Julian Experiments

PHYS 200 Home Lab Manual - Version 2.3

Course Coordinator: Farook Al-ShamaliAuthors: Farook Al-Shamali and Martin ConnorsIllustrations and Photography: Farook Al-ShamaliLaboratory Technicians: Elaine Goth-Birkigt and Neil SextonLab Coordinator: Rob CarmichaelEditor: Reg SilvesterVisual Design: Jingfen ZhangVisual Presentation: Digital Media Technology Unit

Acknowledgments

Dr. Martin Connors’ original versions (v.1.0-1.5 from 1997-2002) of the PHYS 200Home Lab Guide/Manual were supported by the Office of Learning Technologies(OLT) of Human Resources Development Canada and the Mission Critical ResearchFund of Athabasca University.

The manual was extensively revised by Dr. Farook Al-Shamali in 2004 (v.2.1) andthen further improved in 2008 (v.2.2). This PHYS 200 Lab Manual (v.2.3) is a revisedversion of the PHYS 200 (v.2.2) Lab Guide.

The authors gratefully thank Rob Carmichael, Elaine Goth-Birkigt and Neil Sextonof Athabasca University Science Lab for their efforts to prepare and maintain thelab kits. Appreciation also extends to the physics tutors Jon Johansson, LeonidBraverman, Fuad Sarajov, Arzu Sardarli and Bill Scott for their valuable suggestionsthat helped in developing these laboratory exercises.

Every effort has been taken to ensure that these materials comply with the require-ments of copyright clearances and appropriate credits. Athabasca University willattempt to incorporate in future printings any corrections which are communicatedto it.

The inclusion of any material in this publication is strictly in accord with the con-sents obtained and Athabasca University does not authorize or license any furtherreproduction or use without the consent of the copyright holder.

c©Athabasca University 2012 All rights reserved

Contents

Introduction iii

Experiment 1 Graphical Analysis 1

Experiment 2 Force Constant 9

Experiment 3 Kinematics 14

Experiment 4 Mechanical Energy 21

Experiment 5 Dropping and Bouncing 27

Experiment 6 The Atwood Machine 32

Experiment 7 Motion on Incline 38

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Athabasca University PHYS 200 Lab Manual - V 2.3

Introduction

The lab component provides a hands-on experience of the physical concepts coveredin the course, using suitable tools and equipment. The lab experiments are meant togo with the course material, so you can connect theory and physical phenomena. Allthe experiments in the manual will be done in a place of your choice.

Lab Kit

The material required to perform the lab experiments in this manual are to be bor-rowed from the Athabasca University Library in the form of a lab kit. This includesinstruments and devices (see picture above) that are necessary for performing theexperiments. Each kit contains:

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1. Go! Motion sensor unit

2. USB cable

3. pulley

4. pulley axle

5. pulley support jig

6. fold-back paper clips

7. hockey puck

8. digital stopwatch

9. 100 g brass masses

The lab kit will be mailed to you shortly after you request it. Sometimes you willbe placed on a waiting list if the demand for the kits is high. When you receive thekit, it will contain a card for return postage. Note that this is similar to the bookborrowing procedure, which you are also encouraged to use.

To reach the library, call the toll-free number 1-800-788-9041, ext. 6254. You canalso browse the library website at http://library.athabascau.ca. In addition to yourstudent number and a shipping address, you will have to clearly indicate to the librarystaff which course you are enrolled in. Note that only those enrolled in an AthabascaUniversity physics course may borrow lab kits.

Note: Experiments 1, 2, and 7 can be completed without the lab kit.

Computer and Software

To perform the lab experiments in this manual, you will need to have a computerwith a USB port. The computer will be used for the collection and analysis ofexperimental data, and for the preparation of the lab reports. You will also needLogger Pro software from Vernier Software & Technology (www.vernier.com). Thisis an interface program that allows you to monitor the motion sensor measurementsand save them on your computer. The program is also needed for data analysis.Logger Pro is available for download from the course website.

Evaluation

You are expected to perform seven experiments according to the guidelines presentedin this manual, and you will prepare a lab report for each experiment. All reports

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should be submitted to your tutor for assessment. The lab component counts for 20%of the total course mark, and will be divided among the seven experiments as follows:

Experiment 1 2%

Experiment 2 3%

Experiment 3 3%

Experiment 4 3%

Experiment 5 3%

Experiment 6 3%

Experiment 7 3%

It is important to accumulate a total lab average of at least 50% to pass the course.If you have lab credit from another institution, you may wish to inquire about trans-fer of lab credit. Such transfer and any evaluation associated with it are entirelyat the discretion of Athabasca University. You will be required to submit originallab materials, done by you, for evaluation toward potential transfer of credit. Youmust discuss this possibility with the course coordinator before submitting any suchmaterials.

Lab Report

Lab reports are an effective way of communicating important information, and theiruse is stressed in this course. There is little point in doing a wonderful experiment withgreat results if you cannot effectively communicate your findings to others. Some-times, the results will not seem so great, and organizing yourself to write the reportwill help to understand what went wrong. On the other hand, the results we areseeking are relatively straightforward, so there is no need to make the report overlylong!

The lab report serves several purposes and gives an organized framework for recordingyour procedures and results. Although some students may have encountered labora-tory reports before and may feel that there is a standard format for them, this is notentirely true. However, make sure to include the following sections:

1. Cover Page: On this page you write the course’s name and number, lab manualversion, experiment’s title, your name, student ID, and date completed.

2. Introduction: (10%) Here, provide your theoretical background including allformulas needed in the analysis.

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3. Procedure: (10%) Here you should give a clear and detailed description ofyour steps in performing the experiment. If there is more than one part to thelab, it is usually best to describe the actions and observations separately foreach part.

4. Pictures: (10%) Include clear pictures of your setup.

5. Data: (10%) Organize and present the data collected in the experiment, andprovide a description of the trend and behaviour of the collected data. Nocalculations or analysis should be included in this section of the report.

6. Analysis and Discussion: (40%) This is a very important section of the labreport. In here you are expected to give a clear and detailed analysis of yourdata, as described in the manual. Make sure to include sample calculations,especially for new calculated columns in data tables. You may also need toproduce graphs and perform appropriate fits using the Logger Pro software.Errors in the observations may have a bearing on your analysis and you shoulddiscuss their role here.

7. Conclusion: (10%) Present a brief summary of your findings in this experi-ment, including the final numerical results.

8. Questions: (10%) At the end of each lab you will find a number of questionsrelated to the experiment. Provide a detailed answer to these questions at theend of the lab report.

Lab Safety

Appropriate care should be taken due to moving objects and other potentially haz-ardous situations and materials. The level of risk involved in doing these labs iscomparable to that of day-to-day activities and care has been taken to avoid suggest-ing activities which produce hazards.

It is your decision to proceed with any experiment, and in making that decision youcontrol your own situation and assume any risks involved. It is your responsibility toact in a responsible manner to avoid hazard to yourself or members of the public.

The authors, Athabasca University, or any equipment supplier cannot be held liablefor the consequences of any action undertaken in association with these laboratoryexercises. If you cannot safely do these labs, please withdraw from the course.

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Experiment 1 Graphical Analysis

This experiment introduces significant figures and measurement uncertainty and demon-strates the importance of graphical analysis.

Introduction

Physics is a human effort to study the natural world around us and to understandhow the universe behaves. Therefore any theory (no matter how fancy or attractiveit looks) is actually useless unless it is supported by experimental evidence. Doing anexperiment usually involves making quantitative measurements. However, in orderfor the measured values (or data) to be meaningful, it is very important to understandthe limitations of the instruments used and to recognize the possible sources of error.

Significant Figures and Measurement Uncertainty

Assume that you want to calculate the surface area and volume of your physicstextbook. Of course, you need to measure the book’s dimensions (length, width andthickness). Using a ruler, you first measure the length of the textbook. Since thesmallest division on the ruler is the millimetre, you can only give an upper limit(26.3 cm) and a lower limit (26.1 cm) for the length. Due to the limitations of theinstrument (the ruler in this case) you can then say that the correct length (L) of thetextbook is most probably between these two values. This is written as (L = 26.2±0.1cm), where the first number is called the measured value and the second number iscalled the uncertainty (or error) in the measurement. Similarly, you measure thewidth (W ) and the thickness (T ) of the textbook to be (W = 20.6 ± 0.1 cm) and(T = 3.9 ± 0.1 cm) respectively. In these measurements, the first digit after thedecimal point is uncertain, making it meaningless (or insignificant) to include moredigits. The digits in a measured value, up to and including the first uncertain digit,are called significant figures. So, there are three significant figures in L and W andtwo significant figures in T . Note that we cannot write the length as L = 26.20 cm,since it will then include two uncertain digits.

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In counting the number of significant figures in a measurement, we start from the firstnon-zero digit and end at the first uncertain digit. If the uncertainty of a measurementis known, then it becomes easy to specify the uncertain digit and count the numberof significant figures. However, if the uncertainty of a measured value is not known,then we need to be careful. If there is no decimal point in the number, then the lastnon-zero digit is considered to be the first uncertain digit. If the number containsa decimal point then the last digit after the decimal (whether it is a zero or not) isassumed uncertain. This ambiguity in the number of significant figures is avoided byusing the scientific notation, in which only one digit is kept to the left of the decimalpoint, while the remaining digits, up to and including the uncertain one, are movedto the right. Of course, we must multiply by the appropriate exponent. See Table1.1 below for examples.

Table 1.1

Measured Value Scientific NotationNumber of

Significant Figures

9.80 9.80 3

120 1.2× 102 2

120± 2 (1.20± 0.02)× 102 3

120.0 1.200× 102 4

1.00560 1.00560 6

0.00560 5.60× 10−3 3

375× 10−9 3.75× 10−7 3

Limit Error

Knowledge of measurement uncertainty and error propagation in a series of calcula-tions, is very important in evaluating the accuracy of a certain set of data. Thereare two major types of measurement uncertainties: random and systematic. In mea-suring the textbook’s dimensions, for example, the uncertainty due to your limitedability to read the ruler to better than 1 mm is a random error. If, on the other hand,you mistakenly confuse the inch and the centemetre units, your measurements willinclude a systematic error of scale.

Returning to your exercise, the surface area (A) of the textbook is calculated usingthe equation (A = L × W ). If you multiply the measured values of length andwidth using a calculator you will get A = 539.72 cm2. However, how accurate is this

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value? How many significant figures are there in A, and what is its uncertainty? Asa general rule for combining numbers by multiplication or division, the final resultshould not have more significant figures than the original value with the least numberof significant figures. For addition and subtraction, however, the final result shouldnot have more decimal places than the original value with the least number of decimalplaces. Therefore, since L and W have three significant figures each, A should berounded to three significant figures and written as A = 540 cm2 (or 5.40× 102 cm2).Similarly, the volume of the textbook is written as V = 2100 cm3 (or 2.1× 103 cm3),which should not include more than two significant figures.

What about the uncertainties of A and V ? Since the area and the volume of thetextbook are calculated from the measured dimensions, the errors in A and V arepropagation of the errors of L, W and T . Table 1.2 lists the rules used to calculatethe propagated errors in various arithmetic operations. Notice that whether we areadding or subtracting numbers, the combined error simply adds up. We use thesame rule for combining error whether the operation is a multiplication or a division.Notice that in the last rule, k is an exact constant factor that has no uncertainty (i.e.∆k = 0).

Table 1.2

Relation Equation Limit Error

Addition z = x+ y ∆z = ∆x+ ∆y

Subtraction z = x− y ∆z = ∆x+ ∆y

Multiplication z = xy ∆z = |z|(

∆xx

+ ∆yy

)Division z = x/y ∆z = |z|

(∆xx

+ ∆yy

)Power z = xn ∆z = nxn−1 ∆x

Exact constant z = k x ∆z = k∆x

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So, the uncertainty in area is given by

∆A = |A|(

∆L

L+

∆W

W

)= 540×

(0.1

26.2+

0.1

20.6

)= 4.7 cm2 (1.1)

which is reduced to ∆A = 5 cm2 because it is not an actual measurement. So, wesee that the third digit in A is uncertain, indicating that the number of significantfigures in A is indeed three. The final answer is then written as

A = (5.40± 0.05)× 102 cm2 (1.2)

Similarly, we find that the volume of the textbook is equal to

V = (2.1± 0.1)× 102 cm2 (1.3)

It is often useful to speak of the relative error ∆x/x to compare the size of errorto the size of the value measured. Another advantage is that the relative error of aproduct or dividend is just the sum of the relative errors of what makes them up.

Example: What is the uncertainty of the volume (v) of a cylinder in terms of itsradius (r) and height (h)?

Solution: The volume of the cylinder is given by the formula v = πr2h. Using therules in Table 1.2, the uncertainty in v is calculated as follows

∆v = π ×∆(r2h)

= π × |r2h|(

∆r2

r2+

∆h

h

)= (πr2h)

(2r∆r

r2+

∆h

h

)= v

(2∆r

r+

∆h

h

)(1.4)

Linear Graph

The aim of a typical scientific study or experiment is to investigate how a particularphysical quantity is related to another. In such an experiment, the value of the firstquantity (the independent variable) is varied, and the value of the second quantity(the dependent variable) is measured. The result is two sets of values correspondingto the two quantities. For example, assume that you performed an experiment that

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involved monitoring the growth of a fast growing indoor plant, placed in a sunnylocation in your house. In the experiment, you measured the plant’s height every 12hours and recorded the results as shown in Table 1.3. Since a regular ruler was usedfor measurements, you estimated the error in the plants height to be about 1 mm.

Table 1.3

Elapsed Time Plant’s Heightx (day) y ± 0.1 (cm)

0.5 39.61.0 39.91.5 40.32.0 40.72.5 40.93.0 41.3

Fig 1.1

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Usually, it is very convenient to display tabulated data graphically. The independentvariable is normally (but not necessarily) plotted on the horizontal axis while thevertical axis is reserved for the dependent variable. In Fig 1.1, the circled pointsrepresent the x and y values in Table 1.3. The vertical bar on each data point showsthe range of uncertainty in the corresponding measurement. You probably noticedthat the data points on the graph lie (almost) on a straight line. This indicates thatthe linear function y = mx + b is likely to represent the data very well. The beststraight line through the data points is called the best linear fit to the data. Theparameter m in this equation refers to the slope of the line, and it represents the rateat which y increases with x. The other parameter b refers to the y-intercept, and itrepresents the initial value of y at x = 0.

To find the slope of the linear graph, we need first to mark two points on the straightline and write down their x and y coordinates. Note that these coordinates shouldbe read directly from the linear graph and not from the data table. If the first pointhas the coordinates (x1, y1) and the second point has the coordinates (x2, y2), thenwe can calculate the change in y (rise = ∆y = y2−y1) caused by the change in x (run= ∆x = x2 − x1). The slope is then calculated as the ratio rise-over-run such that

m =rise

run=y2 − y1

x2 − x1

(1.5)

In Fig 1.1, the slope is calculated to be m = 0.7 cm/day, which is a relatively highgrowth rate.

The y-intercept (b) corresponds to the point at which the linear graph intersects they-axis (at x = 0). In Fig 1.1, b = 39.2 cm, which represents the height of the plant 12hours before you took the first measurement. The best linear fit to the data in Table1.3 is then given by the equation

y = 0.7 x+ 39.2 (1.6)

Procedure

Imagine that you and a friend made a trip from Calgary to Edmonton. Since yourfriend was driving the car, you decided to make yourself busy by keeping a recordof the distance travelled as a function of time. So, when you were waiting for thelast traffic light to turn green on the outskirts of Calgary, you reset the odometer tozero. When the green light turned on, you started timing the trip using a stopwatch.Every ten minutes you recorded the odometer reading. At the end of your trip, youhad the two columns of data in Table 1.4.

Since the odometer gave the distance within 100 m, you estimated the uncertainty ind to be (∆d = ±0.1 km). Regarding the time, even though you did your best to take

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the meter reading at equal time intervals of 10 minutes, you still estimated a timingerror of about 12 seconds (or ∆tm = ±0.2 min).

Table 1.4

Time in Minutes Time in Hours Distancetm ± 0.2 (min) th ± (h) d ± 0.1 (km)

0 010.0 17.520.0 34.530.0 52.240.0 76.150.0 87.060.0 105.070.0 122.380.0 144.090.0 152.3100.0 177.0110.0 191.1120.0 209.0130.0 227.4140.0 241.0150.0 263.7160.0 285.2170.0 292.1

Analysis

Now you will analyze the data collected in Table 1.4.

• Since you are using kilometre (km) as the unit of distance, then probably it isa good idea to use hour (h) as the unit of time. So, begin by transforming theelapsed time from minutes to hours and insert the results in the second columnof Table 1.4. After completing the transformation, you need to calculate theuncertainty of the new quantity th. Remember that th = 1

60× tm and that the

factor 160

is an exact number with zero uncertainty.

• On graph paper draw a horizontal axis representing the time in hours (th)and a vertical axis representing the travelled distance (d) in kilometres. Then,represent the data in Table 1.4 by points on the graph paper. Each point consistsof a dot enclosed in a small circle. If you think the error bars are too small to

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be visible, you may ignore them after providing a suitable justification. In thisfirst experiment, please draw the graph by hand and do not use any computersoftware for this purpose.

• Using a ruler, draw a straight line that passes (on the average) through the datapoints. This should represent the best linear fit to the data.

• Calculate the slope of the best-fit line using the rise-over-run relation. Also,determine the y-intercept and write the equation of the best-fit linear graph.

• How is the slope value related to the speed of the car during the trip? Explainyour answer.

• Estimate the instantaneous speed of the car at tm = 75 min. Is this valuegreater or less than the average speed of the car during the trip?

Conclusion

After the analysis, write a brief conclusion in which you summarize the results of theexperiment. You should also include any general comments and suggestions to modifythe experiment and improve accuracy. This also applies to the remaining experimentsin the manual.

Questions

Answer the following questions at the end of your lab report.

1. Write the following numbers in scientific notation: 100.1 m, 987 s, 1000 ± 5 J,0.000005 s, and 9001× 1010 mm.

2. Determine the error formula for y = x5.

3. Calculate the circumference (including uncertainty) of a circle whose measuredradius is r = 7.3± 0.2 cm.

4. Calculate the area (including uncertainty) of the circle in the previous question.

5. Two pieces of wood are glued together to form a long stick. The length ofthe first piece is L1 = 0.97 ± 0.02 m and the length of the second piece isL2 = 1.04± 0.01 m. What is the length (including uncertainty) of the stick?

6. An object of mass m = 2.3±0.1 kg is moving at a speed of v = 1.25±0.03 m/s.Calculate the kinetic energy (K = 1

2mv2) of the object. What is the uncertainty

in K?

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Experiment 2 Force Constant

In this experiment you will study the stretching response of a rubber band (or aspring) and compare the observed behaviour with the predictions of Hooke’s Law. Theexperiment will also introduce the Logger Pro program, which is used for collecting,graphing and analyzing data.

Introduction

Mary noticed that when she sits on her new couch, the seat is depressed by about10 cm. She also noticed that if she holds her cat while sitting on the couch, theseat depression increases to about 11 cm. Since the cat’s weight caused the springsinside the couch to compress by an additional 10%, Mary argued that she is ten timesheavier than her cat. In her argument, Mary made use of Hooke’s law.

To understand this important law, consider the spring shown in Fig 2.1. The initiallength of the spring is l0. After applying a tensile force F , the spring stretches andits length increases to l0 +x. Hooke’s law states that the compression or extension ofa spring (and many other elastic objects) is directly proportional to the force actingon it. Mathematically, this is described by the equation

F = k x (2.1)

where k is a constant that depends on the properties of the spring. This law is trueas long as the spring does not reach its elastic limit. Beyond this limit, the spring (orthe elastic object used) does not follow Hooke’s law and may eventually break.

Procedure

Before starting this experiment, you need to supply a medium size rubber band, thatis approximately 20 cm in length. If you have a suitable spring at home, you may useit instead of the rubber band. You will also need about 10 identical coins (preferablyloonies or toonies) and a container for carrying the coins. See Fig 2.2 for some ideas.

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Fig 2.1

Fig 2.2

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To perform the experiment, first hang the rubber band from a support attaching thesmall container to its lower end, as in Fig 2.2. If needed, you may put an additionalweight in the container until the rubber band is slightly stretched. At this stage,measure and record the initial length (l0) of the rubber band. After adding eachcoin to the container, measure the length (l) of the rubber band and enter yourmeasurements into the first and the third columns of Table 2.1. Use your judgmentto estimate the uncertainty of the measured length. For example, if you are usinga ruler that has millimetres as the smallest division and you think you can clearlydefine the beginning and the end of the rubber band, an uncertainty of 1 or 2 mmshould be a good estimate.

At this stage, you should take a picture of your experimental setup to include in yourlab report. This applies to all remaining experiments in this manual.

Table 2.1

Number Weight Rubber Band Rubber Bandof Coins of Coins Length Stretch

n w (N) l ± (m) x ± (m)

Analysis

In this experiment, you will test whether or not the rubber band follows Hooke’s law.Before starting the analysis, make sure that the Logger Pro program is installed onyour computer. The program can be downloaded from the course website.

Complete the second column in Table 2.1. Note that w is equal to the weight of then coins added after recording the initial length of the rubber band. Therefore, it iscalculated using the equation

w = n mg (2.2)

where m is the mass of a single coin. In the lab report, show at least one detailed

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sample calculation of w ±∆w. You can assume an uncertainty of ∆m = 0.2 g in themass of each coin.

Table 2.2 includes the masses of Canadian coins issued from the year 2000 to date.It is important to note that coins with earlier dates may have different masses. Youcan refer to the Royal Canadian Mint website (www.mint.ca) for more informationand technical specifications of the Canadian coins. For the specifications of Americancoins refer the United States Mint website at www.usmint.gov.

Table 2.2

Coin Mass

(2000 to date) (g)

1 cent 2.35

5 cents 3.95

10 cents 1.75

25 cents 4.4

1 dollar 7.0

2 dollars 7.3

To begin, calculate the stretch (x = l − l0) of the rubber band caused by adding theweight of the coins. This is done by subtracting the new length of the rubber bandfrom the initial length. Show at least one detailed sample calculation.

Now you are ready to graph and analyze your data. In the previous experiment, youdid this manually on a graph paper. Fortunately, there is plenty of software that cando the graphing and data analysis more efficiently. In this lab you will use the LoggerPro (LP) program, which you should have already installed on your computer. Notethat the instructions in this manual are based on version 3.8.3 of this software.

Start Logger Pro and double-click on the header of the first column, since that iswhere you will enter the x value. In the pop-up window, enter the name (rubberband stretch), the short name (x) and the units (m) in the appropriate boxes, andclick OK. Repeat for the second column, which is where you will enter the weightof the added coins. After that, enter the appropriate data from Table 2.1 into thecorresponding columns in LP. You will notice that the graph window is updatedautomatically to display the data points on the graph. See Fig 2.3 for sample entries.

It is a good idea to adjust the axes options and make them appropriate to the rangeand scale of your data set. This can be done by double-clicking anywhere in the

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graph window and then making the appropriate adjustments. To enter measurementuncertainties, click on the appropriate column header, select Options, check the ErrorBar Calculations box and then enter your values for the uncertainties.

Fig 2.3

You will probably notice that the rubber band stretches with the addition of coinsin an almost linear fashion. The Linear Fit function in the LP program, generatesa straight line (y = mx + b) that best fits to the data and calculates the slope (m)and the y-intercept (b). You can fit a straight line either to the whole graph or to aregion of interest. Drag the mouse and highlight all the data points to be includedin the fit. Then, click on the Linear Fit button on the tool-bar or select it from theAnalyze menu.

To display the uncertainties in the slope (∆m) and in the y-intercept (∆b), double-click on the floating box and check the Show Uncertainty option. You can also changethe font size in the floating box by clicking on Appearance. Compare the best-fit linearequation and the prediction of Hooke’s law (see Introduction) and determine the forceconstant k ±∆k of the rubber band.

Note: Be sure not to confuse the slope of the graph with the mass of the coin. Theseare two different quantities that use the same symbol (m).

Questions

1. Based on your fit equation, how many coins are required to stretch the elasticband by one metre? Is this possible using your rubber band?

2. A certain uniform spring has spring constant k. Now the spring is cut in half.What is the relationship between k and the spring constant k′ of each resultingsmaller spring? Explain your reasoning. [Source: Serway and Jewett, Physics for

Scientists and Engineers, 8th ed. 2]

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Experiment 3 Kinematics

In this experiment you will study the kinematics of objects moving with uniformvelocity or uniform acceleration.

Introduction

In the previous experiments, you saw how related changing quantities can be repre-sented graphically. In this experiment, you will use some of the equipment in the labkit to observe and measure the position of an object moving with constant accelera-tion. A familiar example of a uniformly accelerated motion is that of a freely fallingobject. The equipment in the lab kit will allow you to make accurate measurementsof such relatively rapid motion.

We refer to the change in distance per unit time as the speed (or the velocity when thedirection is specified). The average speed, within a specific time interval, is definedas the travelled distance divided by the time interval. In a very small time interval,the average speed approaches that of the instantaneous speed. It is important torealize that both instantaneous and average speeds can be calculated from any graphrepresenting the position as a function of time. The linear equation

x = xi + v t (3.1)

represents the position (x) versus time (t) of an object moving with constant speed(v). Graphically, the equation represents a straight line of slope v and y-interceptequal to xi. In general, the instantaneous speed at a specific time is equal to the slopeof the tangent to the curve at that time.

If the speed of the object is changing at a constant rate, we say that the object hasa uniform (or constant) acceleration. In this case the equation above is modified toaccount for the changing speed and becomes:

x = xi + vi t+ 12at2 (3.2)

where vi is the initial speed of the object at t = 0 and a is its acceleration.

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As mentioned earlier, uniform motion produces a linear graph of position versus time.Uniformly accelerated motion, on the other hand, produces in a slightly more complexcurve known as parabola. The equation of a parabola is an example of a type ofequation called the quadratic equation. The acceleration (ay = −g = −9.80 m/s2) ofan object moving under the influence of gravity is constant and directed downwards.So, for a freely falling object the quadratic equation above becomes

y = yi + vyi t− 12gt2 (3.3)

Procedure

For this experiment, you need to supply a medium size ball (a basketball for example).From the lab kit you will need the Go! Motion motion sensor. Connect the squareend of the cable to the USB port on the left side of the Go! Motion. Then connectthe other end of the cable to a USB port on your computer. Start the LP programand familiarize yourself with the different menus and screens. Also, it is a good ideato make a number of tests by moving different objects in front of the sensor.

Motion with Uniform Velocity

In this part of the experiment, you will examine the motion of the ball while it ismoving in a straight line at a constant speed (i.e. no acceleration). Begin by preparingan open area for the ball to roll smoothly on the floor for several metres, with severalmetres clearance to the sides. Place the sensor on the floor, approximately 50 cmbehind the ball, pointing in the direction of motion. Make sure that the sensorremains stable during the experiment (see Fig 3.1). When ready, click the greenCollect button in the LP program, and you should hear the sonic pulses emitted bythe sensor. Give the ball a gentle push to start it rolling steadily in a straight line,away from the sensor. The program will run for five seconds before the pulses stop.

If the experiment is done properly, you should get a graph similar to Fig 3.2. Notethat, you can change the data collection time and rate by clicking on Experiment inthe menu bar and then selecting Data Collection from the scroll down menu. Savethe LP file containing the distance and velocity measurements versus time under aproper name.

Repeat the procedure above, but this time allow the ball to move faster away fromthe sensor. In a third trial reverse the direction of motion and allow the ball to movegently towards the sensor. In this case, you will need an assistant at the other endof the room to push the ball towards you near the sensor. Your assistant should staymotionless while the sensor is collecting data. Be sure to stop the ball before it hitsthe sensor. After each of the three trials, save your data file under a proper name.

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Fig 3.1

Fig 3.2

16


Fig 3.3

Fig 3.4

17


Fig 3.5

Fig 3.6

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Motion with Uniform Acceleration

Using the same connection as before, place the sensor on the floor facing upward andstart the LP program. Click the Collect button. Hold the ball about 50 cm abovethe sensor, toss it upward in the air, and catch it (see Fig 3.3). If the experiment isdone properly, you should get a graph with a parabolic section, similar to Fig 3.4.If you need to repeat the experiment, click the Collect button again and repeat theprocedure above. Make sure to save your data file before starting the analysis.

Note: Do not forget to take pictures of your setup to include in your lab report.

Analysis

Motion with Uniform Velocity

Open the first data file that you saved using the LP program. The interesting sectionof the graph, which corresponds to the position of the ball while in motion, should bealmost linear. Drag the mouse to highlight the data points of this section and performa linear fit. The program will generate the best fit linear graph of the selected datapoints and calculate the slope and the y-intercept, as in Fig 3.5. Compare the slopevalue with the ball’s speed in velocity versus time graph. Save the graph and includeit in your lab report. Repeat this analysis for the other two graphs of motion withconstant velocity. Comment on your results.

Motion with Uniform Acceleration

Open the saved file corresponding to this part of the experiment using the LP pro-gram. The interesting section of this graph is a parabola corresponding to the heightof the ball while in free flight. The remaining points on the graph are not part of themotion being studied, and should not be included in the fit.

Drag the mouse and highlight the data points of interest. Click on Analyze in themenu bar and select Curve Fit from the scroll down menu. Select the Quadraticgeneral equation and click Try Fit. A parabolic curve will be generated showing thevalues of the three parameters in the fit equation Y = A t2 + B t + C (see Fig 3.6).Save this graph and include it in your lab report.

Recall that Equation (3.3) above is the appropriate equation for a freely falling object.

• Interpret the meaning of the fit coefficients A, B and C.

19


• What is the value of the acceleration due to gravity predicted by your experi-ment? Compare to the expected value and calculate the percentage difference.

• Calculate the quantity B/(2A) showing units. Can you give a physical meaningto this quantity?

Questions

1. If the velocity of a particle is nonzero, can the particle’s acceleration be zero?Explain.

2. If the velocity of a particle is zero, can the particle’s acceleration be zero?Explain.

3. If a car is traveling eastward, can its acceleration be westward? Explain.[Source: Serway and Jewett, Physics for Scientists and Engineers, 8th ed. Page 48]

20

Experiment 4 Mechanical Energy

In this experiment you will test the principle of conservation of mechanical energy inan oscillating pendulum.

Introduction

According to the Merriam-Webster Online Dictionary, a pendulum is “a body sus-pended from a fixed point so as to swing freely to and fro under the action of gravity.”The motion of a pendulum is expected to conserve mechanical energy in the absenceof external influences, aside from gravity. Over the first few swings, we do not expectair resistance or other frictional sources to convert a substantial percentage of thetotal mechanical energy (E) to other forms. The potential energy (U) at the top ofthe swing should be fully converted to kinetic energy (K) at the bottom, and viceversa.

Figure 4.1 shows a pendulum consisting of a ball hanging from a fixed support. Thelength L of the pendulum is equal to the distance from the point of support to thecentre of the ball. When there is no oscillation, the ball hangs motionless from thevertical string, which indicates a zero net force. This is called the equilibrium positionof the pendulum.

When the pendulum is displaced from the equilibrium position, the ball is elevatedby a vertical distance y above the bottom of the swing. As a result, the gravitationalpotential energy increases by an amount equal to mgy. When released, the restoringgravitational force accelerates the ball back towards the equilibrium position until allpotential energy is converted to kinetic energy at the bottom of the swing. The ballcontinues to move to the other side, converting the kinetic energy to potential energyuntil it, momentarily, stops and starts heading back towards the initial position.Ideally, if there is no energy loss due to friction or air resistance, the total mechanicalenergy,

E = U +K (4.1)

= mgy + 12mv2

21


is conserved and the pendulum continues to oscillate indefinitely.

In this experiment the motion sensor is used to measure the horizontal position ofthe oscillating ball. There is no need to directly measure the ball’s vertical positionbecause, from Pythagorean Theorem, we can show that:

y = L−√L2 − x2 (4.2)

So, relative to the lowest point of the swing, the gravitational potential energy of thependulum is equal to

U = mgy

= mg(L−√L2 − x2

)(4.3)

Fig 4.1

Procedure

Before starting the experiment, you need to supply a medium size ball (e.g. a basket-ball) to be the oscillating mass. You will also need to supply a strong piece of stringor rope that can support the mass without breaking.

22


Construct a pendulum about one metre long by tightly wrapping a robe aroundthe ball. If necessary, you may use masking tape to prevent the ball from slipping.Possible places to support the ball are a basement beam, a ceiling hook, or a showercurtain rod (see Fig 4.2). If you cannot find a proper support, you may get anotherperson to hold the top end of the string and remain motionless while you makeyour measurements. Ensure that the ball swings freely in a vertical plane withoutobstruction or interference from other objects in the vicinity. Before proceeding,record the pendulum’s length L.

Note: These are only suggestions. You must decide on your own how to constructa safe setup, and the university assumes no responsibility for possible damage due toyour procedure.

Fig 4.2

Mount the motion sensor in the oscillation plane, making sure that it is at approxi-mately the same height as the centre of the ball when it is at the bottom of the swing.Back and forth oscillations of approximately 50 cm, on each side, should suffice, andthe swing should be directly towards and away from the sensor. Swing the pendulum,and when you see steady oscillations, click the Collect button in the LP program. A

23


smooth sinusoidal curve (see Fig 4.3) should be displayed. Save the data file under aproper name. At this stage, take a picture of your experimental setup to include inyour lab report.

Fig 4.3

Analysis

Open the saved data file using the LP program. As in the previous experiment,you should see three sets of data representing the horizontal components of position,velocity, and acceleration of the oscillating object versus time. Since the accelerationdata is not needed in this experiment, click on the head of the corresponding columnto highlight it, and delete it.

Note that the horizontal position of the oscillating object is measured from the motionsensor and corresponds to the distance R in Fig 4.1. Double-click on the head of theposition column and change the name (and also the short name) of the column toR. Similarly, double-click at the head of the velocity column and change the name(and the short name) to vx. Save the two graphs for R versus t and vx versus t to beincluded in your lab report.

Horizontal position versus time

As mentioned above, the collected data gives the horizontal position of the objectrelative to the motion sensor (i.e. R vs. t). However, we are more interested in theposition of the ball relative to the equilibrium position (i.e. x vs. t). From Fig 4.1,

24


you should convince yourself that

x = R−Rave (4.4)

where

Rave =

(Rmax +Rmin

2

)(4.5)

The maximum and the minimum R values can be obtained from the data table.

Now, you should be able to create a new data column corresponding to the horizontaldisplacement (x) of the ball from the equilibrium position. To do that, click on Datain the menu bar and select New Calculated Column from the scroll down menu. Usex for the name (and also for the short name) of the new column and use Equation(4.4) above to calculate it. For example, if you found that Rav = 0.825 m, then enter“R” − 0.825 (with R between quotes) in the appropriate space. Save the resultinggraph and include it in your lab report.

Vertical position versus time

Using Equation (4.2) above, create a new calculated column y, for the vertical positionof the ball. Note that in the LP program the equation should be entered in the formL − sqrt(L∧2 − “x”∧2), where L is the numeric value of the pendulum’s length and“x” (in quotes) refers to the values in the x column. Save the resulting graph andinclude it in your lab report. Why do you expect the y column to include positivevalues only?

Velocity versus time

The horizontal velocity (vx) of the oscillating object was automatically calculated bythe program. To find the vertical velocity (yy), create a new calculated column usingthe equation derivative(“y”). The derivative (or dy/dt) is the rate of change of thevertical position with time. You should be able now to calculate the magnitude (v) ofthe resultant velocity. What is the appropriate equation for this calculation? Displayvx, vy and v together on the same graph and include it in your lab report. Commenton the relationship between the three graphs.

Mechanical energy versus time

After determining the position and velocity of the ball as a function of time, youshould be ready to calculate the mechanical energies. Create a new calculated column,

25


named U , using the equation 9.80 ∗ y. Then, create another column for the kineticenergy, named K, which is calculated using the equation 0.5∗“v”∧2. Finally, create anew column for the total mechanical energy, named E, using the equation “U”+“K”.

Since the ball’s mass was omitted, these three columns actually correspond to thepotential, kinetic, and total mechanical energies per unit mass. Therefore, the ap-propriate unit for the data in these columns is J/kg. The absence of the balls massshould not affect your goal to test the law of conservation of mechanical energy. Thisis because both potential and kinetic energies are directly proportional to the mass,which can be factored out from the calculations.

Include the graphs of U , K, and E versus time in your lab report and describe thebehaviour of these energy values during a complete oscillation. Does your observationsupport the principle of conservation of mechanical energy? If not, discuss to whatextent there is a problem, and whether or not there is a reasonable explanation. Donot forget to include a copy of the data table containing all new columns created inthis analysis.

Note: It is important to show sample calculations for every new calculated column.

Questions

1. From Fig 4.1, show that the relation between the horizontal and vertical com-ponents of the balls position is given by Equation (4.2). In other words, derivethis equation.

2. A block is connected to a spring that is suspended from the ceiling. Assumingair resistance is ignored, describe the energy transformations that occur withinthe system when the block is set into vertical motion. [Source: Serway and Jewett,

Physics for Scientists and Engineers, 8th ed. Page 224]

26

Experiment 5 Dropping and Bouncing

In this experiment, you will study the dynamics of a bouncing ball.

Introduction

Bouncing motion involves free fall under the influence of gravity, with sharp upwardforces at each contact with the ground. When the ball hits the ground, a large butbrief upward force is applied on it. This force arises from the compression of theball and, to some extent, from the compression of the ground. While the ball iscompressed, it exerts a downward force on the ground which reacts with an upwardforce on the ball, according to Newton’s third law. Therefore, for the duration of theimpact the ball experiences a very large upward acceleration that slows the ball veryquickly and then reverses its direction of motion.

After the ball loses contact with the ground, it continues to move upward, but underthe influence of gravity only. So, between impacts the ball has a downward accelera-tion equal to g, and its vertical motion is described by the equation:

y = y0 + v0t− 12gt2 (5.1)

The total mechanical energy of the ball while in free flight is conserved and it is equalto

E = U +K

= mgy + 12mv2 (5.2)

Note that with each impact the ball loses some of its mechanical energy (in the formof heat, sound, . . . etc.) which causes a decrease in the maximum height of eachsubsequent bounce.

The magnitude of the linear momentum of the ball at any time during the motion isequal to

p = mv (5.3)

Remember that momentum is a vector quantity and its sign is important.

27


Procedure

Before starting the experiment, you need to supply a medium-sized ball with goodbounce, such as a well-inflated basketball. Support the motion sensor (safely) about2 metres above a hard floor. For example, the sensor can be taped to the top of adoorway or a basement beam or placed on a closet shelf (see Fig 5.1). If you cannotfind a suitable support, you may ask another person to hold the sensor at a suitableheight and remain motionless during the experiment. Keep the cables out of the wayto avoid interference with bouncing or echo detection.

Practice dropping the ball below the sensor. When you feel that you can get aminimum of three clear bounces, start the LP program, click the Collect button anddrop the ball directly below the motion sensor. Make sure that there is at least 20 cmbetween the sensor and the ball at all times to remain within the sensitivity range ofthe sensor. If the experiment is done properly, a graph similar to that shown in Fig5.2 will appear on the computer screen. Save the data file under a proper name.

Analysis

Open the saved data file using the LP program. You should have three sets of datarepresenting the vertical components of position, velocity, and acceleration of thebouncing ball versus time. Note that the measured position represents the verticaldistance between the ball and the sensor. Double click on the head of the positioncolumn and change the name (and the short name) to R.

Vertical position versus time

As mentioned above, the collected data gives the vertical position of the ball relativeto the motion sensor (i.e. R versus t). However, we are more interested in the positionof the ball relative to the ground (i.e. y versus t). From the R data find the maximumvalue (Rmax). Create a new data column corresponding to the height of the ball abovethe floor using the equation

y = Rmax −R (5.4)

28


Fig 5.1

Fig 5.2

29


Motion between bounces

Using the LP program, perform quadratic fits for at least three intervals of free flight.This means fitting the data points corresponding to each interval, using the equationy = A t2 +B t+ C (see Fig 5.3). Write a clear interpretation of the meaning of eachparameter in this equation.

Construct a table in which you summarize the results above. In the table, indicatethe gravitational acceleration predicted by each fit and calculate the percentage dif-ferences from the expected value. Add a column to your table that includes themaximum height reached by the ball in each interval of free flight. This can be easilydetermined from the graph and the y-column in the data table.

From the fit results of each interval, you should notice that the B (parameter) in-creases as the ball makes a new bounce. If B is interpreted as the initial velocityof the ball for the corresponding bounce, this seems to contradict the observed lossof mechanical energy after each bounce. Provide an explanation of this apparentdiscrepancy.

Fig 5.3

30


Dynamics of ball bouncing

From the maximum height values, you should be able to calculate the maximumpotential energy per unit mass (U/m) of the ball in the middle of each interval. Fromthe principle of conservation of mechanical energy this should be equal to the totalmechanical energy per unit mass (E/m) of the ball during an interval of free flight.Determine the amount of mechanical energy (per unit mass) that the ball loses duringthree selected impacts with the ground.

The downward velocity of the ball is available as a separate column in the originaldata set. Determine the change in the ball’s linear momentum (per unit mass) duringthe three selected impacts with the ground. Note that momentum is a vector quantity,and therefore the sign is important.

Write a brief discussion, supported with graphs, in which you comment on the relationbetween vertical position, velocity, acceleration, and force during all parts of themotion including impacts. Remember that the ball is accelerated while it is in flightand also during the short periods of contact with the ground. However, the forcesacting vary significantly.

Questions

1. A ball of mass mb is dropped and bounces off the floor. If the impact is viewed asan elastic collision between the ball and the earth, show that the kinetic energyof the earth after the collision can be ignored. [Hint: Start your argument fromthe two equations in Problem 30 in Chapter 7 (page 189) in the textbook]

31

Experiment 6 The Atwood Machine

In this experiment you will observe and analyze the motions generated in the Atwood’smachine system.

Introduction

The Atwood machine is a simple device in which two masses are joined by a string orcable looped over a pulley. The system is accelerated due to the difference in weighton both sides of the pulley. An interesting application of this machine is found in theoperation of elevators.

Consider the schematic diagram in Fig 6.1. To understand the motion of the system,we need to analyze the forces acting on each object. From the corresponding free-body diagrams, we can write the following equations of motions for the two massesand the pulley:

m1 g − T1 = m1 a (6.1)

T2 −m2 g = m2 a

(T1 − T2) R = I α

where I refers to the moment of inertia of the pulley and α to its angular acceleration.Note that we assumed a frictionless pulley and a massless string. By solving theequations above simultaneously, we get the following equation describing the motionof the system in Fig 6.1:

(m1 −m2) g = (m1 +m2 +I

R2) a (6.2)

32


Fig 6.1

Fig 6.2

33


Procedure

Fig 6.2 shows the items needed in this experiment. Most of these items (pulley, pulleyaxle, pulley support jig, 100 g masses, paper clips, and stopwatch) are provided inthe lab kit. You are expected to provide the remaining items.

To perform the experiment, you need to find a place, or a piece of furniture, thatcan support the pulley at least 1.5 m above the floor. The top of a refrigerator, astorage cabinet, or a bookshelf are few examples. The pulley support can be securedby masking tape or using the clamp included in your lab kit (see Fig 6.3).

Note: Before proceeding, make sure that the pulley is securely mounted and is in asafe and stable position.

Fig 6.3

34


Fig 6.4

Now, you are ready to construct the Atwood machine sketched in Fig 6.1. The aimis to measure the acceleration of the system as a function of the mass difference(m1 −m2), while keeping the total mass (m1 +m2) constant.

Prepare a piece of string that is approximately 20 cm longer than the height of thepulley above the floor. The string should be strong enough to hold the masses withoutbreaking and, at the same time, light enough that its mass can be neglected. Regulardental floss should be a good choice in this case. Attach one paper clip (to be used asa coin holder) to each end of the string. Then suspend a 100 g mass from each paperclip, as shown in Fig 6.4. Use scotch tape to make sure that each mass is attachedtightly to the clip and does not fly off when it hits the ground. The purpose of thisadditional mass is to increase the inertia of the system, which in turn reduces itsacceleration, allowing for more accurate time measurements.

In the m2 paper clip insert six dimes (or pennies). This will clearly create a massdifference equal to the mass of the six coins. However, the system may not show anyacceleration or motion due to friction in the pulley. To create a larger mass differencethat can overcome the force of friction, attach (using scotch tape) a few nickels tothe bottom of the 100 g mass in m1 (see Fig 6.4). Test the setup by holding m1 at

35


about 1 m above the floor and then releasing it. If no acceleration is observed, evenafter adding the nickels, you may want to put few drops of oil on the bearings of thepulley to lubricate it and reduce friction. If this does not help, add more nickels tothe bottom of m1 until a small acceleration can be observed in the system. The initialmasses of m1 and m2 are then (see Table 2.2)

m1 = 100 g + mass of nickles + mass of clip (6.3)

and

m2 = 100 g + mass of dimes + mass of clip (6.4)

Before you start the experiment, make few trials to practise the release and the timingof the system. It is a good idea to put a thin cushion on the floor below the massesto allow a softer impact. You also need to measure the distance (h) traveled by theheavier mass when it falls to the ground. To do that, hold m1 at its maximum heightsuch that m2 is just hovering above the floor, and measure accurately the distancefrom the bottom of the m1 mass and the floor. Record this value since it is requiredfor calculating the acceleration later in the analysis.

Note: Do not allow any unsupervised children near your setup, since the hangingobjects may form an attractive toy that may not be safe.

Table 6.1

n m1 m2 t1 t2 t3 t4 t5 t a (m1 −m2) g

6543210

Using the stopwatch, measure the time it takes m1 to fall from its highest elevationto the ground. Repeat this measurement five times, recording the time of each trialin the appropriate column in Table 6.1. Transfer one dime from the m2 clip to the m1

clip. Enter the new values of m1 and m2 in the second row in the table and make fivetrials to measure the acceleration time. Notice that the mass difference (m1 −m2) isnow greater by an amount equal to the mass of two dimes. Repeat this process untilno dimes are left in the m1 clip.

36


Analysis

Since your goal in this experiment is to test the validity of Equation (6.2), it isconvenient to start by linearizing the equation. This can be done by taking the weightdifference (m1−m2)g as the y variable and the acceleration a as the x variable. Whatis the slope and the y-intercept of the resulting linear equation?

Calculate the average time (of the five trials) for each mass difference and enter thevalue in the appropriate column in Table 6.1. The time uncertainty (∆t) can betaken as the maximum deviation from the average value. From the average timeand the travelled distance (h), calculate the acceleration of the system for each massdifference. Note that since the system is released from rest, we can write

h = y − y0 = 12at2 (6.5)

Plot (m1 − m2)g versus a, and perform a linear fit to determine the slope and they-intercept. From the slope value, calculate the moment of inertia (I) of the pulley.Equation (6.2) predicts a zero value for the y-intercept. However, you will probablyget a non-zero value! What could be the reasons behind this deviation? Did we misssomething when deriving Equation (6.2)? Provide a clear argument regarding thispoint in your lab report.

Questions

1. Starting from Equation (6.1) show the derivation steps of Equation (6.2).

2. Answer Question 17 at the end of Chapter 8, page 218 in the textbook.

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Experiment 7 Motion on Incline

In this experiment you will study the acceleration of an object sliding and rollingdown an incline, and investigate the role of friction in such motions.

Introduction

An incline is a flat surface making an angle θ with the horizontal (see Fig 7.1). Anobject placed on the incline experiences a number of forces. One of these forces isthe weight (mg) acting vertically downward. Another is the normal force that acts toprevent the object from penetrating the surface. As a result, motion is restricted sothat the object can only move in the direction parallel to the surface of the incline.

Consider an object resting on a horizontal surface (θ = 0). The weight of this objectis canceled completely by the normal force and, in the absence of any other forces,the object remains at rest. If the angle θ is slightly increased, a small force (mg sin θ)appears and acts on the object down the incline. However, the object is expectedto remain at rest because a force of friction of equal magnitude also appears andacts up the incline. As the inclination angle increases, so does the force opposingthe motion. However, there is an upper limit to the magnitude of the force of staticfriction generated. It cannot exceed (µsmg cos θs), where µs is the coefficient of staticfriction between the object and the surface and θs is the inclination angle at whichthe object is on the verge of sliding. Therefore, at this angle the following equationapplies:

mg sin θs = µsmg cos θs (7.1)

Any further increase in the inclination (i.e. θ > θs) will result in a non-zero net forceon the object, causing it to accelerate down the incline.

While the object is sliding down the incline, the force of kinetic friction opposing themotion is generally smaller than the maximum force of static friction just before themotion started. The magnitude of the force of kinetic friction is equal to µkmg cos θ,where µk < µs. The net force acting on the object, in this case, is equal to

Fnet = mg sin θ − µkmg cos θ (7.2)

38


From Newtons second law, we see that such force produces the following accelerationdown the incline

a = g sin θ − µk g cos θ (7.3)

If a round object is placed on an incline, it starts to roll. The force of static frictionin this case acts to prevent the object from sliding. Rolling involves both transla-tional and rotational motions. Assuming smooth rolling, it can be shown that theacceleration of a solid cylinder down the incline depends on the inclination angle andis given by the equation

a = 23g sin θ (7.4)

Note that the acceleration of a rolling cylinder down an incline is less than its ac-celeration while sliding without friction down the same incline. This is because as itrolls, part of the initial gravitational potential energy is transformed into rotationalkinetic energy. The theoretical derivation of Equation (7.4) is left as an exercise atthe end of your lab report. In this experiment, you will investigate the validity of thisequation.

Fig 7.1

Procedure

For this experiment, you need to provide a relatively long board (about 1m in length),to be used as the incline. Make sure that the board is thick enough so that it doesnot sag in the middle when placed in an inclined position. For the moving object, ahockey puck is a good choice, since it can be used for both the sliding and the rollingparts of the experiment. If you do not have your lab kit and it is difficult for youto provide a hockey puck, you may use something similar, such as an unopened foodcan.

39


Static Friction

In this part of the experiment you will determine the coefficient of static frictionbetween the hockey puck and the board. This is achieved by gradually increasing theinclination angle and measuring the angle θs at which the puck begins to slide downthe incline.

Start with the board lying horizontally on the floor and the puck placed near thecentre. Hold one end of the board and raise it slowly increasing the inclination angle.When the puck starts to move, stop and measure the elevation (hs) of the boardsedge above the floor. Repeat this process five times. Enter the measurements intothe first column of Table 7.1. Do not forget to record the length of the board, whichis necessary for calculating θs.

Table 7.1

hs θs(m) (Degrees)

Kinetic Friction

To find the coefficient of kinetic friction (µk) you need to study the sliding motionof the hockey puck. To do that, first make an incline by raising and supporting oneend of the board (see Fig 7.2). The inclination should be steep enough such that thepuck continues to slide smoothly once it is set in motion. Make two marks close tothe two ends of the board and measure the distance (d) between them.

Hold the puck (flat surface touching the board) near the upper mark. Release thepuck and, using a stopwatch, measure the travel time between the two marks. Repeatthis measurement five times for the same board inclination. Measure the height h ofthe elevated end of the board, which you will need to calculate the inclination angle θ.Repeat for three different inclination angles and record your measurements in Table7.2.

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Fig 7.2

Table 7.2

Inclination Time Average time Accelerationθ t t±∆t a±∆a µk

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Rolling Motion

The goal of this part of the experiment is to test the validity of Equation (7.4) fora rolling cylinder. For this you will need to measure the acceleration of the rollinghockey puck down the incline as a function of θ. The procedure is similar to whatyou did before, except that the puck is now rolling instead of sliding.

Do three trials of the experiment for at least six different inclination angles. Notethat the rolling motion can be achieved with relatively small inclinations (See Fig7.3). This is recommended since it increases the travel time and, therefore, reducesthe timing error. When you have all your data, enter your measurements into thefirst two columns of Table 7.3.

Analysis

Static Friction

In the first part of this experiment, you will determine, from your data, the coefficientof static friction between the hockey puck and the board. The inclination angle atwhich the puck loses balance and starts sliding down the incline is referred to as θs.At this angle, the force of static friction reaches its maximum value (µsmg sin θs) andbecomes equal to the component of the gravitational force down the incline, as statedin Equation (7.1). From this equation, derive a simple relation (independent of theweight) between µs and θs.

Next, use the measured heights hs (from the five trials) and the length of the board,to calculate the corresponding inclination angles θs. The average of these five trialsis the experimental measurement of θs and the largest deviation from the average isan estimate of the experimental error (∆θs). After that, it should be straightforwardto calculate the coefficient of static friction µs between the puck and the board.

To estimate the uncertainty (∆µs) you need to use the general theory of errors, whichstates that if y is a function of x, then

∆y =dy

dx∆x (7.5)

Note that for the tangent function, the uncertainty is given by

∆(tan θ) =d

dθ(tan θ) ∆θ

=

(1

cos θ

)2

∆θ (7.6)

where ∆θ is in radians (not degrees).

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Fig 7.3

Table 7.3

Inclination Time Average time Accelerationθ t t±∆t g sin θ a±∆a

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Kinetic Friction

To calculate the coefficient of kinetic friction µk between the sliding puck and theboard, you need first to calculate the acceleration for a particular inclination angleand then substitute into Equation (7.3). The acceleration can be calculated from themeasured time (t) and distance (d) of the sliding puck (i.e. d = 1

2at2). From the five

timing trials, calculate the values of t and ∆t for each inclination angle θ. Then,calculate the acceleration corresponding to each inclination angle. Complete Table7.2 and comment on your results. For example, does µs appear independent of θ?Compare your values for µs and µk.

Rolling Motion

As mentioned earlier, your goal in this part of the experiment is to test the validityof Equation (7.4). For this purpose, you need to find the acceleration of the hockeypuck at different inclination angles. As you did previously, calculate the accelerationof the rolling cylinder (the hockey puck) using the average time and distance travelledby the puck down the board. Then, complete Table 7.3.

Theoretically, Equation (7.4) suggests a nonlinear relation between a and θ. Since alinear graph is a very convenient method of testing theoretical equations, it is a goodidea to first linearize Equation (7.4). This can be done by designating (g sin θ) as thex-variable and the acceleration (a) as the x-variable. What are the expected valuesof the slope and the y-intercept of the resulting linear graph?

Enter the data from the last two columns of table 7.3 into the LP program and plot aversus g sin θ. Perform a linear fit to the data and determine the experimental valuesof the slope and y-intercept. Compare with the expected values from Equation (7.4).

Questions

1. Show that the acceleration of a solid cylinder rolling down an incline, makingan angle θ with the horizontal, is given by Equation (7.4). Hint: You can usethe principle of conservation of mechanical energy.

2. Answer Question 19 at the end of Chapter 8, page 218 in the textbook.

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Julian Experiments

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