details of yourexperimental results. Examples of what questions you should be asking yourself areprovided in each lab outline, but do not feel limited by these questions. If you haveother observations or conclusions that you would like to discuss, then do so. Thissection is the most important, yet most difficult.

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Conclusion: A final sentence(s) stating the outcome(s) of the experiment written in thecontext of the objective of the experiment. Your conclusions should be supported byarguments made in your discussion.

If you glance through a scientific journal you will observe that the outline listed above is asimplified version of most journal formats. Items such as introductions and procedures havebeen provided for you within the manual. It is important to note here that presentationmatters; the sections of your report should appear in the same order listed above.

The difference between a "good" lab write-up and a poor one is often attention to detail. Asyou write your report you must assume that the person who will read it knows little aboutyour experiment. You are presenting a case upon which you will base your conclusion. Guidethe reader through the report by clearly labeling tables and graphs with descriptive titles,succinctly presenting your discussion of the experiment and results, and unambiguouslystating your conclusion(s).

Marking scheme

Each group is required to submit one lab report for each experiment unless directed oth-erwise. Some labs are designed to be done individually. Groups can be no more than 2students. When experiments are done in groups one lab report for the group is sufficient,although individual group members may submit their own report if they prefer. Whengroup reports are marked each member of the group receives the same mark.

In some cases, several groups will team up on a given apparatus to obtain their results.Each group will then share the same data, but ALL analysis must be done by each groupindividually. This includes tabulating the original data; you are not allowed to ‘share’spreadsheets.

Lab reports are typically due one week after each experiment is completed, unless instructedotherwise.

Lab reports will generally be marked out of 10 (longer labs may be marked out of 15). Yourlab reports will be marked on the following criteria:

Completeness: - Have you completed all the necessary tables, graphs, and calculations?- Are all the required sections present, i.e. title page, conclusion, etc.?

Correctness: - Do you have the proper labels and units for all graphs and tables?- Are your numerical calculations done correctly?

Presentation: - Are the sections of the report in correct order?- Are the results presented in a clear and concise manner?- Are the spelling, grammar, and punctuation correct?

Reason: - Do your conclusions and arguments seem reasonable, based on your results?

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Note that the list of comments for each section given above is not exhaustive. There aremany different aspects to a good lab report which are not given here. Refer to the LabReport Outline section above to obtain more guidance on each section.

Your final lab mark is calculated as:

Total Marks Received / Total Marks Possible

Absence from Lab sessions

If you must miss a lab section, contact your lab instructor as soon as possible to let themknow you will be absent, preferably before the lab, if possible. You will need to makearrangements to do the lab at some other time. If you are able to attend a different labsection within the same week, then let your lab instructor know. He or she can make thenecessary arrangements with the instructor of this other lab session. Under no circumstancesare you allowed to put your name on a report for a lab that you did not participate in.

iii

Sample Lab Outline

Objective

The objective of this experiment is to investigate the principle of conservation of energyusing an inclined plane.

Apparatus

• Air Track

• Sonar System

• Carpenter’s Level

• Meter Sticks

Background

The near frictionless surface of an air track represents a close approximation to somethingcalled an isolated system. Such a system would be expected to demonstrate the principleof conservation of mechanical energy.

By definition an isolated system does not exchange mass or energy with its surroundings.Any mechanical energy put into the system will remain there and we should be able toaccount for it as either kinetic or potential energy. If the principle of conservation ofmechanical energy is correct then the total energy in the system should remain constant.

Tipping the air track at an angle creates what physicists call an inclined plane. In theabsence of friction an object placed on an inclined plane will begin to move in the downhilldirection.

iv

Energy is introduced into the inclined plane system by placing the glider at the top ofthe incline. Until released all the energy in the system is in the form of potential energy.Once moving some of the potential energy is transformed into kinetic energy. As the gliderprogresses down the incline the potential energy continues to decrease while the kineticenergy increases. At any instant in time, however, the sum of the two should be constantand equal the initial amount of potential energy in the system.

Procedure

Set the air track at a slight angle to the horizontal (about 2 degrees). Place the glider atthe high end of the air track and allow it slide to the low end. Track the position of theglider during its travel using the sonar equipment. Save your data on a disk. Be sure torecord the mass of the glider and measure the angle of incline with two meter sticks andthe carpenter’s level.

Analysis

Construct a data table like the one below and make the appropriate calculations.

Time Position of Velocity of Kinetic Energy Potential Energy Total Energy(s) Glider (cm) Glider (cm/s) of Glider (J) of Glider (J) (J)

From your calculations plot on a single graph the potential, kinetic and total energies as afunction of time.

v

Discussion

From your results, can you conclude that energy was conserved? If not, where might itbeen lost?

How accurately were you able to measure the angle of incline? What impact(s) might anerror in the measurement of the angle of incline have on the results of the experiment?

vi

Sample Lab Report

CONSERVATION OF ENERGY ON AN INCLINED PLANE

Larry BrownCurly JonesMoe Smith

Thursday, Jan. 8, 2010Lab Section #2

vii

Table 1. Inclined Plane Data Table

Mass of glider (g) 220 ± 0.5∆t (s) 0.0500 ± 0.00025Angle of incline (°) 2.7 ± 0.5

Time Position Velocity Potential Kinetic Total(s) (cm) (m/sec) Energy (J) Energy (J) Energy (J)1.10 198.43 0.316 0.2015 0.0110 0.2131.15 196.85 0.330 0.1999 0.0120 0.2121.20 195.20 0.428 0.1982 0.0202 0.2181.25 193.06 0.402 0.1961 0.0178 0.2141.30 191.05 0.352 0.1940 0.0136 0.2081.35 189.29 0.508 0.1922 0.0284 0.2211.40 186.75 0.450 0.1897 0.0223 0.2121.45 184.50 0.442 0.1874 0.0215 0.2091.50 182.29 0.518 0.1851 0.0295 0.2151.55 179.70 0.590 0.1825 0.0383 0.2211.60 176.75 0.568 0.1795 0.0355 0.2151.65 173.91 0.598 0.1766 0.0393 0.2161.70 170.92 0.632 0.1736 0.0439 0.2181.75 167.76 0.652 0.1704 0.0468 0.2171.80 164.50 0.644 0.1671 0.0456 0.2131.85 161.28 0.690 0.1638 0.0524 0.2161.90 157.83 0.772 0.1603 0.0656 0.2261.95 153.97 0.696 0.1564 0.0533 0.2102.00 150.49 0.812 0.1528 0.0725 0.2252.05 146.43 0.778 0.1487 0.0666 0.2152.10 142.54 0.806 0.1448 0.0715 0.2162.15 138.51 0.826 0.1407 0.0751 0.2162.20 134.38 0.826 0.1365 0.0751 0.2122.25 130.25 0.896 0.1323 0.0883 0.2212.30 125.77 0.880 0.1277 0.0852 0.2132.35 121.37 0.912 0.1233 0.0915 0.2152.40 116.81 0.932 0.1186 0.0955 0.214

Sample Calculations

Notes:

1.) The position data column represents the distance from the sonar unit to the glider.

2.) The frequency of the sonar unit was 20.0 ± 0.1 Hz, which translates into a ∆t of .0500±0.00025 seconds.

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Velocity Calculation:

Velocity is defined as: ∆Position∆time

Each entry in the velocity column was calculated as: Pn − Pn+1∆t

Sample calculation: 198.43cm− 196.85cm0.05s = 31.6 cm/s = 0.316 m/s

Potential Energy:

Potential energy is defined as U = mgh.

Given that the data can be thought of as the hypotenuse of a right-hand triangle, the heightcan be calculated as,

h = (position) sin θ∴ U = mg(position) sin θ

Sample calculation: U = (.220 kg) (9.80 m/s2) (1.9843 m) (sin (2.7°)) = 0.2015 J

Kinetic Energy:

Kinetic energy is defined as K = 1/2mv2

Sample calculation: K = (0.5) (0.220 kg) (.316 m/s)2 = 0.0110 J

Total Energy:

Total energy is the sum of the potential and kinetic energies, E = U +K.

Sample calculation: E = 0.2015 J + 0.0110 J = 0.2125 J = 0.213 J

ix

Figure 1: Potential, kinetic, and total energy of a single glider on an inclined plane.

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Figure 2: The velocity of the glider on the track as a function of time.

Discussion

It can be seen from Fig. 1 that energy was conserved as the glider descended the air track.There are some slight deviations from a constant energy value that appear to be due tosimilar deviations in the kinetic energy values. This is probably due to small errors in thevelocity values, most likely caused by timing errors in the motion sensor.

We checked to see if air resistance was a problem. We found that the velocity of the glider(see Fig. 2) was linear and therefore air resistance did not appear to affect our results. Thefact that the velocity was linear also tells us that acceleration on the air track is constant.

Measurement of the angle of the air track is a concern. If the angle is improperly mea-sured the potential energy calculations will be inaccurate. The result could be an apparentincrease or decrease in the total energy when in reality energy is conserved.

We measured the angle several times and used the average value as the actual angle ofincline for the air track. We found the accuracy of the measurement to be about one halfof one degree. The limiting factor in measuring the angle is the carpenter’s level. Theexperimental calculations are quite sensitive to the value used for the angle of incline. Thisis because this angle is used to calculate the potential energy. Using the same data, andvarying the angle over a range of 1.5 degrees, we were able to get a variety of results. Over

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this range of angles we could show a loss of energy or a gain of energy.

Conclusion

Energy was conserved on the inclined plane.

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LAB 1

Statistics

Objective

The objective of this exercise is to familiarize the student with the use of Gaussian distri-butions and linear regression as they relate to data analysis for experiments in introductoryphysics.

Background

Statistical analysis plays a very important role in experimental work. In this exercise we areconcerned with the connotations of terms such as average, standard deviation, and normaldistribution, as these terms have particular meanings when associated with experimentalwork. In addition, we will develop quantitative tools for judging the validity of data.

Throughout the semester we will associate the term average or mean with the "best estimateof the actual value". Standard deviation will represent the "average value of uncertainty inthe average" or probable error, and a normal distribution will be viewed as a probabilitydistribution. We will use techniques such as linear regression and Chauvenet’s criterion toprovide methods for assessing the validity of data gathered during experiments.

The following sections are brief overviews of the important points as they relate to statisticalanalysis.

Gaussian Distributions

The Gaussian curve, named after Karl Friedrich Gauss, was devised in 1795. Gauss foundthat his experimental data would not repeat itself exactly from trial to trial during exper-

1

2 LAB 1. STATISTICS

iments. After plotting the results of a particular experiment he developed his now famouscurve.

Gaussian distributions are also called normal distributions or bell curves. The term bellcurve obviously arose from the symmetrical curved shape of the Gaussian distribution whilethe name normal distribution is related to one of the mathematical requirements of a Gaus-sian distribution; that it be normalized. In practice all three terms are used interchangeablyand refer to a symmetrical distribution of values caused by random error. There are othertypes of distributions used in statistical analysis which differ from Gaussian distributions(e.g. Poisson distributions or binomial distributions). We will confine our interest solely toGaussian distributions as they are applicable to the type of experimental work we will dothis semester.

Any time measurements are made they are subject to some degree of uncertainty. Thisis known as experimental error. When the error is small and random (as opposed to sys-tematic) then it can be shown that the repeated measurements of the same quantity willcluster around the actual or true value of the measurement. Further, it can be shown thatdistribution of values around the true value will be symmetrical and have a curved shaped(the bell curve).

The "actual", "best", or "true" value of a set of measurements is, as you probably alreadyknow, called the average or mean. It can be shown mathematically that this is the case,however, we will be content at this time to simply accept this fact.

If we denote a measurement as x, then the average of a set of x’s can be calculated as:

Average = x̄ =N∑i=1

xiN

(1.1)

BACKGROUND 3

The width of the distribution curve about the average is also important. If it is narrowthen most values are close to the average. If it is wide then the values are not as closeto the average. A narrow distribution indicates higher precision (lower error) than a widedistribution.

The width of a Gaussian distribution is denoted by its standard deviation. Standard devi-ation represents the average difference of a given measurement from the value of the mean.

If you look closely at the equation for calculating standard deviation you can see that youare taking the average of the square of the differences between the data points in the set andthe average of the set. If the standard deviation is numerically large then the distribution iswide. If it is numerically small then the distribution is narrow. Naturally then the standarddeviation is a measure of experimental error.

A Gaussian distribution (shape of the curve) is expressed mathematically as

N = Nmax exp[−(x− x̄)2

2σ

], (1.2)

where N = the number of times a given measurement is observed,Nmax = the greatest number of times a single measurement is observed,

x̄ = average value of all measurements,σ = standard deviation, defined below.

σ2x = lim

N→∞

1N

N∑i=1

(x̄− xi)2, (1.3)

where N = total number of measurements,xi = results of the ith measurement.

4 LAB 1. STATISTICS

There is an interesting and useful relationship between a Gaussian distribution and itsstandard deviation. In any Gaussian distribution 68% of the total area under the curvefalls in the region from x̄−σ to x̄+σ. 95% of the total area falls between x̄− 2σ to x̄+ 2σ.This tells us that if an experiment is done to measure some quantity, although the result isnot duplicated exactly each time, there will be a definite pattern in the results. First therewill be a tendency toward some central value (x̄). Second, 68% of all values will fall withinone standard deviation of x̄ and 95% of all values will fall between 2 standard deviations ofx̄. Thus, it is reasonable to associate σ with the uncertainty in a given set of measurementsand to view the Gaussian distribution as a probability curve.

For experiments done in the lab, use the following definition of standard deviation σx. (Onyour calculator this function is usually denoted as σn−1.)

σ2x = 1

N − 1

N∑i=1

(x̄− xi)2 (1.4)

It can be shown that the standard deviation of a data set is related to the probable errorin the value of the average for the data set. The derivation of this is too complex to beconsidered at this time, but the result is worth knowing. The standard deviation of themean σm is given by

σm = σx√N

(1.5)

Note the factor of 1/√N present. This tells us that the more measurements we take, the

lower the standard deviation of the mean, and therefore the more confident we are in thatmean. This leads us to a very simple conclusion: More measurements = More accurateresults. All of your results should be presented with uncertainties (where possible), withthe uncertainty being represented by the standard deviation of the mean. Thus

x̄± σm (1.6)

BACKGROUND 5

An example: Consider the measurement of a steel rod using a micrometer (accurate to .001cm). The results are listed below.

length xi (cm) (x̄− xi)× 10−3 (cm) (x̄− xi)2 × 10−6 (cm2)10.351 0 010.354 -3 910.350 1 110.351 0 010.350 1 110.352 -1 110.352 -1 110.349 2 410.352 -1 110.349 2 4

10∑i=1

xi10 = 10.351 cm

10∑i=1

(x̄− xi)2 = 22× 10−6 cm2

The standard deviation is: σx = [2.2×10−5

10−1 ]12 = 1.6× 10−3cm

The standard deviation of the mean is: σm = σx√N

= 0.002√10 = 5× 10−4

Therefore, the length of the rod is: 10.3510 ± .0005 cm.

6 LAB 1. STATISTICS

Table 1.1: Gaussian Probability: Area under the Gaussian Curve

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.00 0.80 1.60 2.39 3.19 3.99 4.78 5.58 6.38 7.170.1 7.97 8.76 9.55 10.34 11.13 11.92 12.71 13.50 14.28 15.070.2 15.85 16.63 17.41 18.19 18.97 19.74 20.51 21.28 22.05 22.820.3 23.58 24.34 25.10 25.86 26.61 27.37 28.12 28.86 29.61 30.350.4 31.08 31.82 32.55 33.28 34.01 34.73 35.45 36.16 36.88 37.590.5 38.29 38.99 39.69 40.39 41.08 41.77 42.45 43.13 43.81 44.480.6 45.15 45.81 46.47 47.13 47.78 48.43 49.07 49.71 50.35 50.980.7 51.61 52.23 52.85 53.46 54.07 54.67 55.27 55.87 56.46 57.050.8 57.63 58.21 58.78 59.35 59.91 60.47 61.02 61.57 62.11 62.650.9 63.19 63.72 64.24 64.76 65.28 65.79 66.29 66.80 67.29 67.78

1.0 68.27 68.75 69.23 69.70 70.17 70.63 71.09 71.54 71.99 72.431.1 72.87 73.30 73.73 74.15 74.57 74.99 75.40 75.80 76.20 76.601.2 76.99 77.37 77.75 78.13 78.50 78.87 79.23 79.59 79.95 80.291.3 80.64 80.98 81.32 81.65 81.98 82.30 82.62 82.93 83.24 83.551.4 83.85 84.15 84.44 84.73 85.01 85.29 85.57 85.84 86.11 86.381.5 86.64 86.90 87.15 87.40 87.64 87.89 88.12 88.36 88.59 88.821.6 89.04 89.26 89.48 89.69 89.90 90.11 90.31 90.51 90.70 90.901.7 91.09 91.27 91.46 91.64 91.81 91.99 92.16 92.33 92.49 92.651.8 92.81 92.97 93.12 93.28 93.42 93.57 93.71 93.85 93.99 94.121.9 94.26 94.39 94.51 94.64 94.76 94.88 95.00 95.12 95.23 95.34

2.0 95.45 95.56 95.66 95.76 95.86 95.96 96.06 96.15 96.25 96.342.1 96.43 96.51 96.60 96.68 96.76 96.84 96.92 97.00 97.07 97.152.2 97.22 97.29 97.36 97.43 97.49 97.56 97.62 97.68 97.74 97.802.3 97.86 97.91 97.97 98.02 98.07 98.12 98.17 98.22 98.27 98.322.4 98.36 98.40 98.45 98.49 98.53 98.57 98.61 98.65 98.69 98.722.5 98.76 98.79 98.83 98.86 98.89 98.92 98.95 98.98 99.01 99.042.6 99.07 99.09 99.12 99.15 99.17 99.20 99.22 99.24 99.26 99.292.7 99.31 99.33 99.35 99.37 99.39 99.40 99.42 99.44 99.46 99.472.8 99.49 99.50 99.52 99.53 99.55 99.56 99.58 99.59 99.60 99.612.9 99.63 99.64 99.65 99.66 99.67 99.68 99.69 99.70 99.71 99.72

3.0 99.73 - - - - - - - - -3.5 99.95 - - - - - - - - -4.0 99.994 - - - - - - - - -4.5 99.999 - - - - - - - - -5.0 99.999 - - - - - - - - -

BACKGROUND 7

Linear Regression

Linear regression or least linear squares fitting provides a method by which the "best"straight line fit for a set of data may be calculated. Linear regression does not determine ifthe relationships in a data set are linear. That must be established by some other means(usually from theory). For example the speed of an object undergoing constant accelerationwill change linearly with respect to time. A plot of speed vs. time would therefore be astraight line. Linear regression could be used to determine the slope and intercept of theplot. The fact that the plot is linear is a consequence of the physics of the situation, not theuse of linear regression. If the plot where a curve rather than linear it would be meaninglessto find a slope and intercept using linear regression.

If we wish to fit a straight line through a set of data in which x is the independent variableand y the dependent variable the following must be true:

• each y value must be governed by a Gaussian distribution with standard deviation ofσy.

• the uncertainty in x must be small.

• the uncertainties in y must be uniform.

• the relationship between x and y must be linear, i.e. y = mx+ b.

Deriving the equations for linear regression is beyond the scope of Physics 1000. They aredeveloped by applying probability techniques to Gaussian distributions. The results of thederivation, called the normal equations, allow the calculation of the best fit slope and theintercept of the plot.

INTERCEPT = (∑xi

2)(∑yi)− (

∑xi)(

∑xi yi)

∆ (1.7)

SLOPE = N (∑xi yi)− (

∑xi)(

∑yi)

∆ (1.8)

where

∆ = N∑

xi2 − (

∑xi)2 (1.9)

You will use these equations in part II of this exercise.

8 LAB 1. STATISTICS

Procedure: Part I

This exercise requires two sets of data, which are the results of measurement. Each setmust be measured or estimated to the indicated tolerance.

Set #1. The height of each student in Physics 1050 (to the nearest cm).

Set #2. The diameter of the metal plate located at the front of the lab (to the nearestmm).

In the first case you will be provided with additional data from previous classes.

Analysis: Part I

1. For each set of data calculate the average, standard deviation, and standard deviationof the mean. Plot a distribution (histogram) of the number of repetitions of a valuevs the range of values.

2. Answer the following questions:

(a) How is the meaning of "average" different for these two sets of data? What doesthe average actually represent in each case?

(b) Should the distribution for set #1 appear Gaussian? Think of the different"types" of people measured, and if their heights are the same. How should thedistribution appear if these different types are considered? Is this represented inyour results?

(c) Locate the definitions of random error and systematic error (see Appendix B).Describe the effects of these two types of errors on a Gaussian distribution.

Procedure: Part II - Linear Regression

The data set below represents the position of a moving object with respect to a fixed point.Beneath each position measurement is the corresponding time of the measurement. Thisrelationship is known to be linear (y = mx+ b).

Time (s) 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00Position (m) 6.50 10.7 19.9 21.3 26.9 32.7 38.2 44.1 49.5

Using a spreadsheet program, plot the data (position vs time) using a scatter plot, makingsure to NOT join the points. Print this plot, and draw on your best estimation of thebest-fit line using a ruler and pen.

ANALYSIS: PART II 9

Analysis: Part II

Determine the slope and intercept of your hand-drawn line, and clearly label them in yourreport as the estimated slope and intercept. To do this, simply pick two points on your linefor which you can easily determine the position and time coordinates. Use m = rise / runto find the slope, and use this m value in conjunction with one of your points to find theintercept b using the equation y = mx+ b.

Now using the equations contained in the section on linear regression (equations 1.8 and1.7), calculate the slope and intercept of the actual best-fit line. You will need to constructa suitable table to find all the necessary sums, which is much easier if a spreadsheet programis used.

Compare your estimated slope and intercept with those calculated by linear regression andcomment.

What are the physical interpretations of slope and intercept in this case?

LAB 2

Free Fall

Objective

The objectives of this experiment are to show that a freely falling body has constant accel-eration, g, and to determine a value for g.

Background

The term free fall implies a one dimensional motion for which the motivating force is solelythat of gravity. An object falling freely under the influence of gravity in vacuum hasan acceleration that is constant, therefore, as the object falls its velocity increases at aconstant rate. Any object that moves in a straight line, with equal changes in velocity inequal intervals of time, is said to be moving with uniform, or constant, acceleration.

It should be noted that the terms velocity and speed are not exactly interchangeable. Ve-locity is a vector, meaning that values for both magnitude and direction are required tofully describe the motion. Speed is by definition the magnitude of the velocity vector. Forthis experiment we will use the term velocity since the object will experience changes inboth speed and direction.

From an experimental point of view the easiest way to determine if the acceleration of anobject is constant is to plot its velocity as a function of time. If the plot is linear thenthe equation which governs the plot is velocity = (acceleration)(time) + initial velocity(recall the equation of a straight line y = mx+ b). The slope of the plot is the value of theacceleration while the intercept represents the velocity of the object when the experimentbegan.

10

PROCEDURE 11

The definition for average velocity is

vave = DisplacementTime = ∆~x

∆t (2.1)

where ∆ means "change in". Equation 2.1 defines "average velocity over an interval oftime". A useful point to remember in situations where the acceleration is constant is thatthe average velocity over some interval of time is equal to the instantaneous velocity at themid point (in time) of that interval.

In this experiment we will toss a large rubber ball vertically into the air and track itsposition with a sonar system. The ball rises about two metres and then falls. The speedsassociated with these displacements are low enough that air resistance should not be asignificant factor.

Regardless of the fact that the ball first travels upward and then downward the only forceacting on the ball after it is tossed is the force of gravity. If the acceleration of the ballis constant then we should see a linear plot of velocity versus time despite the change indirection.

The motion of the ball is tracked by a sonar system that can accurately measure the locationof the ball by measuring the time of flight of an ultra sonic sound pulse reflected off theball. Since the sound pulses are sent at regular intervals of time, ∆t is a constant.

Procedure

The sonar apparatus will be set-up at the front of the lab. Practice tossing the ball verticallyupward such that it rises and falls in a straight line above the sonar device. When you areready, start the DataStudio program and proceed as follows:

1. Click the "Setup" button, and ensure that the motion sensor is displayed. If not, click"Add Sensor" and find the motion sensor in the list.

2. Click the "Start" button to begin collecting data. You should hear a ticking soundwhile the sonar system is collecting data.

3. Position the ball over the sensor and drop it from rest. Make sure you are preparedto catch the ball before it hits the sensor. It may help to have one lab partner dropthe ball while the other catches the ball.

4. Toss the ball into the air, allowing it to rise and fall, catching it before it hits thesonar system (keep your arms to the side so as not to interfere with the sonar beam).Repeat this step several times to get many trials in your data file.

5. Once you are satisfied with the data use the "Export Data" entry in the "File" menuto save your data to a suitable medium.

12 LAB 2. FREE FALL

Open your file in a spreadsheet program, plot all the data, and select the portions of datawhich correspond to the flight of the ball. The portion you use should appear as a smoothparabola, with few erroneous points. Ultimately, the data that you keep will be decided bythe velocity graph, so don’t be too picky about where the data starts and ends. Transferthe points you select by copying and pasting the appropriate entries into a new spreadsheetfile or worksheet within the same file.

Analysis

We wish to construct a graph of velocity vs. time for each case. The first step is to constructa data table like the one below:

Time Position Velocity Acceleration0 y1 v1 = (y2 − y1)/∆t a1 = (v2 − v1)/∆tt1 y2 v2 = (y3 − y2)/∆t a2 = (v3 − v2)/∆tt2 y3 v3 = (y4 − y3)/∆t a3 = (v4 − v3)/∆t...

......

...tn−2 yn−2 vn−2 = (yn−1 − yn−2)/∆t an−2 = (vn−1 − vn−2)/∆ttn−1 yn−1 vn−1 = (yn − yn−1)/∆ttn yn

The position and elapsed time columns in this data table are the data you selected for theflight of the ball. Reset the time to start at zero. Each successive time value will then be ∆tgreater than the previous one, since the change in time will be constant in this experiment.∆t is determined by the frequency of the sensor, and should be easy to determine from yourinitial data. Calculate the velocity as indicated above by calculating the displacement of theball as the difference in two successive positions of the ball and dividing by the correspondingchange in time. The acceleration column will be calculated in a similar manner. Be sureto include appropriate units for all columns, and sample calculations for your velocity andacceleration columns. Also note that there are blank spaces in the bottom right cells of thistable; you should attempt to understand why this is the case.

Create one table for each case done (2 total), and clearly label which table corresponds tothe dropped ball, and which corresponds to the tossed ball. Each table is created in anidentical manner, using different data.

Plot the velocities against elapsed time and determine the acceleration due to gravity fromthe slope of the graph (use linear regression). Your graph should appear as having a constantdownward slope. If there are portions of your graph which have an upward slope, these donot correspond to free fall; simply remove the necessary points from your table until thereare no upward sloping portions (the graph will automatically update when removing thesepoints). Your graph should have 2 series (lines), one for each case (dropped and tossed).

DISCUSSION 13

Make sure each case is clearly labelled on your graph. If you need help adding 2 series to agraph, ask your instructor.

Discussion

Compare your experimental values for g with the accepted value and compute the per-centage difference in your results. The percentage difference is defined as %difference =|actual - measured|/actual, where actual and measured refer of course to the actual valueand the measured value. Comment on the disparity in your result and the actual value;what do you feel may have affected your results? Which case gave better results? Theanswer to this question may help you identify any errors in the procedure.

Without repeating the experiment, can you determine from your data if air resistance is aproblem in this experiment? Think of how air resistance would affect your graph, notingthat the acceleration due to gravity is towards the ground, but air resistance always opposesthe direction of motion.

LAB 3

Projectile Motion

Objective

The objectives of this experiment are to show that motion in two dimensions can be treatedindependently in each dimension and to determine a value for g, the acceleration due togravity.

Background

Projectile motion, or trajectory motion as it is sometimes called, refers to the two dimen-sional motion of an object projected into the air at an angle where gravity is the only force(e.g., throwing a baseball or basketball). Analysis of this motion is based on the premisethat the vertical and horizontal motions are independent of each other. In order to verifythis premise, the trajectory of a steel ball will be analyzed in both the vertical (y) and thehorizontal (x) directions. By way of illustration we consider the following example (that isdifferent from the experiment that is actually performed).

Figure 3.1: Independent Motion

A ball is thrown into the air (Fig. 3.1, point A) having velocity in both the horizontaland vertical directions. At the exact moment it reaches its maximum height (point B) and

14

PROCEDURE 15

begins to fall earthward, an identical ball is dropped from the same height (point C).

If the horizontal motion is actually independent of the vertical motion, the balls will strikethe ground at the same instant. One expects this to be the case since the only force acting onthe balls is gravity (neglecting air resistance) and as a result, Newton’s second law predictsthat the horizontal motion of ball 1 will remain at its initial constant velocity while bothballs will accelerate downward with acceleration g.

A strobe photograph of the falling balls would therefore reveal the following.

Procedure

All necessary equipment is assembled in the dark room. You will be instructed in itsoperation and will obtain a digital photograph of a projectile.

1. Record the flash rate (flashes/minute) from the dial of the strobe light.

2. Have one group member take the picture while the other drops the ball through thetube. With the strobe light flashing, depress the shoot button on the camera. Youwill hear a beep shortly thereafter, indicating that the multi-second exposure is nowbeing taken. Immediately upon hearing this beep, the ball should be dropped intothe tube. The camera will beep again when the exposure has finished.

3. Check your picture to ensure that the path of the ball was in fact recorded. Hit the

Figure 3.2: A sample strobe picture depicting the parabolic flight of a ball in projectilemotion.

16 LAB 3. PROJECTILE MOTION

replay button and use the zoom buttons on the camera to do this. Your picture shouldappear similar to Fig. 3.2. If it does not, try again.

4. Once satisfied, note the file name of your picture.

5. Once each group has taken a picture, your lab instructor will distribute your pictureto you in some fashion (either directly downloading it to your computer, or emailingit to you).

6. Open the picture using the Paint program. Find the first image of the ball as itleaves the tube and place the mouse pointer over it. The coordinates of the pointerwill appear just underneath the bottom right corner of your picture, as an x,y pair.Record these numbers.

7. Repeat the previous step for each successive image of the ball. You should get at least12 points. Make sure to record the coordinates in sequential order (as the ball travelsfrom left to right)!

8. Place the mouse pointer at the top of the metre stick shown in your picture. Recordjust the y coordinate, the x coordinate is inconsequential.

9. Repeat the previous step at a point 10 cm below the top of the metre stick. Thestick should have alternating bands of colour for every 10 cm section to facilitate thismeasurement. Repeat this procedure until you have 6 total coordinates.

Analysis

Enter your data into a spreadsheet as 2 columns. Create 2 new columns that have thefirst point scaled to (0,0). To do this simply subtract the first x coordinate from all thex coordinates, and subtract the first y coordinate from all the y coordinates. This is veryeasy to do using a spreadsheet equation; don’t resort to using a calculator! Your instructorcan give you assistance should you require it.

Construct the following table using your scaled set of coordinates as the first 2 columns.You do not need to include your unscaled set of data in your report.

x (pixels) y (pixels) x (cm) y (cm) vx (cm/s) vy(cm/s) ax (cm/s2) ay (cm/s2)0 0 0 0 v1x v1y a1x a1y

In order to convert your coordinates from pixels to cm, you will need to find a scale factor.To do this, use your results from the measurement of the metre stick. You should have6 different y-coordinates corresponding to the ends of 5 separate 10 cm sections of ruler.The difference between these coordinates is the number of pixels per 10 cm. Simply find

DISCUSSION 17

the difference between successive coordinates (which should give you 5 similar numbers),average all your results together, and use this result as the scale factor. Include all yourcalculations. Once you have the scale factor, you should be able to easily convert your pixelvalues into cm. Writing out the correct unit conversion might help.

The velocity and acceleration of the projectile are calculated by dividing the change in twosuccessive positions of the projectile by the corresponding elapsed time. The time betweenflashes is gained from the strobe rate that you recorded. Note that this strobe rate is givenin flashes/minute, which is an example of a frequency. What you require is seconds/flash; asimple unit conversion and a little ingenuity will give the correct ∆t. Also note that similarto the free fall lab, you will have 1 less velocity value than position, and 1 less accelerationthan velocity. This will create blank spaces in the bottom right of your table.

From your data and calculations construct the following graphs:

Graph 1 Horizontal position vs. elapsed time.Vertical position vs. elapsed time.

Graph 2 Horizontal velocity vs. elapsed time.Vertical velocity vs. elapsed time.

Graph 3 Horizontal acceleration vs. elapsed time.Vertical acceleration vs. elapsed time.

Perform linear regression on both series of your velocity graph. What do the slopes tell youin each case?

Discussion

From your results, what can you conclude about the independence of motion in the verticaland horizontal direction? Consult your free fall lab results. Do your results in the verticaldirection appear similar (or identical) to the free fall lab? What exactly does this prove, ifanything?

You should have found a value for g in your analysis; how does it compare to the actualvalue? Is this result better or worse than previous values you have found in the lab? If yourvalue for g is substantially different from the actual value, what do you think is responsiblefor this? Think about how your scale factor and value for ∆t are involved in this value;what if they’re incorrect? What effect would this have on your results?

LAB 4

Vectors, Forces, and Equilibrium

Objective

The objective of this exercise is to show that when a system is in static equilibrium (i.e. notmoving), the conditions for translational and rotational equilibrium,

∑ ~F = 0 and∑~τ = 0,

are indeed satisfied.

Apparatus

• String or thread

• At least 300g of weights

• Spring Balances

• Force table

• Large Wooden Protractor

• Rotational Bar

• Wooden Plank

• Bathroom Scales

Background

Vectors are composed of two distinct and equally important quantities; magnitude anddirection. How you choose to account for these quantities is to some extent a question ofstyle and is often dictated by the nature and complexity of the physical situation you are

18

BACKGROUND 19

trying to describe. Since this exercise involves analysis in two dimensions we will use the"x ,y" notation. Your textbook develops the subject of vectors in great detail, we will onlylist the highlights here.

One always needs a reference grid in which to do vector analysis (for two dimensions this isthe familiar x− y graph). The most important consideration for any reference grid is thatthe reference axes be mutually perpendicular. If they are not they are useless. Considerthe following situation.

Figure 4.1: Components of a vector.

Notice that the same change in position can be accomplished by moving along the x axisa distance Ax and then parallel to the y axis a distance Ay or vice versa. If the axes werenot perpendicular to each other we could not generate completely separated componentssuch as Ax and Ay. Remember, we are accounting for both the magnitude of the change inposition and its direction. In a mathematical sense then we can write ~A = ~Ax+ ~Ay becauseeither path yields the same result, implying that they are in fact equal. In order to arriveat the correct value for the magnitude of the change we must add the components of ~A ( ~Axand ~Ay) as vectors, not algebraically.

Suppose the vector ~A took us from the origin to the point (5,5). That would mean that thevector ~Ax went along the x axis from the origin to the point (5,0) while ~Ay went from (5,0)to (5,5). It is clear that ~Ax and ~Ay each have a magnitude of 5. What is the magnitudeof ~A? If we add the magnitude of the components in an algebraic fashion the answer is 10.Clearly 10 is not the correct answer. In the x− y reference system of Fig. 4.1, ~A, ~Ax, and~Ay form a right hand triangle with ~A as the hypotenuse. The geometry of the situationallows us write the relationships between the three sides as

AxA

= cos θ, AyA

= sin θ, (4.1)

where θ is the angle between ~A and the x axis.

To find the relationship between the magnitudes of the three sides, recall Pythagoras’theorem, which states that √

A2x +A2

y = A. (4.2)

For the example given above the magnitude of ~A would be | ~A| = A =√

52 + 52 =√

50 = 7.1.

20 LAB 4. VECTORS, FORCES, AND EQUILIBRIUM

When there are several vectors involved in an analysis the proper approach is to resolveeach vector into is components add all the x components and then all the y components.The sums of the components yield the components of the resultant vector which can bethought of as the "net effect" of all the vectors in the problem.

Consider the following example.

Figure 4.2: Addition of vectors by components.

An ideal way to use vectors and vector analysis is in the representation of forces and torques.Let us first define the difference between a force and a torque.

If you wish to slide a box across the floor (translational motion) you apply a force. If youwish to spin a wheel about its axle (rotational motion) you apply a torque. You may wishto argue that you apply a force in both cases. This is almost true. However, there is animportant difference between a force and a torque. Imagine as you try to spin a wheel youuse a force that passes from the edge of the wheel through the center of the wheel wherethe axle is located. Would the wheel spin? If the force you applied to the wheel did notpass through the center of the wheel would the wheel spin?

Figure 4.3: A force causing pure translation, and a force causing a combination of translationand rotation (torque).

Hence the difference between a force and a torque. A torque involves a force whose line ofaction does not pass through the axis of rotation. Note that a torque is in fact a vector

PROCEDURE 21

Figure 4.4: The relationship between the angle θ and the ~r and ~F vectors.

quantity (it has direction as well as magnitude). The vectors used to represent torquesare different than those for forces. However, in two-dimensional cases, there are only twopossibilities for the direction of rotation, clockwise or counterclockwise. Thus it sufficesto simply set one of these directions as positive and the other negative, similar to one-dimensional translational motion. In general, counterclockwise is labelled as the positivedirection.

The magnitude of a torque is calculated by multiplying the applied force by the perpendic-ular distance from the axis of rotation or, force multiplied by the distance to the axis ofrotation multiplied by the sin of the angle(smaller) between the force and distance vectorswhen placed tail to tail. As an equation, this becomes

τ = Fr sin θ (4.3)

In conjunction with Newton’s second law ΣF = ma, a definition for equilibrium is easilystated, i.e. the sum of the forces on an object must be zero (ΣF = 0) and the sum of thetorques must be zero (Στ = 0). We must, however, make a distinction between static anddynamic equilibrium. In both cases ΣF = 0 and Στ = 0, but, in the static case the objectis at rest while in the dynamic case the object is moving and/or spinning with constantvelocity.

Procedure

Part I - Translational Equilibrium

1. Obtain 2-4 lengths of string, a force table, some weights, and 2 spring balances.

2. Tie the pieces of string on to the metal loop provided with the force table, and placethe loop on the central column of the force table.

3. Place pulleys at the 0o and 180o marks of the force table, and run the strings over

22 LAB 4. VECTORS, FORCES, AND EQUILIBRIUM

the pulleys. Attach 200 g of weights on to the string over the 180o mark, and 100 gof weights on to the string over the 0o mark.

4. Attach 2 spring balances to the loop by hooking them directly on to the loop, orattaching them with string.

5. One balance will be held at a constant angle of either 90o or 270o, measured from the100 g weight. This will be referred to as the compensating force.

6. The other balance will be known as the sweeping force. This force will be appliedat a number of different angles ranging from 20o to 70o, measured from the 100 gweight and opposite the compensating force. Refer to the diagram below to see thefinal setup.

7. The object of this experiment is to keep the loop directly over the coloured circle inthe centre of the force table. At this point, all the forces will be in equilibrium. Oncethis is achieved, one member of the group records the force on each of the balancesand the angle of the sweeping force.

8. Begin with the sweeper at an angle (θ) of 20o. Proceed in increments of 10o until youreach 70o. Do at least 2 trials for each angle.

Part II - Rotational Equilibrium

This exercise is similar to part I except that we are interested in torques rather than forces.Note that between 4 and 6 people are required per apparatus, so try to team up with othergroups to collect your data.

1. Attach ropes or string on to any 3 posts of the rotational bar. Measure and recordthe distance to each post using a metre stick.

2. Attach a spring balance to the end of the strings. Have 3 separate group memberspull on each balance to apply a force on to each string at varying angles, and tryto attain equilibrium in the system. Keep the forces low, approx. 20 N max. Oneadditional member of the group should ensure that the bar does not tip. The anglebetween each force and the bar must be at least 30o.

PROCEDURE 23

3. Once the system is in equilibrium, have one member of the group record the anglebetween the bar and each string, and the reading on each spring balance. Also makesure to include the direction of each force; a diagram of the situation may be theeasiest way to record this situation.

4. Repeat steps 1-3 for 2 additional situations, varying radii, force and angle (3 situationsin total).

PART III - Calculating the Position of Your Center of Mass (An applica-tion of equilibrium)

In this part of the experiment you will use the principles of equilibrium to calculate thevertical height of your center of mass. If you are not familiar with the concept of centerof mass simply stated it is the point at which the distributed weight of an object can berepresented by a single force.

In order to determine your center of mass arrange the plank, and scales as shown in Fig.4.5. Lie on the plank with the bottoms of your feet even with point C. Have someone readthe scales and measure the distance between points A and C. Make sure to record whichscale measurement was for your feet, and which was for your head.

Figure 4.5: Determining the centre of mass of a person.

24 LAB 4. VECTORS, FORCES, AND EQUILIBRIUM

Analysis

Part I

Prepare a table to present your data.

Angle Sweep Force Comp. Force ΣFx ΣFyTrial 1 Trial 2 Avg Trial 1 Trial 2 Avg

20°30°...

70°

In order to calculate the final two columns, we must first decide on a coordinate system.The simplest choice would be to let the 0° mark be the positive x axis, and the 90° markbe the positive y axis. This is depicted in Fig. 4.6. Using this coordinate system andeverything you know about vectors, find an equation for the sum of the forces in the x andy directions. Note that the tension created along the strings connected to the 2 hangingmasses is in fact the weights of those masses, which you will need to calculate. Use theseequations to evaluate the values for the final 2 columns. Plot a graph of the sweeper’s forcevs angle.

Figure 4.6: The forces present on the force table, with an appropriate coordinate system touse.

Part II

Present your data for this section in a table, also. For each trial, the table should appearas follows:

r θ Force Direction TorqueTrial 1 Force 1 r1 θ1 F1 ±1 τ1

Force 2 r2 θ2 F2 ±1 τ2Force 3 r3 θ3 F3 ±1 τ3

DISCUSSION 25

A note about the direction; simply assign one direction to be positive and the other negative.If the direction is positive, then record +1, if negative, record -1. Using this method, youcan use a spreadsheet equation to calculate τ and still obtain the correct sign for the torque.To find the torque, use equation 4.3. Find the sum of the torques in each case.

Part III

Draw a free body diagram of the plank. Include all forces and distances on this diagram.There are 3 forces in total, the weight of the person, and the 2 normal forces exerted by thescales on the plank. Ignore the weight of the plank. Using translational equilibrium, findthe weight of the person (this should be very easy).

Use point C on the plank as your pivot point. Find the torque due to each force on yourfree body diagram by using 4.3, with all the r values being measured from your pivot point.Using the condition that the sum of the torques (with respect to any point along the plank)must be equal to zero for equilibrium (Στ = 0), calculate the value of x as per Fig. 4.5.x represents the location of the center of mass with respect to the point C, i.e. from yourfeet up.

Discussion

Do your results for Parts I and II represent equilibrium in each case, within a reasonabledegree? If any cases do not, explain what may have caused these trials to deviate fromequilibrium, and how to improve the results.

Should the plot of sweeper force vs. angle be a straight line? Why or why not? Use anequation to justify your answer.

Does your value for the centre of mass seem correct? To help you answer this, use a metrestick to find what part of the body this height corresponds to. Also compare your resultto the height of the person measured by expressing the position of the centre of mass as apercentage of their height. Find this same percentage for someone in the lab of the oppositegender; how do your results compare?

LAB 5

Mechanics of the Human Arm

Objective

The objective of this experiment is to use a model of the human arm to determine thelifting angle of the arm that creates the least stress in the elbow joint, and find the relationbetween the amount of force lifted to the amount of force required in the bicep muscle.

Background

Any object that is completely stationary can be said to be in equilibrium. This particularstate of equilibrium can be characterized by noting that the object remains stationary onlywhen there is no net force and no net torque acting upon it. To hold an object motionlessin your hand for example requires your arm muscles to provide the forces necessary tocounterbalance the loads (forces and torques) placed on the arm by the object. Holding anobject motionless in your hand is an example of static equilibrium.

Figure 5.1: Arm of a physics lab instructor, and a simple diagram of the apparatus used tomodel it.

Analyzing a situation for static equilibrium is an easy task. Since there must be no net forcenor net torque, the vector sum of all the forces present must be equal to zero (ΣF = 0),

26

BACKGROUND 27

Figure 5.2: The forces on a human forearm held level with the ground, assuming the presenceof a load in the hand.

likewise the vector sum of all the torques must be zero (Σ τ = 0). Analogously if an object isin equilibrium it is possible to calculate an unknown force or load using the same equations.

The forearm can be treated as a lever acted upon by forces which tend to produce rotationabout the elbow.

Referring to Fig. 5.2, the forces acting on the forearm are:

~T The tension in the string (or biceps muscle).~WA The weight of the arm.~WL The weight of the load.~R The reaction force in the elbow joint.

Here we can think of the ~R force as the force with which the bone in the upper arm pushesdown at the elbow joint, or the strain produced in the elbow. If we think of the elbow jointas a "pivot" point, then the weights attempt to rotate the forearm in a clockwise fashionwhile the muscle tension tries to rotate the forearm in a counter clockwise fashion. Thereaction force produces no rotation. Do you know why?

The rotation of an object about a pivot point can be described in terms of torque. As youmay remember, torque (τ) is the rotational analogue to force. Recall that its magnitude is

Figure 5.3: The magnitude of a torque can be found by multiplying the magnitude of F⊥by d.

28 LAB 5. MECHANICS OF THE HUMAN ARM

given by the product of the distance d to the pivot point and the perpendicular componentof the force F (see Fig. 5.3, and equation 4.3).

The sign of the torque is determined by convention. A torque that causes a counterclock-wise rotation is considered to be positive, and a torque that causes a clockwise rotation isconsidered to be negative (the torque in Fig. 5.3 is positive).

For the forearm to remain stationary, two conditions must be satisfied:

1. The forearm is not accelerating out of its elbow joint. Hence the net force on the fore-arm must be zero. Resolving all of the forces on the forearm into x and y componentswill result in two scalar equations:

ΣFx = 0, ΣFy = 0

2. Since the forearm is not rotating about its elbow joint, the net torque on the forearmmust be zero. If we choose any pivot point, the sum of the counterclockwise torques(positive) and clockwise (negative) must be zero about that point:

Σ τ = 0

Procedure

Calibrating your Apparatus

In order to ensure that the weights are at a 90o angle to the arm itself, the arm must beparallel with the ground. This requires that we calibrate your apparatus. This calibrationwill require 3 objects; 2 protractors, one of which has the origin cut out, and a level. Obtainthese items, and complete the following steps:

1. Clamp the upper beam of the apparatus to a bench. Tape the protractor (the onethat does not have a hole) to the lower beam such that the origin of the protractor(the spot where the 0°, 90°, and 180° lines coincide) is aligned with the point wherethe string exits the top of the beam. Also ensure that the 0° lines on the protractorare aligned with the top of the beam.

2. Check that your biceps beam is difficult to move back and forth, and stays fixed ata particular angle when moved. If this is not the case, ask your instructor to tightenthe upper bolts for you.

3. Apply enough tension in the string so that the lower arm becomes horizontal (use alevel). While the lower beam is held level, adjust the biceps beam so that the stringaligns with the 90o mark on the protractor. Remove the tension in the string so thatthe lower arm simply hangs, but ensure that the biceps (upper) beam does not move.

PROCEDURE 29

4. Using tape, attach the protractor to the upper (shoulder) joint so that the 90o markaligns with the black mark on the biceps beam. Try to make sure that the locationthat the origin of the protractor would be at lines up with the axis of rotation (thecenter of the bolt).

Your apparatus is now calibrated so that the arm is level whenever the angles on theprotractors are identical.

Part I: Weight of the Model Arm

After calibration, you will mount weights on the end of the arm using a mass hanger.

1. Obtain and weigh a mass hanger. Record the mass.

2. Place the mass hanger at the end of the arm apparatus by inserting the hooked endinto the hole provided.

3. Hook a 20N spring balance to the end of the string (tie a loop in the string). Thestring should pass over both pulleys of the apparatus before being attached to thebalance.

4. Set the upper arm so that the black line is aligned with the 90° mark of your upperprotractor.

5. Pull down on the balance until the string aligns with the 90° mark of the lowerprotractor. At this point, the arm should be level with the ground if calibratedproperly. Record the force on the spring balance as the string tension T .

6. Mount 10g of weights on to the hanger, and repeat the process. Continue repeatingthis process, adding 10g each time, until you have 6 measurements (the last weightused should be 50g). Note that any amount of mass less than 120 g is appropriate touse, so if you are unable to find a 10g mass, simply use 5 different masses.

NOTES:

1) Friction gives the forearm a tendency to "stick" near equilibrium. To account for thiswhen making measurements, you should move the forearm to the horizontal position andsee if it moves away from this position. (A light tap is also helpful).

2) The distance between the elbow joint and the tendon is 4.1 ± 0.1 cm, the distancebetween the elbow joint and the load is 31.4 ± 0.1 cm and the distance from the elbow jointto the centre of mass of the arm is 16.0 ± 0.1 cm.

30 LAB 5. MECHANICS OF THE HUMAN ARM

Part II: Muscle Tension as a function of angle

1. Remove any load from the end of the arm. Set the angle of the arm to be 50o byaligning the black mark on the bicep with this angle on the upper protractor.

2. Find the tension needed to keep the forearm horizontal (parallel to the ground) byusing a spring balance attached the end of the string, as in Part I. Record your resultin a table. Recall that the forearm is level when the angle on each protractor is thesame.

3. Repeat this procedure for angles between 60o and 140o, in steps of 10o.

CAUTION: For obtuse angles (>90o) the apparatus is unstable and the arm may moveupwards quickly and in the process break the string, and any fingers in the way, etc. Youwill observe that the forearm loses its balance very easily and seems to move either wayfrom the horizontal position; this is called unstable equilibrium. To prevent damage to theapparatus, keep your hand above the lower arm while applying tension. The closer yourhand, the better.

Analysis

Put your results from Part I into a simple table with columns for the mass of the load andthe tension. You will need to find the weight of the load (including the hanger), and createa separate column for this. Note that the weights should be recorded in Newtons (you willhave to calculate the weight using your mass values).

From your data from Part I, for each of your measurements, compute WA using rotationalequilibrium. To do this, refer to the forces shown in Figure 5.2. For each force, write thetorque it creates by using the distances provided in the procedure (recall that τ = Fd if theforce and the radius are perpendicular). Substitute each torque into Σ τ = 0, making sureyou use the correct sign for each torque. This should give you an equation that containsWA, WL, and T (note the absence of the R force; why is this?). Solve this equation forWA and use your equation to create a column in your spreadsheet to find WA for each WL.This will give you 6 values for WA, which should all obviously be similar. Find the averageof these and use this as your value for the weight of the arm.

From your data for Part II, use the translational equilibrium conditions ΣFx = 0 andΣFy = 0 to find Rx and Ry, the components of the reaction force ~R. You may wish to referagain to Fig. 5.2 to see which forces point in which direction. These components can thenbe used to find the magnitude of the reaction force ~R. Present these results as columns inyour table of tension and angle. Present your results in a graph of force vs. angle with twoseries, one for the tension, and one for the reaction force.

θ T Tx Ty Rx Ry R

DISCUSSION 31

Discussion

From your results for Part I, what can you conclude about the force in the bicep musclecompared to the weight of an object lifted in the hand? What does this tell you about theefficiency of the human arm for lifting heavy objects?

Fron your results for Part II, what is the angle that causes the least amount of strain in theelbow? Does this make sense, physically? What do you predict the tension T to be at θ =180o? Your graph of tension vs. angle may help you answer this question. Do not attemptto test this experimentally!

Are the values of T the same (within error) on the obtuse side of 90o as those on the acuteside? For example, does the tension recorded at 70o equal that at 110o? Should these valuesbe equal? Explain your answer. Hint: Think of where the string actually connects to thearm. Are we accounting for all the torques present?

LAB 6

Angular Acceleration of a Disk

Objective

The objective of this experiment is to determine the relation between torque and angularacceleration, and to verify that the moment of inertia of a complex body can be found bysumming the moments of inertia of the individual parts.

Background

The analogy of Newton’s second law ~F = m~a as applied to rotational motion states that theangular acceleration of a body (α) is proportional to the torque τ and inversely proportionalto the moment of inertia or rotational inertia I of the body.

τ = Iα, (6.1)

where [τ ] = N m, [I] = kg m2 and [α] = rad/s2.

Just as the mass of a body is a measure of the tendency of the body to resist linear acceler-ation, so the moment of inertia is a measure of a body’s resistance to angular acceleration.If a body is free to rotate about an axis, then a torque is required to change the rate ofrotation. The resulting angular acceleration is proportional to the torque as per Equation6.1 and exists only during the time the torque acts on the body.

The moment of inertia of a body is dependent on the mass of the body and the distributionof the mass about the axis of rotation. To determine I, the body is divided into a number ofinfinitesimal elements mi, each at some distance ri from the axis of rotation. The momentof inertia is given by the sum of all the products mir

2i calculated for each element.

I =N∑i=1

mir2i (6.2)

32

BACKGROUND 33

Figure 6.1: The forces and distances present in the simple disk-mass apparatus.

If the body is a simple geometric shape, e.g., a disk, sphere, or cylinder, I can be easilycalculated; results for such calculations are given in tables. For a solid disk the expressionis

I = 12mr

2, (6.3)

where m is the total mass of the disk and r its radius. For a ring of inner radius r1 andouter radius r2 (imagine a disk with a central hole cut out), the moment of inertia becomes

I = 12m(r2

1 + r22). (6.4)

In this experiment the linear acceleration a of a mass m suspended on a cord is measured.This cord is wound around the hub of the disk (see Fig. 6.1) and as a result, the magnitudeof the linear acceleration of the rim of the hub is also a.

By calculating the linear acceleration on the rim we can determine the angular accelerationα of the hub (which is also the angular acceleration of the disk) through the equation

a = rα, (6.5)

where r is the radius of the hub.

The torque ~τ (a vector) resulting from a force ~F applied about an axis is given by theexpression:

τ = rF sin θ. (6.6)

34 LAB 6. ANGULAR ACCELERATION OF A DISK

In this experiment the torque is supplied by the tension in the cord from which the mass issuspended (see Fig. 6.1). The torque can be calculated as

τ = rT sin 90o = rT, (6.7)

and is considered to be a vector pointing into the paper.

Combining Eqs. 6.1 and 6.7, we obtain

T = Iα

r(6.8)

Applying Newton’s second law to the linear motion of the falling weight we can write

T = mg −ma (6.9)

Eliminating T in Eqs. 6.8 and 6.9 yields

mgr −mr2α = Iα (6.10)

This equation is linear, so a plot of mgr−mr2α vs α should yield a straight line with slopeI, assuming the frictional torque is constant. In order to find α, we will use stopwatches tomeasure the time required for the mass to fall a certain distance from rest. Using kinematics,this time and distance are related to the acceleration of the falling mass by the equation

d = 12at

2. (6.11)

Since a can be related to α using Eq. 6.5, we can substitute for a in Eq. 6.11 and find that

α = 2drt2

. (6.12)

Procedure

There are 2 different types of apparatus used for this experiment, the wall-mounted appara-tus at the front of the lab, and a number of similar desktop apparati which must be placednear the forced air outlets. The procedure for each apparatus is slightly different, consultthe appropriate section below in order to obtain your data. Note that you are only requiredto take data for ONE of the two types of apparatus, not both. Each apparatus requires 3to 5 people, so you should team up with another group to take your data.

Wall-Mounted Apparatus

1. Using a metre stick, measure the distance between the ground and the bottom of themass hanger as it rests on the platform provided.

PROCEDURE 35

2. Record the masses of the hanger, the disk, and the hub. These are printed on theapparatus.

3. Using calipers, measure the inner radius of the hub, the outer radius of the hub (wherethe string in actually wrapped around, NOT the lip), and the radius of the disk itself.Note that it is generally much easier to measure diameter with a caliper, so this isprobably the most suitable technique. The disk itself is too large for this method,however; you’ll most likely have to simply do your best to measure the radius of thedisk with the calipers.

4. Place an amount of mass between 20g and 200g on the mass hanger as it rests on theplatform.

5. Arm 3 group members with stopwatches, and let the other be in charge of droppingthe mass. Use the lever or the string attached to the lever to remove the platformfrom beneath the hanger, and time the hangers descent using the stopwatches.

6. Reset the apparatus and repeat the previous step for the same mass. This should giveyou 6 times per mass.

7. Repeat the previous 3 steps for 5 additional different masses between 20g and 200g(6 masses total).

Desktop Apparatus

1. First ensure that your apparatus is connected to the forced air supply. Your labinstructor can assist you if this is not the case. These first few steps should be donewith the air supply off.

2. Remove the hub by first removing the central screw. Measure the radius of the hubusing calipers, and the mass using one of the scales. Note that there is a lip on thehub around the outer radius; be sure to measure the radius to the inner surface, notthe lip.

3. Remove the top disk and measure its radius and mass in a similar manner. Reassemblethe apparatus after you are done, including the central screw. Note that you do notneed to make any measurement for the lower disk, as this disk will spin independentof the top disk, and thus is not part of the experiment.

4. Tie a piece of thread around the central screw and lead the thread through the grooveon the hub. Wrap the thread around the hub a few times, then lead the thread overthe pulley attached to the apparatus and to the ground. Cut the thread so there itwill easily reach the ground, with at least a foot to spare.

5. Obtain and weigh a mass hanger, then attach the hanger to the end of the thread.

6. Check to see if the other group has completed their measurements and is on this samestep, if not, then simply wait until they are.

36 LAB 6. ANGULAR ACCELERATION OF A DISK

7. Turn on the air supply so that the pressure valve reads between 9 and 12 PSI, thenwind the thread around the hub by rotating the disk with your hand until the top ofthe mass hanger is just below the pulley, and hold the disk in place.

8. Using a metre stick, measure the distance between the ground and the bottom of themass hanger as it hangs below the pulley. This height can technically be any distanceyou choose, but you must place the hanger at this same height with each trial, somake sure that this height is repeatable.

9. Place an amount of mass between 10g and 100g on the mass hanger while you holdthe disk still.

10. Arm 3 group members with stopwatches, and let the other be in charge of dropping themass. Simply let the disk spin by removing your hand, and time the hangers descentusing the stopwatches. As soon as the hanger hits the ground, stop the disk fromspinning with your hand. If you don’t, the hanger will lurch upwards after reachingthe end of the thread, which may cause your thread to break.

11. Reset the apparatus by spinning the disk to raise the hanger and repeat the previousstep for the same mass. This should give you 6 times per mass.

12. Repeat the previous 3 steps for 5 additional different masses between 10g and 100g(6 masses total).

Analysis

Construct a table similar to the following, using equations 6.10 and 6.12 to calculate valuesfor τ and α. Note that the equations for both of these variables contain r, which in all casesrefers to the outer radius of the hub (where the string is actually in contact with the hub).Also be sure to include the mass of the hanger in your mass column!

Mass Time(s) α τ(kg) t1 t2 t3 t4 t5 t6 t̄ σt (rad/s2) (N·m)

Plot a graph of τ vs. α. Determine the slope of this graph, and label this value as theexperimental value of the moment of inertia (Iexp).

The theoretical moment of inertia Ith can be calculated using your measurements of themass and radius of the disk/hub system and Equations 6.3 and 6.4. The apparatus consistsof a ring attached to a disk, both of which share the same axis of rotation. When objectsare attached in such a way, their combined moment of inertia is simply the sum of theindividual moments of inertia of each object. By combining equations 6.3 and 6.4 in sucha manner, it can be shown that

Ith = MdR2d +Mh(R2

h + r2h)

2 , (6.13)

DISCUSSION 37

where

Md = mass of diskRd = radius of diskMh = mass of hubRh = outer radius of hubrh = inner radius of hub.

Note that if using the desktop apparatus, the value for rh is 0.

Discussion

Compare Iexp with Ith, and record the percentage difference between the 2. Do these twovalues agree within reason? Given that the equation for Ith given above is in fact foundby summing the moment of inertia of the ring and disk, do your results appear to confirmthat the moment of inertia of a complex body can be found by summing the moments ofthe individual parts?

On your graph of τ vs. α, does the line pass through the origin of your plot? In otherwords, is your y-intercept equal to zero? Should it be? Note that when we were discussingthe rotational form of Newton’s 2nd Law, we said that τ = Iα, but really this should beΣτ = Iα, just like the translational version. We only considered one torque in our analysis,that created by the tension in the string; is this correct? If another torque were present,how would it affect this graph?

LAB 7

Principles of Mechanics

Objective

The objective of this experiment is to illustrate several principles of mechanics, primarilyconcerned with conservation of energy, using a simple collision as an example.

Apparatus

• Ramp

• Clamp

• Ball Bearing, or Mouse Ball

• Newsprint

• Meter Stick(s)

• 2 small pieces of Carbon Paper

Background

This simple experiment provides an opportunity to apply a variety of mechanical principles.

We will roll a ball down an incline and examine its path as it flies, strikes the floor, andbounces. With a few simple distance measurements we should be able to examine energy,projectile motion, kinematics, and collisions.

38

PROCEDURE 39

Figure 7.1: A simple experiment to test kinematics and energy, with all the appropriatemeasurements.

Procedure

Figure 7.1 depicts the experimental set-up and measurements to be made.

1. Place the ramp on a lab bench with the end of the ramp parallel to the floor (shimthe stand if necessary), and clamp it in place.

2. Measure the heights h and H and record them. Refer to Fig. 7.1 to see exactly whatthese heights represent.

3. Measure the mass (m) of the ball using the scales at the back of the lab.

4. Roll the ball bearing down the ramp, allow it to hit the floor and bounce. Place 1 or2 sheets of newsprint on the floor (tape them down) such that the bearing will strikethe paper on impact and first bounce (points C and D in Figure 7.1). In some cases,only 1 sheet is required, in most others 2; you’ll simply have to experiment in orderto determine if 1 sheet will be adequate.

5. Place a small piece of carbon paper face down at each point of impact. These do notneed to be taped down.

6. Place the ball on the ramp and release it. The ball should strike the back of eachpiece of carbon paper and therefore leave a mark on your newsprint. Verify that thisdoes indeed occur, if not, then repeat the trial.

7. Repeat the experiment 5 more times, making sure that the ball is released from thesame point on the ramp each time. Measure the horizontal distance to each pointusing a metre stick (c and d in Fig. 7.1), and record the distances.

40 LAB 7. PRINCIPLES OF MECHANICS

8. The height of the bounce (e) can then be found by placing a piece of paper mounted ona book mid-way between points C and D. Place the leading edge of the book exactlyat the midway point, place a piece of carbon paper on the surface of the book (a pieceof tape may be useful in this case), and do 6 more trials. The ball will leave a markon the paper at the maximum height. Measure and record these heights.

Analysis

Using kinematics, calculate the following quantities:

1. The time required for the ball to strike the ground after leaving the ramp. Hint: Whatis the vertical velocity of the ball as it leaves the ramp?

2. The speed of the ball as it leaves the ramp (point B). Note that this speed is thehorizontal velocity of the ball, which can be calculated using the time from question1 and the distance the ball travelled before reaching the ground (c).

3. The vertical velocity of the ball as it strikes the ground (point C).

4. The speed at point C.

5. The speed at point D. By examining the path of the ball from the maximum heightof the bounce to point D, this problem is solved in an identical manner to questions1-4.

Using these values for the speeds, construct a table that lists the kinetic, potential, andtotal mechanical energy at points A, B, C, and D. Your table should resemble the following:

Height Speed KE PE Mech EnergyABCD

Calculate the speed of the ball at point C using energy considerations. Does it agree withthe result you found using kinematic methods?

Discussion

Is the mechanical energy in this experiment conserved? If not, where is it lost (between Aand B, B and C, etc.)? Is the collision at point C elastic or inelastic?

DISCUSSION 41

To this point we have not considered the energy of rotation as the ball rolls down the ramp.The kinetic energy of rotation (Krot ) can be calculated as,

Krot = 12Iω

2,

where I is the moment of inertia ( I = 25mr

2 for a solid sphere), and ω = vr is the angular

velocity. By substituting these expressions for I and ω into the equation for Krot, findan equation for Krot that is a function of mass and speed. Compare this equation to theusual kinetic energy equation. Using the speed as the ball leaves the ramp as v, calculatethis rotational energy. Now add this number to the energies at points B and C. Doesthe inclusion of the energy of rotation in your calculation reconcile the differences in themechanical energies at A, B, and C?

LAB 8

The Physics of Bicycles

Objective

The objective of this experiment is to examine how work, power, input force, and speed areaffected by the gear system of a bike.

Apparatus

• Multi-geared bicycle

• 20 N Spring Balance

• Ruler

• Weights

• Stopwatch

Background

Simple machines such as levers and pulleys have been used by societies for centuries, longbefore the postulation of Newton’s Laws. Engineers knew of the relationships between theinput and output forces of such machines, even though they had no formal definition of whata force was. Today, Newton’s Laws provide a very useful tool for analysing such machines,while engineers use these laws to design and construct all kinds of machinery. In additionto the technological aspects of simple machines, it should be noted that the human bodyconsists essentially of a variety of levers and pulleys that allow it to move as it does. Studyof the mechanics of the body itself can therefore be undertaken through the use of somesimple applications of Newton’s Laws.

42

BACKGROUND 43

(a) (b)

Figure 8.1: A simple lever. By balancing the torque on each side in (a), it can be shownthat Fin/Fout = r2/r1. In this example, the input force would be larger than the outputforce, but a speed gain would be achieved as seen in (b). The distances d1 and d2 would betravelled in the same time since the bar is rigid. Since d2 > d1, v2 > v1.

A simple example to consider is that of the basic lever. By examining the torques in Fig. 8.1,it can be shown that Fin r1 = Fout r2 if the system is in rotational equilibrium. Rearrangingthis equation gives

FoutFin

= r1r2, (8.1)

where Fin represents the input force, and Fout the resistive or output force, in this case theweight of the object to be lifted. The ratio Fout/Fin is known as the mechanical advantage,and represents the factor by which the input force is multiplied due to the machine. Manytypes of machines are designed to maximize this number, indicative of a work machine thatattempts to lift heavy loads. A system of pulleys designed to lift a piano, for example,has a high mechanical advantage. Other forms of machine actually work to minimise thisnumber, resulting in a system that requires a great deal of force yet provides a speedadvantage. Medieval siege equipment such as catapults and trebuchets are based on thistype of design, as are the joints of the human body. Fig. 8.1b shows this speed increase, asit depicts a situation with a mechanical advantage of less than 1.

The system we will choose to examine within this lab to understand the effects of simplemachines is that of the bicycle. The gear system found on many bicycles provides an idealopportunity to study simple machines not only in terms of Newton’s Laws, but also work-energy relations, conservation theorems, and other physical principles that you have learnedduring the semester. A gear system is yet another type of simple machine.

The system present on a bike is depicted in Fig. 8.2. A force Fp is applied to the pedal atradius rp, which causes the pedal crank and attached front sprocket to rotate at a rate ωf .Note that since these two objects are fused together, they represent one body which mustrotate at the same rate. The force that the front sprocket applies to the chain depends uponthe radius of the sprocket. Similar to the rotation rate, the torque on the pedal crank andthe front sprocket must be the same. If Fc is the force on the chain, and rf is the radius ofthe front sprocket, then

Fprp = Fcrf . (8.2)

Similarly, the torque on the rear sprocket must be the same as the torque on the rear wheel,

44 LAB 8. THE PHYSICS OF BICYCLES

Figure 8.2: Depiction of the chain-gear system of a bicycle. By finding the torque on thefront and rear systems and relating the two using the force on the chain (which is equal inboth cases), a relation between Fp and Ft can be found.

since they are also fixed. This implies that

Ftrt = Fcrb, (8.3)

where Ft and rt are the force and radius of the rear tire, respectively, and rb is the radiusof the rear (back) sprocket. Fc is again the force on the chain, which neglecting the weightof the chain, should be equal in both equations. Combining these relationships gives therelation between Fp and Ft. A similar relation can be found for the speed of the bike fora given rotation rate of the pedals. For the front sprocket/pedal, vc = ωfrf , where vc isthe linear velocity of the chain, while for the rear sprocket/wheel, vc = ωbrb. By combiningthese equations and using the fact that ωbrt = vbike, we can relate vbike to ωf (the pedalingrate).

It is interesting to note that the physics of bicycles is still an area of active research. Manytheories exist as to the reasons for the stability of the bicycle and therefore its ease of ride.Other studies focus on the maximum efficiency at different work rates or cadences (pedalingrates). This information is of tantamount importance to professional cyclists.

There are one or two important pieces of information we will need for this experiment. Theresults of physiological testing reveal that the average bicycle rider can comfortably exertone fifth their body weight as force on the pedals of a bicycle, and that a pedaling rate of1 to 1.33 revolutions per second (60-80 rpm) is the most efficient or optimal rate for lowerpower outputs.

PROCEDURE 45

Procedure

There are a limited amount of bicycles, so up to 3 groups can work on a single bike. Notethat Part IV is to be done individually by group, so while 2 groups collect data for Parts I-III,the 3rd should collect their data for Part IV. When this 3rd group is finished with collectingthe Part IV data, another group should begin collecting this data while the returning grouphelps collect the remaining data for the other parts. The data for Parts I-III is to be sharedbetween all groups on a given bike, while the Part IV data is not. Before collecting data,turn your bicycle upside-down so that it rests on the seat and the handlebars.

Part I

Gather the basic information about your bicycle:

1. Count and record the number of teeth on each sprocket.

2. Count and record the number of spokes on the rear tire.

3. Measure the length of the pedal crank, from the centre of the axis to the centre of thepedal.

4. Measure the radius of the rear wheel from axle to tire contact surface.

Part II

In this section we will examine the distance travelled by the bike in each gear for onecomplete revolution of the pedals. This can be accomplished in the following manner:

1. Choose a front sprocket to work with and change gears until the chain is on thatsprocket. Record the number of teeth on this sprocket.

2. Mark a spoke on the wheel with masking tape, near the edge of the wheel.

3. Record the number of teeth on the current rear sprocket.

4. Align the marked spoke with some fixed part of the bike, i.e. a mudguard, a piece ofthe frame. Do a similar alignment for the pedal.

5. Have one group member push the front pedal with their hand while keeping the bikestable. Ensure that the rear wheel is not allowed to move freely by applying a smallamount of force opposite to the direction of motion. Simply gripping the tire lightlybetween 2 fingers should suffice. Push the pedal for one complete revolution, countingthe number of times your marked spoke passes your start point.

46 LAB 8. THE PHYSICS OF BICYCLES

6. It is likely that the rear tire rotated through an angle that is not an integer numberof revolutions (for example, 2.37 revolutions). In order to determine this fractionof a revolution, count the number of spokes between your fixed starting point andthe current location of the marked spoke. Divide this number by the total numberof spokes on the wheel. This will give the fraction required. For example, if youcount 20 total spokes on the tire, and count 5 spokes between your starting point andthe current location of the marked spoke, then the tire has experienced 5/20 = 0.25revolutions. Record this number of revolutions.

7. Repeat this procedure (from Step 3 onward) for each rear sprocket.

Part III

In this section we will examine the forces involved with the bike and therefore the mechanicaladvantage. Measure the force exerted by the rear wheel of the bicycle on the ground ineach gear. This can be done using a spring balance, 3 kg of weights, and some string.

1. Ensure that the front sprocket is the same used for Part II, and record the number ofteeth on the current rear sprocket.

2. Tie the string around the rear tire so that the string hangs from the outer edge. Tiea loop in the free end of the string, and hook a spring balance on to the loop.

3. Adjust the front pedal until it is level with the ground (you may use a level if youwish). Of the 2 front pedals, test which one will cause the rear wheel to rotate ina clockwise direction when pushed towards the ground. Hang a 3 kg mass from thispedal.

4. Pull upwards on the spring balance to oppose the motion caused by the hanging mass,ensuring that you are pulling tangential to the surface of the tire. Record the forceshown on the balance.

5. Repeat for each rear sprocket. Note that you may have to untie and retie the stringwhen you change gears if the string happens to catch on the braking pads. To avoidthis, you can unclip the braking mechanism and allow the brake pads to rest far awayfrom the surface of the wheel. Your instructor can assist you if needed.

Part IV

In this section we will examine power output of a human being and attempt to apply thisdata to a biking situation. Each group on an apparatus should perform this step separatelyand record their own data, there is no need to share. You will need a stopwatch and a rulerfor this section.

ANALYSIS 47

1. Locate a staircase such as those in E section. You will be travelling up several flightsof stairs, you may choose any number of flights between 2 and 6.

2. Determine the total vertical rise of the stairs by measuring the height of one step andmultiplying by the number of steps. You only have to calculate the total vertical dis-tance covered, do not include any horizontal distance from travelling across a landingbetween flights.

3. Using a stopwatch, have one group member measure the time required for the otherto walk SLOWLY up the stairs (the climber may time themself, if preferred). Tryand maintain a pace that you think would be sustainable for a long period of time; inother words, ask yorself if you could maintain this pace for 15 minutes. If you haveto stop midway to catch your breath, do NOT stop the stopwatch, just let it run.

4. Repeat the previous step, this time walking up the stairs at a quicker pace.

5. Swap roles with your group member and repeat the procedure.

Analysis

From your data from section I, find the gear ratios for your bicycle. This ratio is found bydividing the number of teeth on each rear sprocket by the number of teeth on each frontsprocket (not just the one you used in part II/III). Present your results in a simple table.

From your data for parts II and III, find the torque at the front pedal τf created by the 3kg weight. Also calculate the theoretical work done at the front pedal due to the weight forone complete revolution of the pedal. Recall that W = Fd = τθ, where θ is in radians (1revolution = 2π radians).

We wish to examine the amount of work done at the rear tire and compare it to the workdone at the pedal. This can be accomplished using a table. Construct a table like the onebelow to find the work done at the rear tire for each gear.

Rear Sprocket Revolutions Ft Torque Work rb/rf τb/τfr1 θ1 F1 τ1 = rtire × F1 W1 = τ1 × θ1 r1/rf τ1/τf...

......

......

......

Note that the r values are from Part I, and can be given as the number of teeth (the radius isproportional to the number of teeth). The θ values are from Part II, and the F values fromPart III. Note again that in order to find work using W1 = τ1 × θ1, θ1 must be convertedto radians. This should be simple enough to do within your spreadsheet equation.

Calculate the work required to climb the stairs in Part IV, for each group member. Alsofind the power developed (in Watts) for both the walking and running cases.

48 LAB 8. THE PHYSICS OF BICYCLES

Using these power values, and some limited knowledge of the optimal pedalling rate andforce output, we will find the optimal velocity for a given power output. We will assumethat your slow walking power is sustainable for a long period of time, hence we will use thisvalue as the power being applied to the pedals. Your results will be presented in a table asfollows:

# of Teeth Force Required Power Output SpeedFront Back

Solve equations 8.2 and 8.3 for Fp, the force at the pedal, in terms of Ft, the force at therear tire. For each gear, use this equation to find the Fp required to balance a 30 N resistiveforce at the surface of the rear tire (Ft). This magnitude of resistance corresponds to aslight uphill slope coupled with a small amount of rolling resistance (friction). Note thatthe force values in this table have nothing to do with your Part III results.

Assume a constant pedalling rate of 1 revolution/second. Using this rate, find the poweroutput for each gear using the force you found in the previous step. Note that P = W/t =Fd/t, where d is the distance travelled in time t. The pedalling rate should give youinformation on what values to use for d and t. From the information in the backgroundsection, find an equation that relates the speed of the bike vbike to the pedalling rate ωp(note here that ω should given in radians/sec, not revolutions/sec; you should be able toconvert between the two). By looking at your power values and comparing them to yourpower output from the stairs, determine which gear is the optimal gear for this amountof power and resistive force. Does the speed in this gear seem right from your experienceriding a bike?

As an interesting comparison data from the final stage of the 1989 Tour de France showsthat Greg Lemond averaged 34 MPH with a 5 MPH tailwind. This corresponds to anaverage power output of 513 Watts.

Discussion

Compare the gear ratios to the ratio of the torques at the front and rear sprocket in eachgear (last 2 columns in the first table). Is there a relationship? What exactly does thistell you about the relation between the gear ratio and the mechanical advantage? Alsocomment on the gear ratios themselves; are they greater than or less than 1? Do this implythat your bike is built for speed, or power?

Compare the work values from your table to the work done by the 3 kg weight on the frontpedal for one complete revolution. Are they similar? Why should they be? Explain whatthis tells you about machines in general, and what exactly a machine does.

Use the equation P = Fvbike, where F is the resistive force and P is your input power,to find the velocity for one of the power outputs you found in your table. How does this

DISCUSSION 49

compare with the velocity from the equation you used in the table? Try a few more valuesand see how they compare. If the resistive force is fixed, how do you change velocity? Doesthis have anything to do with the gear ratio or mechanical advantage? Does this make sensebased on your answers from the previous paragraph? What is the point of the gears, then?

LAB 9

The Speed of Sound

Objective

The objective of this experiment is to determine the speed of sound in air by using theprinciples of resonance and standing waves.

Apparatus

• Function Generator

• Oscilloscope

• BNC Cable

• Wires, with alligator clips

• Microphone

• Resonance Tube w/ 2 ends, one containing a speaker

Background

Waves

The repetitive motion of waves breaking on a beach is probably the most familiar conceptwe have of waves. This familiar concept can be extended to represent many processes thatoccur in nature. Physical terms such as frequency (f in cycles/sec, or Hz), wavelength (λin cm or m), phase (φ in radians), and amplitude (magnitude of process represented by the

50

BACKGROUND 51

Figure 9.1: Depiction of a typical transverse wave, showing the characteristics associatedwith the wave, such as amplitude and wavelength (λ).

wave) can be employed to quantify and measure properties in other cyclical processes whichbehave like the conceptually familiar water waves.

Terms such as node and antinode are also used to identify particular positions on the wave.Nodes represent points on the wave where the value of the process depicted by the waveis at a minimum. Conversely, antinodes are points depicting the maximum ( positive ornegative) value of the process depicted by the wave. For example if the wave shown aboverepresented 110 volt household AC electricity, then the amplitude would represent voltage(maximum value of about 155 volts at the antinodes). The frequency of the wave would be60 Hz and the value of the voltage at the nodes would be 0 volts.

Waves propagate from one place to another at a speed which can be defined as the frequencyof the wave multiplied by the wavelength of the wave, v = fλ.

Sound Waves

Sound waves are longitudinal waves, meaning that the direction of the ’oscillation’ is par-allel to the direction of propagation of the wave. The speed at which the wave travels(propagation velocity) is dependent on the medium through which the wave is travelling.In a simple sense the stiffer the medium the faster the speed of sound.

One indicator of ’stiffness’ in materials or media is the bulk modulus (B). When combinedwith the density of the medium (ρ), the speed of sound (v) may be calculated as:

v =√B

ρ(9.1)

52 LAB 9. THE SPEED OF SOUND

Typical values of B and ρ :

Material Bulk Modulus (Pa) Density (kg/m3)Water 0.21× 1010 1.0× 103

Steel 20× 1010 7.8× 103

Aluminum 7× 1010 2.7× 103

Table 9.1: Physical properties of water, steel, and aluminum.

Propagation of sound in a gas is a slightly more involved subject. Ignoring the details, theequation for calculating the speed of sound in a gas is:

v =

√γRT

M(9.2)

where:

γ = a constant particular to individual gasesR = the gas constant (8.314 J

mol·K)T = the absolute temperature (K)M = the molecular mass of the gas (kg/mol)

Typical values of γ and M :

Gas γ M (kg/mol)Air 1.40 0.0288

Helium 1.67 0.0040CO2 1.30 0.0280

Table 9.2: Various properties of gases.

If one considers the speed of sound in air then factors such as altitude, humidity, andpressure also enter into the calculation for the speed of sound.

A sound wave consists of alternating pressure changes called compressions and rarefactions.As the wave moves it compresses the air to a pressure greater than atmospheric pressure.Between successive compressions there is a region of low pressure called a rarefaction. Thecompressions and rarefactions occur along the path the wave is travelling.

Audible sounds travelling through air will have wavelengths ranging from approximately 3to 300 cm. Wavelengths of this size are easy to study. We will employ sound resonance ina tube to determine the wavelength of the sound waves (λ), determine the frequency (f)of the sound wave with an oscilloscope frequency, consequently calculating the speed of thewave as the product of f and λ .

BACKGROUND 53

Superposition

Observations of waves in nature reveal that waves interact with each other in special ways. Ifyou throw two stones into a pond for example, and watch the wave pattern created by each,you will observe that the patterns interact with each other in passing, but the individualpatterns remain intact and continue to propagate after the interaction.

These observations are indicative of the principle of superposition. Multiple waves mayinteract with each other without losing their individual character. The interaction is gen-erally referred to as interference. For our purposes the interaction of sound waves in thisexperiment will be considered to be linear.

Superposition of sound waves moving through air can be very complicated. The air moleculesare already moving in a random motion. As a sound wave(s) travels it imposes an orderedvibrational motion on the already moving molecules. The motion of a molecule at anyinstant will be a combination of random motion and the oscillations caused by any numberof sound waves present at that instant.

Resonance

The principle of superposition leads to a phenomenon known as resonance. In this experi-ment sound resonance occurs when the wavelength of the sound "fits" the cavity in whichit is travelling. Consider a long tube closed at one end. A sound wave enters the open endand travels along the tube until it hits the closed end. Upon striking the closed end thesound wave reverses direction (reflection) and travels back down the tube toward the openend. When a wave is reflected in this manner there is a 180o phase change in the wave; thisis a very important factor for resonance.

Figure 9.2: Two examples of standing waves in a tube with one closed end and one openend. The top wave has a wavelength of L/4, where L is the length of the tube, while thebottom wave has a wavelength of 3L/4.

54 LAB 9. THE SPEED OF SOUND

Figure 9.3: Resonace tube setup

By adjusting the frequency, and therefore the wavelength of the sound, it is possible tocreate standing waves in the tube. As you can see in Fig. 9.2, if a node occurs at the closedend of the tube then the reflected wave is 180o out of phase with the incoming wave. If thewavelength is such that an antinode is located at the mouth of the tube then we have theproper conditions for a standing wave. Each wave travelling in the tube will reflect backupon itself without altering the positions of the nodes and antinodes.

The conditions for a standing wave are also the conditions for resonance. The resultinginteraction of the sound waves in the tube is the addition of their respective amplitudes(sound) thereby making a much louder sound than that of any single wave in the tube.

You can also see from Fig. 9.2 that only odd multiples of λ/4 can produce resonance in thetube and distance between successive resonances for a tube of length L will be λ/2.

Procedure

Setup

Assemble your apparatus as directed by the instructor using the set-up at the front of thelab as a guide. Your instructor will assist you with any difficulties and demonstrate how touse the oscilloscope and function generator.

The experiment will comprise three trials at different frequencies. For each trial, start withthe moveable piston at the zero mark on the scale. With the microphone located next tothe speaker slowly move the piston away from the speaker and observe the oscilloscopescreen. The wave on the oscilloscope screen is the signal from the microphone. Its heightrepresents the magnitude of the sound inside the tube. The height of the wave will rise andfall depending on the length of the tube. When the tube is at a length that corresponds toa resonance for the frequency in question the wave will have its maximum height. Recordeach length for which a maximum occurs.

ANALYSIS 55

Since this is a closed tube each resonance occurs at nλ/4, where n = 1,3,5,7,... . Thechange in the measured distances is half a wavelength. The frequency of the signal can bemeasured with the oscilloscope.

Pick three different frequencies in the range of 1000 to 3000 Hz. For each frequency alterthe length of the tube using the moveable piston. Each time the tube comes into resonancerecord the length (L) of the tube by noting the position of the piston on the scale.

Analysis

Construct a table like the following:

Frequency Length Wavelength Speedf1 L1 λ1 v1

L2 λ2 v2...

......

f2 L1 λ1 v1L2 λ2 v2...

......

To find the wavelength, note that the difference between lengths will always be λ/2. Ex-pressing this as an equation and rearranging for λ gives λ = 2∆L. Note that the value for λshould be approximately the same for each frequency. The speed is found by using v = λf .Report the speed of sound for each trial as the average of your calculations ± the standarddeviation.

The speed of sound is in fact a function of many variables, as discussed in the backgroundsection. One of these variables is temperature. Using the information from the backgroundsection of the lab (specifically, table 9.2 and equation 9.2), plot a graph of the speed of soundin air from -40oC to +40oC. To do this, construct a simple table in a spreadsheet consistingof a variety of temperatures within this range and the speed at these temperatures. Plotthis data as you normally would. You will need to convert your temperatures to Kelvin inorder to use the equation provided, but note that the x-axis of your graph must be in oC.

Using the information from the background section (specifically, equation 9.1), calculatethe speed of sound in steel and water. How many times faster does sound travel in steelthan in water? In water than in air (at 20°C)?

56 LAB 9. THE SPEED OF SOUND

Discussion

The ambient temperature in the lab is a little over 20oC. Calculate the speed of sound inair using this temperature and compare to your results. Do they agree? Which of yourfrequencies actually results in the best data? Is there any pattern in your results, such ashigher frequencies giving better results?

In practice the diameter of the tube used in the experiment also has an effect on thecalculated speed of sound. The actual wavelength of the sound is calculated as L+ 0.4d =nλ/4, where L is again the measured length of the tube and d is the diameter of the tube.If this equation was used instead of the one provided in the manual (L = nλ/4), whateffect would this have on your results? Specifically, would the results for the speed of soundincrease, decrease, or remain the same? Justify your answer.

LAB 10

Nuclear Half-life

Objective

To determine the half-life of 137Ba.

WARNING: No food or refreshments are allowed in the lab during this exper-iment.

Apparatus

• Ba-137 Liquid Sample

• Geiger Tube

Background

Most elements present in nature exist in more than one nuclear configuration. The slightlydifferent configurations of an element are known as isotopes. Consider neon which hasseveral different natural forms:

2010Ne 21

10Ne 2210Ne

The atomic number (10) identifies each as being the element neon, while the mass numberdenotes differing numbers of neutrons in the respective nuclei (20, 21, 22). Other isotopesof neon exist, i.e.: 18

10Ne 1910Ne 24

10Ne. These have half-lives ranging from 1.5 sec to 3.38 min.

Unstable isotopes rearrange themselves into stable configurations primarily through nuclearemission. In this experiment we will study some of the emission characteristics of various

57

58 LAB 10. NUCLEAR HALF-LIFE

isotopes, shielding properties of various materials, and examine different detection methods.

Types of Nuclear Radiation

We are interested in three types of radioactive nuclear emission:

Alpha (α) These are positive particles which are identical to helium nuclei except for theirnuclear origin. They travel about 0.1 the speed of light and possess great ionizingpower because of their relatively slow speed and great charge. All the alpha particlesemitted from a given species of nucleus are mono-energetic (identical in energy). Alphaparticles may be stopped by a thin sheet of paper.

Beta (β) Unlike alpha particles, they are negatively charged and are identical to the elec-tron except for their nuclear origin. Commonly encountered beta particles travel up to0.9 the speed of light. Unlike alpha particles, beta particles are emitted in a wide rangeof energies, hence the variation in their speed. Beta particles are more penetratingthan alpha particles but may be stopped by several thin sheets of metal.

Gamma (γ) Gamma emission is similar to x-rays and light rays but have a shorter wavelength. These rays are emitted because of energy transitions within the nucleus itself,while x-rays are emitted because of energy transitions in orbital electrons. Sincegamma rays have no charge, they do not react with matter in the same manner asalpha and beta particles. They travel at the speed of light and have great penetratingpower.

Nuclear Emission

While it is impossible to predict which nucleus in a collection of radioactive nuclei will decaynext, it is possible to establish the rate at which the collection of nuclei decays. It can beshown that a collection of N radioactive nuclei will decay at a rate proportional to thenumber of nuclei yet to undergo emission in the sample. The rate of decay is characteristicof the isotope in question and can be expressed either as the half-life (T1/2) or the mean life(τ) of the isotope.

Mathematically the rate of decay (R) is expressed as,

R = λN (10.1)

where N is the number of nuclei yet to undergo decay and λ is the decay constant char-acteristic of a particular radionuclide. The rate of decay as a function of time follows anexponential pattern expressed as

R = R0 e−λt, (10.2)

where R0 is the rate of decay at t = 0.

BACKGROUND 59

Figure 10.1: Radioactive activity (R) of a decaying species as a function of time.

By taking the natural log of both sides of equation 10.2 a linear relationship results whichcan be used to determine λ.

lnR = −λt+ lnR0 (10.3)

λ can therefore be found by calculating the slope of a plot of ln R vs. t.

The rate at which a sample decays is called the activity and carries the SI unit of becquerel(Bq). One becquerel is one decay per second. Another unit of nuclear decay rate is thecurie (Ci). One curie is 3.7× 1010 Bq. When it is not clear whether a detector is recordingall the disintegrations from a sample it is common practice to use units of counts per unittime.

Nuclear Half-Life

The half-life (T1/2) of a radionuclide is a measure of how long a particular isotope willcontinue to decay. The value of the half-life of an isotope is the time required for the initialrate of decay (which can be any value) to drop to one half that rate.

The half-life of a radionuclide is calculated by relating the decay constant to the time forone half of the nuclei in a sample to disintegrate.

12R0 = R0 e

−λT1/2 (10.4)

Taking the natural logs of each side of this equation yields an expression for the half-life ofa radionuclide.

T1/2 = ln 2λ

(10.5)

60 LAB 10. NUCLEAR HALF-LIFE

Figure 10.2: The decay process of Cs-137. 2 emissions are possible, both γ and β. TheBa-137 compound which is leached from the sample decays only by γ emission, and with avery short half-life.

The barium isotope used in this experiment is generated from the beta decay of Cs-137 witha half-life of 30 years. At this point the Ba-137 nucleus is unstable and quickly undergoesa decay by gamma emission (662 KeV) to become stable. The half-life determined in thisexperiment is the half-life of the gamma decay to form stable Ba-137.

Procedure

You will be given a liquid sample of Ba-137 on a planchette. This isotope decays quickly soyou must be ready to work as soon as the sample is received.

1. Place the sample under the Geiger-Muller tube.

2. Set the DataStudio program to display readings every 30 seconds by clicking on‘Setup’, then selecting the sensor in the setup window. There should be a windowshown below the sensor where you can set the amount of time between each reading,simply type in 30.

3. Hit ‘Start’. A table should be displayed that gives the number of counts every 30seconds. Let the program run to 600 seconds then click ‘Stop’. Copy this data into aspreadsheet.

ANALYSIS 61

Analysis

Construct a data table containing the number of counts within each 30 second intervaland the time. The first time entry in this case should be 30 sec, with the counts thenrepresenting the number of counts registered after 30 seconds. Create an additional columnfor the natural log of the counts.

Construct 2 plots from your data, the number of counts vs. time, and the natural log ofthe counts vs. time. Perform linear regression on your log plot. The slope of this line is thedecay constant (λ) for this isotope. Using equation 10.5, find the half-life of Ba-137.

Discussion

Look up the half-life of Ba-137. Do your results agree? As discussed in the backgroundsection, 2 types of decay are possible from Cs-137, both γ and β, each with their ownhalf-life. The Geiger-Muller tube will detect both kinds, but is much better at detecting βradiation. If some Cs-137 happened to be introduced into your sample, what effect, if any,would this have on your results?

The exponential decay relationship is ubiquitous in the sciences, but it can also be used todescribe many different kinds of real life processes that occur outside science. In general, thisrelation describes any process where the rate of the process is proportional to the currentamount of the reactant, and once a particular reactant has participated in the process, itcan no longer participate. Can you think of another process outside of physics that wouldexhibit this behaviour?

APPENDIX A

Reading Vernier Scales

A vernier scale consists of a stationary scale (main scale) and a sliding scale (vernier scale).The divisions on the vernier scale are smaller than those on the stationary scale. N divisionsof the vernier scale equal N − 1 divisions of the main scale, (see Fig. A.1, where N = 10)

Figure A.1: Divisions on a Vernier scale.

In Fig. A.1, it may be seen that both the zero and the 10 mark on the vernier coincide withsome mark on the main scale. The first mark beyond the zero on the vernier is 1/10 of amain scale division short of coinciding with the first line to the right of the main scale zero.This difference between the lengths of the smallest divisions on the two scales represents theleast amount of movement that can be made and read accurately. This amount is called theleast count of the instrument. If the vernier scale is moved a distance 1/10 the distance ofthe difference between the divisions of the main scale, the next mark of the vernier coincideswith a mark on the the main scale. Only one mark on the vernier will align with a markon the main scale for a given measurement. See Fig. A.2.

62

63

Figure A.2: Reading a Vernier scale. Note how only a single pair of lines on the main scaleand sliding scale line up perfectly.

As you can see accurate measurements of 1/10ths of a main scale division are possible.Consider the scale in Fig. A.3 to have units of millimeters. In Fig. A.3a we see the zero ofthe vernier lies between the 6 and 7 mm marks and the second mark on the vernier coincideswith a mark on the main scale. The reading is therefore 6.2 mm. By similar reasoning thereading in Fig. A.3b is 3.7 mm.

(a) (b)

Figure A.3: Reading a Vernier scale. The left reading is 6.2mm, while the right is 3.7mm.

Nearly all verniers have N divisions of the vernier scale equal to N −1 divisions of the mainscale and the method of determining the reading is similar to that described above. Theleast count is always 1/N of the length of the smallest main scale divisions. In the exampleabove, the least count is 0.01 cm.

APPENDIX B

Error Analysis

Error analysis is the evaluation of precision in measurement. As such, the word "error"takes on the connotation of "uncertainty" rather than "mistake". Since no measurement canbe completely free of error it becomes the experimenter’s task to minimize all uncertaintiesas far as reasonably possible. In our experiments this will be a three-fold process:

• Minimization of uncertainties in experimental apparatus.

• Optimization of experimental methods and procedures

• Calculation of uncertainties in final results.

Every experiment should begin with a thorough assessment of the apparatus including:condition, resolution, precision, calibration, etc. The apparatus should then be assembled,and procedures developed that will provide the most accurate results possible. Finally,the precision of the outcome should be calculated. Following this procedure allows theexperimenter to assess the worth of the experiment even before it is actually done.

The experimenter must have a reasonable knowledge of the types of uncertainties that existand how they will propagate in a given experiment. The remainder of this section will dealwith those two subjects.

Types of Errors

Systematic Errors

A systematic error is one which repeats itself to the same degree in each trial of an exper-iment. As such it cannot be discovered simply by repeating a measurement several times.Due to their reproducibility, systematic errors are very difficult to trace.

Systematic errors usually involve some aspect of the experimenter’s apparatus or methodol-

64

SUMMARY 65

ogy (improperly calibrated equipment, poor experimental design, and/or procedure are themost likely places to find systematic errors). Consider an experiment to measure voltagein an electrical circuit. If the voltmeter used in the experiment reads 1.5 volts when novoltage is applied (calibration error), each reading of the voltage will be 1.5 volts too large.This is typical of the most common types of systematic errors.

Random Errors

Simply stated, random errors are errors which can be found by repeating a measurementseveral times. Random errors can be treated statistically (systematic errors cannot) andtherefore repeated measurement allows one to minimize the effect of random error in exper-iments. Random error arises from our inability to hold all the variables constant in additionto the random character inherent in nature.

Reading Errors

Reading error is associated with any directly measured quantity and reflects the limits ofthe devices used in the measurement process. For example, a ruler graduated in millimetreshas much less reading error than a ruler graduated in centimetres. The resolution of thescale plus the interpolation of the measurement between the finest graduations of the scaleconstitute the major contributors to reading error.

In the case of digital readings the error is considered to be ½ of the final digit. For example,if a display reads 1.09, and there is no other information, the reading would be consideredto be 1.09 ± .005.

Summary

All the factors discussed, systematic, random, and reading errors combine to yield theoverall error in a measurement. Experimenters strive to minimize error through carefuldesign and thoughtful execution of experiments.

Data gathered from experimental work must carry with it some indication of uncertainty.This estimate of uncertainty follows the data through any calculation or manipulation andis always quoted with the final results. The next section deals with the propagation of errorduring calculations. It is more involved than the previous discussion and thus demandsmore attention.

66 APPENDIX B. ERROR ANALYSIS

Propagation of Errors

Definitions

Absolute Error: The total uncertainty in a quantity, x; usually expressed as δx (i.e.x± δx).

Relative Error: The uncertainty in a quantity, x in fractional form. Usually expressed asδx/x, where δx/x is dimensionless. The relative error can be expressed as percentageerror by multiplying δx/x by 100%.

Example: An iron block is found to weigh 3.44 ± .03 kg.Absolute error δm = .03 kgRelative error δm/m = .03/3.44 = .01 = 1%

Possible Error: The maximum possible uncertainty in the outcome of a mathematic op-eration involving values and their uncertainties.

Example: Consider the addition of two values A and B.

If A is known to a precision of ±δA, and B to a precision of ±δB, the sum of the twovalues could be as large as (A+ δA) + (B + δB), or as small as (A− δA) + (B − δB).If C = A+B then the possible error in C, δC would be δC = δA+ δB.

Possible error assumes the most extreme values in the uncertainty combine in theworst possible combination. For example, if A = 10±1 and B = 20±2 then for A+Bthe largest possible value would be 11 + 22 = 33 while the smallest value would be9 + 18 = 27, in other words 30 ± 3. Using the rule δC = δA + δB we arrive at thesame value for error in C namely, δC = 1 + 2 = 3.

Probable Error: A more conservative estimate than possible error. Probable error as-sumes that quantities used in calculations do not combine in the worst possible fash-ion (applies to mutually exclusive events) but rather with a statistical probability.Estimates of probable error will be lower than those of possible error.

PROPAGATION OF ERRORS 67

Rules for the Propagation of Errors in Calculations

Addition / Subtraction: A+B = C, A−B = C, where A± δA,B ± δB

Possible Error: δC = δA+ δB

Probable Error: δC =√

(δA)2 + (δB)2

Multiplication / Division: A×B = C, A/B = C, where A± δA,B ± δB

Possible Error: δC/C = δA/A+ δB/B

Probable Error: δC/C =√

(δa/A)2 + (δb/B)2

Exponents: C = An , where A± δa

Error: δc/C = n δa/A

General Expression for Error Propagation

The relationship for error propagation can be expressed and manipulated as the total dif-ferential for a dependent variable which is a function of one or more independent variables.

Suppose we had a function f(x, y, z), its total differential would be expressed as

df = ∂f/∂x dx+ ∂f/∂y dy + ∂f/∂z dz,

if we exchange the differentials df, dx, dy, dz for the uncertainties δf, δx, δy, δz we havean expression which allows us to calculate error propagation for more than the simplearithmetic operations shown previously.

δf(x, y, z) =

√(∂f

∂xδx

)2+(∂f

∂yδy

)2+(∂f

∂zδz

)2