physics 140l laboratory manual -...

Physics 140LLaboratory Manual

byH. Butner, A. Fovargue, K.Giovanetti,

L. Lucatorto, G. Niculescu, T. O’Neill, B. Utter

James Madison University

Harrisonburg, VA 228072009

c©2009-2010Department of Physics and Astronomy

James Madison UniversityAll rights Reserved

Contents

1 A Mean Lab (Introduction to PHYS140L) 3

1.1 The value of a Measurement(In Class activity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Picturing Motion(In Class activity) 17

3 Spreading the Data(At Home activity) 27

4 Gauging the Force(In Class activity) 37

5 Dropping the Ball(At Home activity) 45

6 Atwood’s Machine(In Class activity) 51

7 Sliding along(At Home activity) 63

8 Crashing Carts(In Class activity) 71

9 Happy and Sad Balls(At Home activity) 81

10 Poe’s Pendulum(In Class activity) 87

11 Functions/Air Drag(At Home activity) 95

12 Comedy of Errors(Final Lab Part I) 99

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13 Tale of Woe(Final Lab Part II) 113

14 Appendix 1: Curve Fitting 120

15 Appendix 2: Excel Spreadsheet 124

16 Appendix 3: Establishing Uncertainty 128

17 Appendix 4: Suggestions for Data Handling 138

4

List of Figures

1.1 Vernier Caliper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Reading: 2.64 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Micrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Metric micrometer reading equals 23.15 mm. 23 whole divisions (= 23 mm);.

0 mm divisions are uncovered (= 0.0 mm);15 0.01 mm divisions line up onthe thimble (= 0.15 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Position versus time plots for four situations. . . . . . . . . . . . . . . . . 172.2 Velocity versus time plots for the situations described in Fig. 2.1. . . . . 182.3 Position, velocity, and acceleration versus time (blank) plots. . . . . . . . 25

3.1 Sample Plot with Labels . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Sample Plot with Trendlines . . . . . . . . . . . . . . . . . . . . . . 33

5.1 Experimental setup for the Bounce Procedure . . . . . . . . . . . . . . . 49

6.1 Atwood machine: A sliding mass is connected to a falling mass via a pulley 536.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1 Friction and Normal Forces . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.1 Case 1: Both carts at rest initially (Note: your setup may be the mirrorimage of this figure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.2 Case 2: Inelastic collision with objects moving with the same final velocity.) 768.3 Case 3: Elastic collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.4 Cart positions after elastic collision, Case 3A. . . . . . . . . . . . . . . . 788.5 Cart positions after elastic collision, Case 3B. . . . . . . . . . . . . . . . 788.6 Cart positions after elastic collision, Case 3C. . . . . . . . . . . . . . . . 79

10.1 Pendulum cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.2 A warning about plots – pay attention to the scales on your axes! . . . . 8810.3 Experimental setup for the pendulum experiment. . . . . . . . . . . . . . 93

12.1 Calorimeter Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10012.2 Calorimeter with boiler . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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List of Tables

1.1 Measurements of a Cylindrical mass with a vernier caliper. . . . . . . . . 131.2 Measurements of a Cylindrical mass with a micrometer caliper. . . . . . . 141.3 Calculated volume of a cylindrical mass . . . . . . . . . . . . . . . . . . . 141.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Sample Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Force Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Setup and Performance of Force Probe Experiment . . . . . . . . . . . . 434.3 Analysis of the First Data (10N Scale) . . . . . . . . . . . . . . . . . . . 434.4 Analysis of the Second Data (10N Scale) . . . . . . . . . . . . . . . . . . 434.5 Analysis of the Third Data (50N Scale) . . . . . . . . . . . . . . . . . . . 444.6 Analysis of the Fourth Data (50N Scale) . . . . . . . . . . . . . . . . . . 44

5.1 Dropping the ball: Sample table for the raw experimental data. . . . . . 48

6.1 Example Cart Mass Table - YOUR NUMBERS WILL BE DIFFERENT 576.2 Example Case A Table - Remember Units! . . . . . . . . . . . . . . . . . 606.3 List of Required Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.1 Experiment: Sliding Along. Data table. If needed, feel free to make copiesof this table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Experiment: Sliding Along. Results table. . . . . . . . . . . . . . . . . . 69

8.1 Instructor check off table for “Crashing Carts” experiment. . . . . . . . 80

9.1 CR Measurements using the happy and sad balls. . . . . . . . . . . . . . 849.2 CR Averages and SD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

11.1 Description of activities and assignments for functions/air drag . . . . . . 97

12.1 Example - Mass of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . 10512.2 Barometric Pressure vs Boiling Temperature of Water . . . . . . . . . . . 111

1

Chapter 1

A Mean Lab (Introduction toPHYS140L)

Welcome to Physics 140L

In this laboratory course, you will explore how to perform experiments, and learn how toaccount for experimental uncertainties. Along the way, you will be exposed to materialcovered in lectures, and hopefully have a better sense of how the physics works.

Look around. In the first class you may see that the total number of students is morethan 16. Ideally we want to have only two students per lab station. To achieve that, wehave broken the labs into different types - those that are done in the lab, and those thatare done at home.

• There are two lab sections - called Group A and Group B. Each group will alternatedoing labs in the laboratory (every other week). On weeks that your group is notmeeting in the laboratory, you will be doing your laboratory “at home” with yourlab partner. This means that you and your lab partner will meet somewhere otherthan P&C 2286 to do the lab.

• To make the lab sections balanced, your instructor will identify which group (A orB) that you are in. If you are in A group, you will start off with the laboratorythe second week. If you are in B group, you come in for the lab in the third week.Group A will finish the labs before Thanksgiving, Group B will finish the weekafter Thanksgiving. See Table 1 for the breakdown of the schedule.

• You and your lab partner will be getting a lab kit which will be used for “at home”labs. Keep track of it and its materials. If you fail to turn it in, or turn it inmissing materials, you will charged a fee. Your course grade will be markedas incomplete until you and your lab partner turn the lab kit in or paythe replacement fee.

• When doing at home labs, you can meet anywhere or anytime, so long as you turnthe lab in by the due date. By the way, you and your lab partner have the block

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of time assigned to the lab period free that week - so you ALWAYS have at leastone meeting time free.

• Before each lab, you will be expected to read the lab. Since the time in the lab-oratory is short, you can’t waste time coming in without reading the lab. Toencourage reading before the lab, there will be a reading quiz on that lab - usuallyadministered through blackboard. Check with your instructor for details. It willbe due before the lab starts - and it will count toward your grade. Note that it isopen-book. You are encouraged to have the lab manual with you when you take it.

Table 1 shows the lab schedule for each Group.

Activity

Week Week of Group A A Lab Group B B Lab

1 Aug 24 In lab A Mean Lab In class A Mean Lab

2 Aug 31 In lab Picturing Motion

3 Sep 07 At home Spreading the Data In lab Picturing Motion

4 Sep 14 In lab Gauging the Force At home Spreading the Data

5 Sep 21 At home Dropping the Ball In lab Gauging the Force

6 Sep 28 In lab Atwood’s Machine At home Dropping the Ball

7 Oct 05 At home Sliding Along In lab Atwood’s Machine

8 Oct 12 In lab Crashing Carts At home Sliding Along

9 Oct 19 At home Sad or Happy In lab Crashing Carts

10 Oct 26 In lab Poe’s Pendulum At home Sad or Happy

11 Nov 02 At home Flight of the Filter In lab Poe’s Pendulum

12 Nov 09 In lab Comedy of Errors At home Flight of the Fillter

13 Nov 16 At home Tale of Woe In lab Comedy of Errors

14 Nov 23 No class Thanksgiving No class

15 Nov 30 At home Tale of Woe

Grading

The grading for the labs will be broken down as follows:

• 15% - Online Quizzes - These will be “reading quizzes” that will be due before eachlab (both in-lab and at home labs).

• 30% - In-class Labs - The lab generally will be complete by the end of the lab.

• 30% - At-home Labs - The lab will be due by the end of the lab period the weekyou are scheduled for the lab, or whatever other time your lab instructor requires.

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Your instructor will let you know if they prefer hard-copies or electronic copies ofyour reports.

• 25% - Final lab report (includes the work done on the final two labs - Comedy ofErrors and Tale of Woe). While you will work with your partner on the experiment,the actual write-up will be your own.

As with any course, if you are having trouble getting the work done, talk to yourinstructor!

The instructors will let you know their grading requirements, and also what to doif you miss a class or have an excused emergency. Just skipping a lab without a validexcuse will get you a zero for the lab, so always check with your instructor as soon as youcan. Failure to do assigned quizzes will also lower your grade. In the event of a conflictor problem with a scheduled lab, the student must make prior arrangements with theinstructor. Otherwise a documented medical excuse is required.

Purpose

For these laboratory experiments, there are three main goals:

1. Become familiar with experimental procedures, including how to identify and solveproblems that arise with real measurements.

2. Become familiar with how to include uncertainties in the analysis of an experimentand how to estimate the overall uncertainty in your experimental results.

3. Become familiar with how to present your results in a form that others can under-stand.

Note: While we will be reinforcing concepts you will learn in your introductory physicscourses, our focus will be on developing your experimental skills - not trying to demon-strate every equation you might see in the course. Although we will see topics in parallelto the Phys 140 an Phys 240 lecture courses, there will be times when we will exploreareas you might not have seen in your lectures.

Introduction to the Mean Lab

Experiments are often portrayed in movies and TV as requiring that the scientist bebrilliant, wear white lab coats, and/or be as obscure as possible when talking to meremortals. Actually most good science starts off with common sense methods and simplequestions about how the technique can be improved at every step of the process. Ifyou are cooking a soup and carefully adding in various ingredients and monitoring thecooking, and then at the last moment pour it on the floor, you will not get a very goodtasting soup. Experiments require that a scientist pay attention every step along the wayuntil the experiment (and analysis) is completed, and identify if the results are reasonableas you proceed.

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The phrase “results are reasonable” is one that students often misinterpret.Getting reasonable results does not mean that your experiment agrees with the ”ex-

pected result” to as many digits on your calculator as possible.In many of the experiments you will be doing, it is very likely that you will not agree

with the “expected result”. That doesn’t imply that your experiment is wrong - but itdoes mean that you need to identify and account for any possible sources of uncertainty(or errors) in your experiment. If you were to repeat the experiment, you would work toreduce the sources of uncertainty that you identified.

Uncertainty and error have different meanings:

• Error - In physics, we use the term “error” to refer to the difference betweena value and its correct or true value. The true value, of course, is often not known.

• Uncertainty - In physics, we use the term “uncertainty” to estimate the differ-ence between a calculated or measured value and the true value.

We measure some physical quantity with an instrument. The values reported by theinstrument are in error by a certain amount. Since we do not actually know the exacttrue value of a quantity, we do not know the instrumental error. Instead, we realizethat the instrumental value is an estimate of the true value and the uncertainty in ourmeasurement is an estimate for the error. Thus the uncertainty is based on the techniquethat you are using for the lab. If you use an instrument with poor resolution, then youwill have larger uncertainties. For example, if you measured length of a field with a meterstick that only meters and decimeters marked on it, you would have more uncertaintyin your measurement than if you used a meter stick with millimeters marked on it. Youwould have “more resolution” with your measuring device in the second case.

A key point is that every measurement has associated with it an uncertainty. Thatuncertainty needs to be recorded as you are taking measurements. In addition, thatuncertainty is quantified. You are not allowed to just say “I think the uncertainty is...”.You have to have a way of estimating the uncertainty. This lab will illustrate how anexperimenter might do this for a simple measurement.

No uncertainty can be introduced into any discussion unless you can definea quantitative estimate of its size. All such estimates need to be justified.

Appendix 3 goes into a more in-depth discussion of the difference between error anduncertainty.

Formulas

Mean, Standard Deviation, and Standard Deviation of the Mean

Statistics are a way for an experimentalist to estimate quantities of interest from theexperimental data.

The mean (average of the data points) usually is a good estimate of the quantitymeasured. If the data has a random component, then averaging several samples togethershould act to cancel out that random fluctuation. That results in the mean being a better

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estimator of the experimental result than any single data point would. Here the bar overthe x indicates the mean of x.

x̄ =x1 + x2 + x3 + x4 + ... + xN )

N= mean(M) (1.1)

Standard deviation (SD) is one statistic measure that can be used to estimate the un-certainty of an experiment. It is an estimate of the error for any one of the measurementsaveraged.

σ =

√

(x1 − x̄)2 + (x2 − x̄)2 + .... + (xN − x̄)2

N − 1= standard deviation(SD) (1.2)

The standard deviation σ is usually a property of the measurement technique. Itdescribes how spread out the data points are around the mean. As you collect moredata points, σ tends to approach a value that is roughly the width of the spread inmeasurements. It seems reasonable that the measurement should become more reliableas the number of trials N increases. The standard deviation of the mean (SDM) can bethought of as the statistical uncertainty in x. We can therefore equate the experimentaluncertainty ∆x with the SDM

σx =σ√N

− standard deviation of the mean(SDM) (1.3)

We can adopt the following:

1. The best estimate of x, is x̄

2. The statistical uncertainty of x is ∆x = σx.

3. We can then write x = x̄ ± ∆x

In the case where we only take one measure, then the resolution of our observation canbe used to define the uncertainty. I.e. it is a educated guess based on the measurementtechnique.

Again - a more detailed discussion of these concepts is presented in Appendix 3 andAppendix 4.

Let us see how to apply these ideas in practice.

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1.1 The value of a Measurement

(In Class activity)

Introduction

This lab serves as an introduction to measurement taking and experimental statisticsthrough the use of calipers.

Accurate measurement requires appropriate tools. When measuring a tabletop, wecould use a meter stick to produce a suitable measurement. The meter stick has grad-uations small enough to attain a measurement to within a millimeter. One can make ameasurement accurate to within a thousandth of a meter. This is good accuracy if thetable is roughly a meter or longer.

To use a meter stick to measure the thickness of a pencil would be inappropriate.Assuming a pencil is roughly 5 mm in diameter; one would want a tool that could givemeasurements accurate to a fraction of a millimeter. The vernier and micrometer caliperswere developed to perform such measurements.

The vernier caliper (Fig. 1.1) is a fairly simple measurement tool. It has two parts:a stem with the fixed main scale (cm) and the vernier, a secondary scale. Each part ofthe caliper forms a jaw to grasp the item being measured. Ten vernier scale divisions fitwithin nine stem divisions (remember the stem is the fixed part), so each vernier divisionis 9/10 as long as a stem division (refer to Fig. 1.2). When the jaws of the caliper areclosed, the first line of the vernier, the zero line, coincides with the zero line of the mainscale.

Figure 1.1: Vernier Caliper

To make a measurement with the vernier caliper, the jaws must be tightly closedaround an object. Wherever the zero line of the vernier falls on the main scale indicates

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the number in the tenths place of measurement. The next line on the vernier that alignswith the main scale indicates the hundredths place, as shown in Fig. 1.2.

Figure 1.2: Reading: 2.64 cm

The micrometer caliper (Fig. 1.3) is another tool for measuring short lengths. It ismore precise than the vernier caliper because it can measure within thousandths of amillimeter.

Figure 1.3: Micrometer

To use the micrometer caliper, an object must be placed between the screw and theframe. The thimble is then turned to advance the screw until the object is touched.The ratchet may click to let one know enough force has been applied and to prevent

9

over tightening. Like the vernier caliper, there are two scales on the micrometer caliper,a circular scale on the thimble and a longitudinal scale along the barrel containing thescrew. The longitudinal scale is divided into half millimeter increments, and the circularscale has fifty divisions. Rotating the circular scale through one full revolution advancesthe screw by 0.5 mm (the distance between two marks on the longitudinal scale). Rotatingthe thimble through one scale division (the distance between marks on the circular scale)advances the screw 1/50th of 0.5 mm or 0.01 mm.

To read the micrometer, first observe the position of the circular scale on the longi-tudinal scale. This yields the number of millimeters to the nearest 0.5 mm. Next, notewhich line on the circular scale aligns with the axial line on the longitudinal scale. Thisgives the fractional portion of the millimeter reading.

Figure 1.4: Metric micrometer reading equals 23.15 mm. 23 whole divisions (= 23 mm);.0 mm divisions are uncovered (= 0.0 mm);15 0.01 mm divisions line up on the thimble(= 0.15 mm).

Formulas

For this experiment, in addition to the statistical quantities discussed above it would begood to remember the following definition/formula:

Volume of a cylinder:

V = πr2h (1.4)

Where r is the radius and h is the height of the cylinder.

Equipment/Materials

For this experiment you will need the following: vernier caliper, micrometer, penny orslug, magnifying lens

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• a vernier caliper

• a micrometer

• a penny or slug to be measured

• a magnifying lens

Experimental Procedure

1 Draw a 4 inch line using a ruler on a piece of paper.

2 Measure the line in centimeters to the greatest precision the ruler will allow.

3 Record the number of centimeters.

4 Calculate the conversion factor between inches and centimeters (divide the twonumbers).

Use the golden rule for reporting measurements: Report all of the digits thatyou know with certainty, plus the first digit that you must estimate.

Length of line: inHow many significant figures are in your measurement?

(this is determined by your ruler).Which is the uncertain digit?Length of line: cmHow many significant figures are in your measurement?

(this is determined by your ruler).Which is the uncertain digit?Calculation of the conversion factor:

(take the ratio of your measurements and include units)

5 Now calculate (see eq. ??) the percent error between the actual value (look it up)and the value you came up with:

6 Take a cylindrical mass (penny) and measure its diameter and height with thevernier caliper. Record this in Table 1.1.

7 Repeat step 6 at least five more times. Be sure to take the caliper off the massbetween measurements.

8 Repeat steps 6 and 7 using the micrometer caliper. Record your results in Table1.2

9 Compute the mean and the standard deviation (eq. ?? ??) of your measurementsand record them in Tables 1.1 and 1.2

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10 Find the volume of the cylindrical mass using your two sets of measurements. Enteryour results in Table 1.3. Remember to use the mean values in your calculationsand use the appropriate number of significant digits.

11 Now that you have the volume, estimate the error in your figures by propagatingthe uncertainty. Record these values in the section below.

Questions:

• Does percent error pertain to accuracy or precision? Explain.

• How could error be improved in this experiment?

• Why are several observations better than one in an experiment?

Main points to remember!!

• All measurements have an associated uncertainty, which should be quantified.

• A calculated result has an associated uncertainty based upon its dependent values.

• The design of an experiment and the skill of conducting an experiment affect theuncertainty in the measurement.

• Uncertainty is used to compare results and draw conclusions.

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No. Height Diameter Radius[mm] [mm] [mm]

1

2

3

4

5

6

StandardDeviationUncertaintyof device

Table 1.1: Measurements of a Cylindrical mass with a vernier caliper.

Data Analysis and Results

VernierVol= [mm3]± [mm3]

MicrometerVol= [mm3]± [mm3]

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No. Height Diameter Radius[mm] [mm] [mm]

1

2

3

4

5

6

StandardDeviationUncertaintyof device

Table 1.2: Measurements of a Cylindrical mass with a micrometer caliper.

Vernier Micrometer

Volume

Table 1.3: Calculated volume of a cylindrical mass

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Lab kit returned by(your name)

on(Date)

Instructor’s Signatureor initials

Table 1.4:

Lab kit Return Page

When you return your lab kit, your instructor will sign below if you so desire:

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Chapter 2

Picturing Motion(In Class activity)

Motion Match Pre-Lab

For the following scenarios, use the coordinate system to sketch a position versus time(x vs. t) graph for the conditions indicated:

x

t

Object at rest.

x

t

Object moving in positive

direction at constant speed.

x

t

Object moving in negative


Object accelerating in positive

direction starting from rest.

x

t

Figure 2.1: Position versus time plots for four situations.

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Notice that the initial position (the x position at t=0) is not specified — only the rateof change of position (velocity) or how the velocity changes (acceleration) are indicated.Any of the curves above can be shifted up or down and still be correct.

Now, for the same situations, sketch a velocity versus time plot (v vs. t) using theaxes below.

v

t

Object at rest.

v

t

Object moving in positive


v

t

Object moving in negative


Object accelerating in positive

direction starting from rest.

v

t

Figure 2.2: Velocity versus time plots for the situations described in Fig. 2.1.

Motion Match

One of the most effective methods for describing motion is to plot graphs of distance,velocity, and acceleration vs. time. From such a graphical representation, it is possible todetermine in what direction an object is going, how fast it is moving, how far it traveled,and whether it is speeding up or slowing down. In this experiment, you will use a MotionDetector to determine this information by plotting a real time graph of your motion asyou move across the classroom.

The Motion Detector measures the time it takes for a high frequency sound pulse totravel from the detector to an object and back. Using this round-trip time and the speedof sound, you can determine the distance to the object; that is, its position relative to thedetector. Logger Pro will perform this calculation for you. It can then use the change inposition to calculate the object’s velocity and acceleration. All of this information can

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be displayed either as a table or a graph. A qualitative analysis of the graphs of yourmotion will help you develop an understanding of the concepts of kinematics.

An object’s velocity is determined the rate of change of position:

v =dx

dt=

∆x

∆t(2.1)

A positive velocity indicates a position that is moving in the positive direction. In thiscase, that means away from the Motion Detector. A negative velocity indicates an objectmoving in the opposite direction.

Similarly, the acceleration is the rate of change of velocity:

a =dv

dt=

∆v

∆t(2.2)

These definitions lead to a couple useful consequences for velocity and position plots.The slope of a position versus time graph is the velocity of the object. The slope of avelocity versus time plot is the acceleration. (We won’t focus on two additional importantrelationships: The integral, or area under the curve, for an acceleration versus timeplot is equal to the change in velocity. The integral of a velocity versus time plot is adisplacement, or change in position.)

Since the positions are measured, there are experimental errors associated with them.Since the velocity is calculated by subtracting two positions at two different times, youwill find that the experimental velocities will typically have larger errors than the positionmeasurements. That is to be expected.

Equipment/Materials

For this experiment you will need the following:

• Logger Pro

• Vernier Motion Detector

• Meter stick

• Masking tape

• Cardboard tube

• Racquetball


Part I. Preliminary Experiments

1. Connect the Motion Detector to DIG/SONIC 1 port on the Lab Pro Interface.

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2. Place the Motion Detector so that it points toward an open space at least 4 m long.Use short strips of masking tape on the floor to mark the 1 m, 2 m, 3 m, and 4 mdistances from the Motion Detector. Be sure to remove the tape when you are donewith the lab.

3. Prepare the computer for data collection by opening Exp 01A from Intro Physicsfolder (see icon on the desktop). One graph will appear on the screen. The verticalaxis has distance scaled from 0 to 5 meters. The horizontal axis has time scaledfrom 0 to 10 seconds.

4. Using Logger Pro, produce a graph of your motion when you walk away from thedetector with constant velocity. To do this, stand about 1 m from the MotionDetector and have your lab partner click “Collect”. Walk slowly away from theMotion Detector when you hear it begin to click. Carefully examine the graph toinsure you understand the measurement. Choose “Experiment Menu” then “StoreLatest Run” to save a good run. Repeat the motion, if it’s better then “StoreLatest Run” if it’s not better, try again.

5. Be prepared to explain what the distance vs. time graph will look like if you walkfaster. Check your prediction with the Motion Detector.

6. Check the distance vs. time graphs that you sketched in the Preliminary Questionssection (Fig. 2.1) by walking in front of the Motion Detector. Once you get anice graph save the data so you can show it to your instructor (4 graphs total).Discuss with your instructor how well your pre-lab sketches match your MotionDetector graphs. Explain any differences. Now, click on the vertical axis andchange “position” to “velocity”. Compare the resulting plots to what you answeredin Fig. 2.2. Again, comment on the results and explain any differences.

Similarities/differences in position plots:

Similarities/differences in velocity plots:

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Part II. Distance vs. Time Graph Matching

7. Prepare the computer for data collection by opening “Exp 01B”. A distance vs.time graph will appear.

8. Describe how you would walk to produce this target graph:

9. To test your prediction, choose a starting position and stand at that point. Startdata collection by clicking Collect. When you hear the Motion Detector begin toclick, walk in such a way that the graph of your motion matches the target graphon the computer screen.

10. If you were not successful, repeat the process until your motion closely matchesthe graph on the screen. Use the “Store Latest Run” command to save your bestattempt. Show your instructor when you have a close fit.

11. Prepare the computer for data collection by opening “Exp 01C” and repeat Steps8 – 10, using a new target graph.

12. Answer the questions for Analyzing Part II on the next page before proceeding toPart III.

Part IIl. Velocity vs. Time Graph Matching

13. Prepare the computer for data collection by opening “Exp 01D”. You will see avelocity vs. time graph.

14. Describe how you would walk to produce this target graph:

15. To test your prediction, choose a starting position and stand at that point. StartLogger Pro by clicking Collect. When you hear the Motion Detector begin to click,walk in such a way that the graph of your motion matches the target graph on the

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screen. It will be more difficult to match the velocity graph than it was for thedistance graph. Have your instructor initial your graph when you get a good fit.

16. Prepare the computer for data collection by opening “Exp 01E”. Repeat Steps 14– 15 to match this graph. Match the graph and answer the questions for AnalyzingPart III below.

17. Remove the masking tape strips from the floor.


Analyzing Part II. Distance vs. Time Graph Matching

1. Explain the significance of the slope of a distance vs. time graph. Include adiscussion of positive and negative slope.

2. What type of motion is occurring when the slope of a distance vs. time graph iszero?

3. What type of motion is occurring when the slope of a distance vs. time graph isconstant?

4. What type of motion is occurring when the slope of a distance vs. time graph ischanging? Test your answer to this question using the Motion Detector.

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Analyzing Part IIl. Velocity vs. Time Graph Matching

Return to the procedure and complete Part III.

5. Using the velocity vs. time graphs from Part III, sketch the distance vs. time graphfor each of the graphs that you matched. In Logger Pro, switch the vertical axis toa position vs. time graph to check your answer. Do this by clicking on the y-axislabel and unchecking velocity; then check distance. Click to see the distance graph.

6. What does the area under a velocity vs. time graph represent? Test your answerto this question using the Motion Detector.

7. What type of motion is occurring when the slope of a velocity vs. time graph iszero?

8. What type of motion is occurring when the slope of a velocity vs. time graph isnot zero? Test your answer using the Motion Detector.

A Final Experiment: Motion Under Constant Accel-

eration

In kinematics, one special case that we frequently see is the motion of an object in freefall. For instance, if we drop a ball that bounces up and down, the object is acceleratingdue to gravity (except for the short intervals when it collides with the ground). Below,

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sketch a plot of the height, velocity, and acceleration versus time, when a ball is droppedand allowed to bounce a few times:

Now, construct an experiment to test your predictions. Attach the Motion Detectorto a ring stand, placed on the table, such that it points directly downward into the largecardboard tube resting on the ground. (The tube merely restricts the ball from bouncingout of view of the detector.) Drop a racquetball down the tube, recording with theMotion Detector. Again, change the vertical axis from position to velocity and then toacceleration to compare with your prediction. (Remember that the position is relativeto the detector, so that it will increase as the ball falls to the ground.)

Comparison of data with prediction:

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position

t

t

t

velocity

acceleration

Figure 2.3: Position, velocity, and acceleration versus time (blank) plots.

25

Chapter 3

Spreading the Data(At Home activity)

Purpose

In this lab, we will work with Excel as a way of displaying and processing data. Manyof you are familiar with Excel spreadsheets. For you, this lab might be primarily review.For others, you know only a few excel commands, so much of this will be new.

• You will learn how to set up a basic Excel Spreadsheet (with labels)

• You will learn how to add data into individual cells

• You will learn how to add multiple data points into columns or rows

• You will learn how to name cells

• You will learn how to use named cells and simple Excel functions to calculate newentries

• You will learn how to plot data from the spreadsheet

Why the emphasis on Excel as a way of recording, analyzing , and plotting the data?It provides a relatively quick way to process even large amounts of data. In addition, itis possible to define relationships such that we can estimate new parameters based onthe experimental data as well as estimate uncertainties.

What is a spreadsheet?

A spreadsheet is a way of storing data in tables. In addition, it is possible to use valuesin the tables to calculate new values automatically as the tables are updated.

A typical spreadsheet might start off as follows:Excel calls a spreadsheet (or table) a Worksheet. Open a new worksheet in Excel.

Along the bottom of the worksheet you will notice a number of tabs. You can changebetween different worksheets by clicking on a different sheet.

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Table 3.1: Sample Spreadsheet

A B C D ...

12 X34 12...

In Excel, the spreadsheet typically has columns labeled with letters, and rows labeledwith numbers. To identify a particular table entry, otherwise known as a cell, you simplygive a letter and a number. For example, in the spreadsheet located above, the cell C2contains the letter X. The cell D4 contains the number 12.

Entering Data

If you want to pick a particular cell, you can move your mouse to that cell and click.In your new worksheet, go to sheet 1, click on A1. It is now highlighted. If you entera number such as “34”, it will be recorded in the cell. Click on a different cell, say B2.Here you can enter a phrase. Enter “Test Phrase”. If you hit return, then you will seethat the mouse (highlighted cell) moves down one to B3. You are ready to enter moredata. The cell ID (the column and row) are also present at the top of the tool bar.

Cells can contain data of all types: Numbers, Dates, Labels, Formulas, and Functions.That data can be displayed in a number of different formats, including percent, dollars,integer, among many others.

If you left-click with your mouse, you can select a cell. If you hold the left buttondown, you can select a range of cells. In contrast, the right button will usually revealadvanced features in a pull-down format. For example, if you are working with a graph,you can use the mouse to select a region of a plot (for example the title or data) and thenpull-up options for that region (to change the title format, or the source of the data).

Naming Cells

One of the great advantages of spreadsheets is that you can name cells, which allowsmuch greater flexibility in their use in formulas.

In Excel, choose “Name” under the “Insert” menu. Choose “Define...” in the list ofoptions. A dialog box will appear. Enter “chosen name” at the top of the dialog box.Be aware that if there is a name in an adjacent cell, Excel will use that name by default.The actual cell address that you are naming appears at the bottom of the box - it isalso highlighted on the spreadsheet. In Excel, the cell name will include the worksheetname, and have $ characters added to refer to the column and row entries. If you wantto edit the cell address, then you can highlight the entry in the dialog box and alter it.

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When you type return (or hit ok in the dialog box), the cell will now have the new nameassociated with it.

Another feature is that you can paste this name onto other cells. Do do that, namea cell. Choose a new cell and select “Name” under the “Insert” menu. Choose “Paste”.A different dialog box appears - listing all the names of the named cells. You select one,and then hit OK. The new cell now will have a formula that refers to the named cell.Whatever the contents of the named cell are, they are now also part of the new cell aswell. If you change the value of the named cell, the second cell’s contents also change.

Try making a spreadsheet that contains named cells in A2 (named as distance),A4 (named velocity) and A6 (named as acceleration). Put in values of 2.0, 4.0, 8.0respectively. Paste the names into B2, B4, and B6 respectively. You should see the samevalues as the named cells. Now change the values of A2, A4, A6 to 8.0, 4, 0, 2.0. If thevalues of B2, B4, and B6 are not 8.0, 4.0, 2.0, then you have named one or more themincorrectly. Make sure you can do handle naming cells before proceeding.

Formating Cells

You can control the format of the cells (how many digits are displayed. Select the cellsyou want to change. Go to “View” and click on “Formating Palette”. A box will appearto the side. Under the category “Number”, you will see various options for how youwant your numbers displayed. Usually it defaults to “General”. However, most of thetime, you will want to control the number of digits displayed. To do that, choose the“Number” option, and then click on the buttons below to shift the digits left or right. Itstarts off with two digits past the decimal point. Set it so three digits are displayed, i.e.0.000).

Note that you can also alter the display format of a single cell by clicking on it andthen changing its format using the “Formating Palette’. You are NOT changing theunderlying

Entering Formulas

Now that we know how to create named cells, we can create formulas easily. You cancreate formulas just using cell locations (such as E3) but it is easier to check your workif you use a name (such as distance or acceleration).

The contents of a cell in Excel will be considered a formula if the first character is anequal sign. Two examples:

• =B3 Set the cell’s contents to whatever is in cell B3

• =vo Set the cell’s content to the cell named v0. If you have not yet named a cellv0, then the error message “#NAME?” will appear in the cell.

Excel has a large number of functions defined for you to use. These include commonmath functions like:

• ∗ multiplication

29

• − subtraction

• / divide

• ˆ raise to the power (i.e. 10ˆ 4 === 10,000)

• () set the order of operations

• For example: =36*B3+vo+7 Multiply the contents of cell B3 by 36, and add thecontents of vo and the value of 7 to the total.

• For example: =36*(B3+vo+7) Add the contents of cells B3, v0, and the value 7.Multiply the total by 36.

In addition, there are many special functions that you can use in formulas. To seewhat is possible, choose a cell in your spreadsheet and “insert” ”function” (on the menu).The dialog box that pops up will list many possible functions. Choose one that looksfamiliar and hit OK. The next dialog box that pops up will help you select the arguments(i.e. cells) that you need. You can either enter the cell numbers or click on the arrow- which allows you to go back to the spreadsheet and select the cells using your mouse.Hitting Enter will then complete the selection.

Depending on the function you select, you may have to select a cell that contains anangle (for something like sin() or a list of cells for a function like sum() that requiresseveral cell entries.

You also can type a function directly into the cell. You can use the mouse to choosethe cells you want as arguments for functions directly.

Create a list of cells, containing say 5 numbers. Find the functions (AVERAGE,STDEV, SQRT, and COUNT. We want to find the mean (i.e. average), standard de-viation (i.e. STDEV), and standard deviation of the mean. The first two are easy, asExcel has those functions defined. To find the standard deviation of the mean is a littletrickier. Recall from our first lab that:

σx =σ√N

− standard deviation of the mean(SDM) (3.1)

So, we will want to define a function that takes the result of the standard deviation(the cell containing STDEV) and divides it by the square root (which is SQRT) of thenumber of cells in our list. We could count and enter the number 5. However, there aretimes when it is useful to have Excel keep track of the number of cells. To do that, weuse a function called COUNT, which will see how many cell entries are in our list. Forexample, if our list spanned C4 to C8, we would have 5 entries.

The formula in that case would be: “ =STDEV(C4:C8)/SQRT(COUNT(C4:C8))”You can replace STDEV(C4:C8) by the cell that contains STDEV(C4:C8).

So, create the three cells containing the mean, standard deviation, and standarddeviation of the mean for your list of 5 numbers.

If you change your numbers in your original list to 12, 10, 9, 8, 11, you should findthat you get 10.000, 1.581, and 0.707. If you don’t, check your formula entries.

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Plotting

Often the best way to analyze data is to plot it. We will use plots frequently to examineour data.

To plot data, decide which columns should be plotted.

1. Choose “Insert” in the menu and choose “Chart”

2. Chose “XY (Scatter)” on the first dialog window. Click “Next”

3. Choose the data to plot by switching from “Data Range” to “Series” using thefolder tab near the top of the dialog window.

• (a) Click “Add” to get your first data series.

• (b) Click the button at the right of the “X Values:” entry window. The dialogbox disappears and you highlight the cells in the appropriate column. HitEnter after the box shows all the desired column entries have been chosen.

4. Now use the “Y Values:” entry window and chose your Y data.

5. Click Finish

As you might expect there is a lot of refinements that you can apply to your data.Multiple data sets can be plotted. You can also add labels or colors. Experiment byclicking with either the left or right mouse button on various portions of a graph and seewhat you change.

To play with this, let us create a simple spreadsheet. X numbers = 1, 2, 3, 4, 5, 6,7 , 8 Y numbers = 1, 4, 9, 16, 25, 36, 49, 64. Enter these numbers into the spreadsheetinto two columns X and Y. Create a plot, and label it.

A sample chart can be found on Blackboard or in the desktop folder (Intro PhysicsLab/excel worksheets/2CurvesOn1Chart.xls). The worksheet delves further into chart(plot) making.

You should end up with something like Figure3.1:Note that we did not actually put a chart title - so we got a default chart title name.

Trendlines

A nice feature of Excel allows you to plot a trendline on a plot. To do that, you click onthe graph you have made on a specific data point. Right clicking will bring up a menu.Depending on where you are on the graph, different menus might appear. Choose “addtrendline”. You can add the equation to the graph by setting the appropriate option onthe options page.

For the sample plot, select a polynomial of order two. You will get something likeFigure 3.2.

Note that some trendline options might be blanked out. That usually means thatyou have a zero in your x data, which cause some functions to be ignored by Excel.

31

Figure 3.1: Sample Plot with Labels

32

Figure 3.2: Sample Plot with Trendlines

33

Saving Data

For any spreadsheet that you wish to keep, you will need to save a copy of the file. Youmight wish to save as you go along, so that a computer glitch at the final entry does notwipe out an hour or more or work. To do that, just go to “File” and use the “Save As”entry. Be sure to keep a copy somewhere (like a flash-drive or your own account) wherethe file will not be removed. On the computers in the lab, the files are removed everySunday. Your instructor can set up storage areas for you if needed.

Text Box

To add a textbox, open the “View” menu. Select Toolbars and Click “Drawing”. Toinsert a Text Box, which is the one with the little A on it. Use your mouse to drag thebox to the size you want. Start typing. When you are done, move the mouse outside theText Box.

Some Sample Data

To illustrate what you have learned in this lab, you and your partner will each create anew worksheet. The worksheets will be turned into your instructor when the lab is due.

Below in Table 3.2, you will find 4 columns of numbers. Choose one column. Yourlab partner and you should choose DIFFERENT columns. While you and your partnercan help each other, the spreadsheet you create should be your own work.

Table 3.2: Test Data

Column Wiley E Coyote Chases The RoadRunner All Around(m/s) (m/s) (m/s) (m/s)

10 10 10 1010 9 10 1010 8 10 210 7 10 210 6 15 1

Beep! 2 5 20 1Beep! 1 4 50 -1

1 3 100 -20 2 100 -100 1 100 -20

For your spreadsheet, do the following:

1. Put your name, your lab partner’s name, and your lab section/group into a TextBox.

2. Label one cell as Column

34

3. In the next cell to the right - enter the label from the column you chose.

4. Label one cell as Time (seconds)

5. Enter the numbers 1 through 10 in the 10 cells below the Time label

6. Label one cell as “Velocity (meters/second)”

7. Enter in the 10 cells below that column the data entries from the column you chose.

8. Label one cell as Mean (meters)

9. In the column to the right - calculate the mean of the ten data entries

10. Label one cell as SD (meters/sec)

11. In the column to the right - calculate the standard deviation of the entries

12. Label on cell as SDM (meters/sec)

13. In the column to the right - calculate the standard deviation of the mean

14. In the cell beside the one labeled as Velocity, label that cell as “Distance traveled(meters)”

15. In the cells beside the velocity data, enter a formula (distance traveled) =velocity* 1.0 where the velocity is the value of the velocity cell, and time is one second.

16. Calculate the mean, SD, and SDM of the distance traveled.

17. Plot velocity vs time and label the axes (remember units)

18. Plot distance traveled vs time and label the axes (remember units)

19. Save your work and turn it in.

35

Chapter 4

Gauging the Force(In Class activity)

Purpose

Computers can be used for a host of applications. In this laboratory, the computer willserve to record the data, analyze the data, and display the data. You will be introduced tothe lab’s equipment and methods. Thus, this lab should be seen as a training exercise forfuture labs. One key thing to note is where important information about the equipmentor the software can be found. That will be useful for trouble-shooting later. As you goalong, if you have questions or are unclear on something, consult with your instructor.

• To learn the basics of data collection with Logger Pro software, LabPro (interface)and measurement probes hardware.

• To learn to manipulate and analyze data

• To write out a brief summary of the experiment

Materials

• Logger Pro software

• LabPro interface

• force probe

• ring stand

• weights

37

Introduction

In these labs, the instructor will serve the role of mentor, which means that they will offersuggestions on how you can solve your problem - rather than just telling you what to do.For you to benefit, you need to be thinking about the lab and raising specific questions.If your lab is not working, then you will need to be able to say identify where the problemis. A common example is: check to make sure that the power is on (otherwise LabProwill not work).

At each lab station there is a colored binder that contains notes on the equipment andon the software you are using. The binder will contain diagrams and explanations thataid in the use and understanding of both the hardware and the software. Most equip-ment will come with a manual. A manual will describe the installation, maintenance,and calibration procedures. It has diagrams illustrating proper connections and how toproperly use the equipment. For this lab, the colored binder is your laboratory stationmanual. If you have a problem, then the solution can be found in this binder or yournotes. When you solve a problem, it is important to make notes of what the problemwas and how you solved it. You might encounter a similar problem in the future!

You should become familiar with the contents of the binder, since you will be expectedto refer to the binder from time to time when conducting your experiments.

As you proceed through the labs, if you have suggestions on how the binder can beimproved, please let the instructor know.

Note that as the labs become more advanced, you will take on more responsibilityabout how to conduct the experiment. The labs will discuss a relationship or quantityof interest and let you work out the details of the procedures and the analysis.

In this lab, we will:

1. Hook up a probe through the LabPro to the computer.

2. Set up the software to record the data from the probe.

3. Record a set of data.

4. Analyze the data to verify the relationship between measured quantities.

5. Repeat the measurement several times

6. Explore the meaning of mean, standard deviation, and standard deviation of themean.

If you have trouble, call on your classmates or the instructor for help.However, after you receive help, be sure that you repeat the process on yourown in the future.

38

Making A Measurement

At the end of the lab, there is section for the instructor record their evaluations as youproceed. As you complete each step, have your instructor enter a comment in this table.

The Diagram

FInd in your manual the diagram that shows the basic setup. Examine your connectionsand see that they agree with the diagram. In this lab, your sensor is a device calleda ”Dual-Range Force Sensor”. Look at the manual for the sensor (located later in themanual) and read through its description. Record the range and resolution of the sensorin the table at the back.

Set the force probe to the 10 N scale using the switch on the force probe. Find theLabPro and note where various probes can be attached. At this point you may have tomake an educated guess as to where your probe will attach. Plug the force probe intoCH1. The white connector should slide in easily. Do not force it. You are trying todevelop an overview.

Running the Software

Start up the Logger Pro program. Logger Pro will set up the probes and the interfacefor LabPro. You can then record data according to your specifications. Be sue that theLabPro is on and is connected to the computer (via the USB port). Check the status ofthe sensor connection to the LabPro interface. On the top toolbar of Logger Pro, clickthe LabPro button on the left. A dialog box should open showing the interface and theattached probes. If the correct sensor is not automatically shown in the correct inputwindow, click the “Identify” button. If you apply a force to the probe by gently pullingon the hook, the value of the input window and the top toolbar changes appropriately.

If you have a problem, first try and solve it yourself. If you can’t solve theproblem, ask your instructor for help.

Tutorials for Logger Pro

You need to be able to use the Logger Pro program and you will need to come comfortablewith using the mouse and menu bar. A complete guide to the LabPro interface is in thestation manual (green binder) along with a Quick Reference Manual for Logger Pro.There is also a short description of the force probe (Dual-Range Force Sensor). Scanningthese will be useful for an overview of how the lab equipment will work.

The help files for Logger Pro are complete and easy to use. Use the “file” menuand the “open” command to start the first tutorial in the tutorial folder (01 GettingStarted.xmbl). Since the tutorial will be displaying temperature data, you will needto ignore sensors. Take a few minutes to explore this tutorial. You can return to thetutorials, help files and manuals as you go through the lab. When you would like tocontinue with the force probe, use the “new button” on the “file” menu. This will close

39

the tutorial and the Logger Pro software should automatically sense the force probe andreturn you to your previous configuration.

Calibrating the Force Probe

Calibrate the force probe. Either reach the calibration dialog box by clicking on the probepicture in the LabPro setup dialog window (same window as before) or use calibrate inthe “experiment” pull-down menu.

To calibrate, apply a known force by hanging a weight from the probe.Enter the value (type it). Record the point by clicking the KEEP point. Apply a

different known force. Enter its value. Record this calibration point. The probe shouldnow be calibrated.

Remember that force and mass are different quantities. The force that we are mea-suring is the result of gravity. This is illustrated by the following calculation:

Mass Acceleration Force Comment(gm) (kg) (m/s2) (N)200 0.2 9.8 1.96 F=mass x acceleration

Table 4.1: Force Calibration

Configure the collection mode as Events with Entry. Use the data collection buttonon the toolbar (to the left of the Lab Pro button). Set the entered value to be the massin grams. To perform a measurement, hang a mass from the probe, click the collectionbutton and enter the mass value.

Once the probe is configured, a good experimentalist will check the setup by per-forming test measurements. Add some weight and check that the results agree with yourexpectations.

One nice feature of calibrating the probe is that each experiment can fix the zerovalue. You may recalibrate so that the force on the probe with holder only is centeredas zero. Now, record a second point where the force is calculated using only the massthat was added without including the holder mass. This calibration allows you to havethe mass holder in place but you don’t need to include it as part of the recorded mass.

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Performing A Measurement and Analysing the Re-

sults

Perform an experiment using 10 measurements that vary between 0 and a few (less than10) Newtons.

Copy the data to an Excel spreadsheet using cut and paste. Label the columns x andy. Add one more column “y uncertainty”. Take a guess at the value for the uncertaintyin each y value and enter it in the column. Add a text box to your spreadsheet anddescribe your reasoning for the error you assigned to the y values. You should mentionthat the errors for the x values are assumed to be very small and therefore have beenneglected for this experiment.

All numbers used or measured in any laboratory experiment must havean assigned uncertainty. Develop the habit now of always including an error.

Plot the data using Excel. (see the appendix or the previous lab). Add two trendlinesto the plot. For the first, use the function y = AeBx; for the second, use y=mx+b, whereA, B, m and b are the parameters found in the fit, and y and x represent the variables.You will need to set the trendline to display the equation on the chart. If necessary,ask your instructor or colleagues for help. Note that if you have a zero value for x or y,exclude that data point - some trendlines will not be able to be used.

The trendline function represents an attempt to mathematically describe the relation-ship between the force and the hanging mass. Enter these parameter values into a tablebelow the graph. Be sure to include a column for the errors in the parameters. For themoment, just enter “unknown” for the errors of the fitted parameters. Add commentsand labels where necessary. Add a text box and enter a brief comment on the fits to thedata. Have your name and your partner’s name on the spreadsheet. Save the spread-sheet. Show the instructor your completed spreadsheet. Be prepared to discuss what youthink it means. Feel free to discuss procedures with other students in the lab.

Repeating Measurements

Record ten or more measurements using the same mass.

Copy this data into another worksheet in your spreadsheet. Find the mass, standarddeviation (SD), and the standard deviation of the mean (SDM). Based on your workin previous labs, interpret what these quantities mean. You may wish to consult theAppendix and your notes from the earlier labs.

To highlight these important results, calculate and label these values in the spreadsheet. Be sure you understand the correct results. Put a textbox in your spreadsheetthat clarifies the meaning of the mean, SD, and SDM for this experiment. Have yourinstructor review your work.

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Changing Resolutions

If you change the scale on the sensor from 0-10N to 0-50N, you have poorer resolution inyour measurements.

First put a mass on - does it give the same value you expect?Change the calibration - does it give the value you expect for the test mass?Repeat the previous two sections and see how your results change. Use the identical

masses that you did in the previous sections to minimize uncertainties due to differentmasses.

Record your results on two new worksheets and show your instructor your results.

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Table 4.2: Setup and Performance of Force Probe ExperimentStep Performed Instructor’s Evaluation

Computer and Interface ConnectedProbe Attached and Configures

Calibration Performed and understoodVerified that measurement sensible

Data Recorded

Table 4.3: Analysis of the First Data (10N Scale)Step Performed Instructor’s Evaluation

Data and Errors EnteredData plotted and Fit

Fit Parameters Tabulated

Table 4.4: Analysis of the Second Data (10N Scale)Step Performed Instructor’s EvaluationData Recorded

Statistics Calculated

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Table 4.5: Analysis of the Third Data (50N Scale)Step Performed Instructor’s Evaluation

Data and Errors EnteredData plotted and Fit

Fit Parameters Tabulated

Table 4.6: Analysis of the Fourth Data (50N Scale)Step Performed Instructor’s EvaluationData Recorded

Statistics Calculated

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Chapter 5

Dropping the Ball(At Home activity)

Introduction

In this experiment you will:

• determine the local acceleration due to gravity, g.

• determine the factors that influence the precision of the experiment.

• determine the accuracy of the measured quantity, g.

Formulas

An object that is moving in a linear fashion under constant acceleration can be modeledby the equation of linear motion:

x(t) = x0 + v0t + 1/2at2 (5.1)

Where x(t) is the position of the object at time, t; x0 is the position of the objectat time t = 0; v0 is the velocity of the object at time t = 0; a is (the magnitude of) theacceleration of the object. The acceleration of the object can be found from the sum ofthe forces acting on the object using Newton’s Second Law:

Fnet = ma (5.2)

In Eq. 5.2, Fnet is the net force1 acting on the object and m is the mass of the object.For this experiment we will assume that the force on the object due to air resistance is sosmall that it can be neglected. Thus, the net force on the object is only the gravitation

1Note that force, acceleration, velocity, and displacement are all vectors. Because the motion studied

in this experiment is one dimensional these equations only deal with the magnitudes of these quantities.

45

attraction between the ball and the Earth. This force is given by Newton’s Law ofUniversal Gravitation:

F = GmM

r2= ma = mg (5.3)

Here G is the universal gravitation constant (≃ 6.6730 × 10 −11 Nm2kg−2), m isthe mass of the object; M is the mass of the Earth (≃ 5.9742 × 1024 kg); and r is thedistance between the center of the Earth and the center of mass of the object. For smallobjects at the Earth’s surface this distance is the radius of the Earth at the point of theexperiment (≃ 6.371× 106 m). If the Earth were perfectly spherical and all experimentswere conducted at the surface of the Earth, then the local acceleration due to gravity(symbolized by g) would be

g =GM

r2(5.4)

The “g” of Eq. 5.4 is the acceleration that an object feels at the surface of the Earth.If the object starts with zero initial velocity (v0 = 0) with the initial position is someheight (above the ground), h, and the final position = 0 then Eq. 5.1 becomes:

0 = h − gt2

2(5.5)

Solving this last equation for “g” yields:

g =2h

t2(5.6)

Here, we’ve been careful with our signs since we are considering up to be positive,the downward acceleration due to gravity is a = −g. Solving for t yields:

t = sqrt2h

g(5.7)

Now consider what happens if the object bounces. By symmetry, the time from thefirst bounce to the second bounce is twice the time for the object to fall from the bounceheight to the ground. Using the new bounce height, hb and the bounce time, tb yieldsthe following equation:

tb = 2sqrt2hb

g(5.8)

The local acceleration due to gravity, g can be found from the bounce height andbounce time by the following equation:

g =8hb

t2b(5.9)

Equipment/Materials


• stopwatch

46

• Measuring tape

• Hi-Bounce ball (35 mm diameter).

• Hi-Bounce ball (45 mm diameter).


1 Find a suitable site for the experiment where the bouncing ball will not hit peopleor equipment.

2 Measure the height of the drop to the nearest cm.

3 Place the smaller ball so that the bottom of the ball is at the measured height.Using the stopwatch, determine the time from the release of the ball to the impactwith the floor.

• Suggestion: Have one lab partner hold the ball at a fixed height while the secondlab partner is operating the timer. The first partner should initiate a brief count-down (3-2-1-drop) which the second partner should use to start the timer coincidentwith the release of the ball. The second partner should listen and stop the timerwhen the ball bounces.

4 Repeat 1–3 at least fifteen times.

5 Organize the data in a suitable Excel spreadsheet with appropriate labels. Anexample data table might look like Table 5.1 below.

6 Repeat the sixteen drops from the same height using the larger 45 mm diameterball. Organize the data in a suitable Excel spreadsheet.

7 For each of the quantities measured in the experiment, determine the uncertaintyin the measurement. Use/Add two new columns to the table(s) and label themappropriately, as shown in Table 5.1.

Bounce ProcedureThe experiment will be re-run using a slightly different procedure:

1 Place the smaller ball so that the bottom of the ball is at the given height. Dropthe ball from the given height and measure the height of the first bounce to thenearest cm (bounce height).

3 Using the stopwatch, determine the time from the first bounce on the floor to thesecond bounce (bounce time).

• Suggestion: Have one lab partner hold the ball at a fixed height while the secondlab partner is operating the timer. The first partner should the release the ball. Thesecond partner should listen, start the timer when the ball bounces the first timeand then stop the timer when the ball bounces the second time. The first partnershould mark the height of the first rebound so that it can be reliably measured.

47

Ball Diameter = mm

Trial No. Height Height Time Time[m] Uncertainty [m] [s] Uncertainty [s]

1

. . .

16

Table 5.1: Dropping the ball: Sample table for the raw experimental data.

3 Repeat 1–2 at least fifteen times.

4 Organize the data in a suitable Excel spreadsheet with appropriate labels.

5 Repeat the sixteen drops from the same height using the larger 45 mm diameterball. Organize the data in a suitable Excel spreadsheet.

6 For each of the quantities measured in the experiment, determine the uncertaintyin the measurement.


1 Add a column to the Excel spreadsheet and label it “Experimental g”. Include theappropriate units. Determine the local acceleration due to gravity g, for each of themeasurements. Be sure to round off the reported value of “g” to the appropriatenumber of significant figures. At the bottom of each of the four tables, include threenew lines labeled “Average g”, “Standard Deviation (SD) of g” and “StandardDeviation of the Mean (SDM) of g”. Calculate the average, standard deviation,and standard deviation of the mean for each table of data.

2 Determine a value of the local acceleration due to gravity, g, by using Eq. 5.4.Report this value in the spreadsheet as “Average Earth surface g”. Note that thisvalue is NOT the textbook value of ‘9.81 m/s2.

3 Using the data from Google Earth and the National Geodetic Survey, the valueof the local acceleration due to gravity for the second floor JMU Physics lab is9.79888(2) m/s2 The appropriate applet can be found at

http://www.ngs.noaa.gov/cgi-bin/grav_pdx.prl

.

48

Bounce height

Bounce TimeTime

Heig

ht

Figure 5.1: Experimental setup for the Bounce Procedure

• The reason for the discrepancy in the values of g is that the Earth is not perfectlyround and James Madison University is not at sea level. Report this value in thespreadsheet as “Accepted JMU g”.

4 Determine the uncertainty in the experimental value of g using the standard devi-ation of the mean (SDM). Compare your measurement to the generally acceptedJMU measurement by determining how well the accepted value lies within yourvalue ± your uncertainty (your Standard deviation of the Mean). See Eq. 5.10.

5 If the number calculated above is less than 3, then your value did not excludethe accepted value with at least 95% certainty. In other words, your value isconsistent with the accepted JMU value. Using a text box, add a statement toyour spreadsheet about whether each of your four values is consistent with the“Accepted JMU g”.

6 Did you have systematic error in your data compared with the accepted JMU valueof g? How do you know? How do you account for this error? Using a text box,

49

add a statement to your spreadsheet answering these questions.

7 Were your results consistent across all four experiments? Is the drop or bouncemethod better at determining g? How do you know? Using a text box, add astatement to your spreadsheet answering these questions.

8 Does the size of the ball appear to change the value of g? How do you know? Usinga text box, add a statement to your spreadsheet answering these questions.

|Accepted JMU g − Y our value

Y our Uncertainty| (5.10)

Lab ReportThe spreadsheet should have the following clearly labeled items:

A Four data tables with trial, height, uncertainty in height, time, and uncertainty intime.

B Four experimental averages, experimental standard deviations and experimentalstandard deviations of the mean.

C Accepted Earth surface g and Accepted JMU g.

D Four comparisons of experimental value to the accepted JMU g.

E Four statements about the consistency of the experimental g with the acceptedJMU g.

F A statement about the possibility of systematic error in the experiment.

G A statement comparing the bounce versus the drop method of determining g.

H A statement comparing the size of the ball and the determination of g.

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Chapter 6

Atwood’s Machine(In Class activity)

Newton’s second law (F = ma) is a cornerstone of physics. Given that, how can one testor verify the law. Given that you are in an lab, the answer is that it can be tested - asall laws of physics can be. How one tests physical laws is one of the things this lab willhelp you explore.

To do that, we will use a device known as ’“Atwood’s machine” - which involves apulley. Our machine is modified to use a cart on a low-friction surface to help isolatethe physical processes. The need to minimize friction is one of the major experimentalconcerns when trying to verify Newton’s second law.

Purpose

1. To use a cart track as a system for minimizing the effects of friction.

2. To develop sound methods for insuring that experiments are working.

3. To examine analysis techniques.

4. To test Newton’s second law.

This lab will illustrate for you many of the processes involved in performing an exper-iment. One key process is the act of questioning. A good experimentalist is continuallyexamining their experiment and questioning what can be done to improve the experi-ment and remove or reduce sources of uncertainty. To help you see how an experimenterthinks, there will be questions posed in the lab manual that are designed to alert you toimportant issues. In future experiments, students will be expected to raise (and answer)these sorts of questions on their own.

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Materials

For this experiment we will need the following:

1. Photogate with LabPro

2. LoggerPro software

3. cart tracks

4. carts

5. weight hanger

6. balance

7. assorted masses

Background Theory

In Figure 6.1 you see a very simple system, consisting of two masses (m1, m2). Themasses are connected by a massless, inextensible string, which passes over a massless,frictionless pulley. When you apply Newton’s second law to the system, if you define Tas the tension on the string, you find the following equations:

Forces on mass 1 : m1a = T (6.1)

Forces on mass 2 : m2a = m2g − T (6.2)

If you add Equations (6.1) and (6.2), then the final equation of motion is given by:

(m1 + m2)a = m2g (6.3)

This last equation can be tested experimentally. However, to do so will require theaid of an (almost) frictionless cart track. The “almost” will be something that you as anexperimenter will have to keep in mind.

Constant Accelerating Force

We can explore (6.3) in several ways. The first would be to keep the acceleration forceFa = m2g constant. We can then check and see how the acceleration a depends on thetotal mass m1 + m2. To see how, rewrite Equation (6.3) as follows:

LetFa = m2g (6.4)

then, we can write:

a = Fa

[

1

m1 + m2

]

(6.5)

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Figure 6.1: Atwood machine: A sliding mass is connected to a falling mass via a pulley

53

When written this way, we see that if we plot acceleration vs the reciprocal of thetotal mass (i.e. ( 1

m1+m2

)), it should be a straight line that runs through the origin with aslope equal to Fa.

Constant Total Mass

Another way to check the 2nd law using Equation (6.3) is to keep the total mass (m1+m2)constant and check the dependence of the acceleration a on the acceleration force Fa =m2g. We will write Equation (6.3) as follows:

Let

Fa = m2g (6.6)

then, we can write:

a =[

1

m1 + m2

]

Fa (6.7)

At first glance the equations are the same, but they have been written in a way tohelp you recognize what is being changed. In this case, if we plot a graph of accelerationversus the accelerating force, we should find a straight line through the origin with a slopeequal to the reciprocal of the fixed total mass.

slope =[

1

m1 + m2

]

(6.8)

Of Slopes and Intercepts

To summarize: In the cases above, Equation (6.5) and (6.7) express the acceleration interms of y = mx + b, where y = acceleration, m = the slope of the fit, b is the interceptat x =0 (set to 0) , and x is the thing you are changing.

In the case of the constant accelerating force, the slope is given by Fa since you arevarying the total mass (or more precisely

[

1

m1+m2

]

).

In the case of the constant total mass, the slope is given by[

1

m1+m2

]

, since you arevarying the accelerating force Fa.

Thus, we have two cases (constant accelerating force, constant total mass) that wecan use to test if Newton’s second law.

Experimental Techniques

To do this experiment, you will use a cart track. The cart track is a device that providesan approximately frictionless system for mechanics experiments. The cart rides on wheelswith minimal contact with the track, resulting in very low friction.

To measure the acceleration of the cart, you will use LoggerPro software, the LabProinterface, and a photogate. The spokes of the pulley create blocked and unblocked statesin the photogate, triggering the Logger Pro software.

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Figure 6.2: Experimental Setup

55

To set up the Logger Pro software, open the experimental setup file “Atwood’s Ma-chine”. You will find the file in the Experimental Setup Files / Intro Physics folder onyour laboratory desktop. This file will configure the photogate and set up the collectionmode. The setup uses the photogate to measure the time interval between the arrivalof adjacent pulley spokes at the photogate. From this time and the distance betweenthe spokes (which is already initialized in Logger Pro) the average velocity and averageacceleration are calculated and recorded. Examine the experimental setup, including thesoftware, before proceeding. Now is a good time to ask your instructor if you have anyquestions about how the data is to be taken.

Measurements and Analysis

A word about good experimental techniques. When faced with a new experiment, oneshould think about what is necessary to successfully perform the measurement. The firststep is to understand what will be recorded and why are those quantities being measuredand not other ones?

In this experiment, for example, we want to explore the motion of objects under theinfluence of applied forces. To do that, the experiment is designed to measure time in-tervals. If you know (or can determine) the distance moved by the cart over a measuredtime interval, then you can calculate velocity and acceleration. In a more detailed ex-periment, the methods used to measure time and distance should be explored to verifytheir accuracy. For this lab, in the interest of time, you can assume that the proceduresare adequate.

Understanding the Experimental Setup

The goal of this experiment is compare the acceleration measured to the accelerationpredicted by Newton’s second law from a measured force and a measured mass. Listthese three quantities on your data sheet.

Some questions that should be answered by an experimenter are:

• What forces are being applied ? Did you consider all of them?

• Which forces are relevant according to the ideal design?

• Which forces may influence the experiment because the experiment is not ideal?

• What object is actually experiencing the force and moving? Think about thisquestion before you answer.

• How does one measure the applied force?

Create an Excel spreadsheet, and briefly summarize your response to theabove questions in a paragraph or series of short answers in a text box inyour spreadsheet. If you understand the experiment, you are more likely to performthe experiment correctly and identify your primary sources of uncertainty.

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Testing Your Equipment

A good experimentalist will also test their equipment. Since the photogate is designedto directly measure time intervals, you should test it. Move the cart by hand very slowlyso that the pulley spokes initiate photogate state changes. If you manipulate the motionin simple ways, you can determine what is actually being measured.

First try some movements and note what is recorded by Logger Pro. Then performcontrolled motions where you predict what the results should be. With a watch you canroughly measure time intervals and compare what Logger Pro measures with what youexpect. From your observations of the photogate, describe what starts the data recordingand what is measured. Logger Pro produces a table of values of t, x, v, and a. You shouldbe able to ascertain how the time and position data is measured. (Logger Pro uses thosedata points to derive the velocity and acceleration data. You don’t need to discover thealgorithm used to find v and a).

Summarize in a text box in Excel what and how Logger Pro measures thet and x data.

Testing Your Experimental Procedure

Here we will use the constant accelerating force to explore our experimental procedure.Measure the mass of the cart and record this value, mc on your spreadsheet. Remem-

ber to clear label your units. Also estimate the uncertainty in your measurement. Oneexample is recorded in 6.1

mass of cart 510 gramsuncertainty in mass 16 grams based on reproducibility and ability to read scale

Table 6.1: Example Cart Mass Table - YOUR NUMBERS WILL BE DIFFERENT

In this exercise, you will want to verify that the track is level. There are levelingknobs on both supports. Now place one cart somewhere in the middle of the track sothat it is not moving. If it starts moving, that is a good hint that the track needs someserious adjustment! Be sure that the cart does not have a tendency to roll one way orthe other before you continue.

After the track is level, attach the mass hanger and string to the cart as shown inFigure 6.2. Start with an accelerating mass of m2 = 50 grams. Hold the cart at the endof the track and start the LogPro program. After releasing the cart, the event timer willbe triggered by the pulley spokes passing through the photogate.

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Verify and demonstrate that your measurement works

Once you have acquired your data, you need to examine it.

• Copy the columns of t, x, v, and a into your Excel spreadsheet.

• Label the columns appropriately.

• Plot the data for a trial (x vs t and v vs t).

• Using your Excel spreadsheet, calculate the mean, SD, and SDM for the accelerationcolumn.

• Fit the velocity versus time data to a straight line

• Compare the slope from the fit to the average of the accelerations.

• Repeat the measurements a few times and compare your results.

As you take the data, consider the following questions:

• Are the values reasonable?

• Does the fitted curve pass close enough to all the data points?

• Do similar measurements (trials) give similar results? How close should results beto each other? (Hint—they are not going to be identical.)

• What did you expect? If you predicted incorrectly, do you understand why?

• How do you expect acceleration to depend on time?

• How can you examine the data to verify the expected behavior?

• If verified, what analysis method do you suggest for estimating the accelerationfrom the measurements of acceleration?

• How do you expect the velocity to depend on time?

• How can you examine the data to verify the expected behavior?

• If verified, what method do you suggest for estimating the acceleration from themeasurement of a velocity?

58

To complete the above analysis, you will need to include the following in your spread-sheet:

• 1 sample data table from Logger Pro (do not submit all your data).

• a plot and fit of the velocity versus time

• a plot and comments on the position versus time

• the analysis of the acceleration column (including the mean, SD, and SDM)

• a statement with your overall analysis of these measurements

In your statement, make sure it is clear what conclusions should be drawnand why various tables and plots are included. Imagine the reader is familiarwith the experiment but may not have read the manual. Note: Instructorswill not do your analysis.

At this point, you are ready to record data efficiently. You will remain alert to preventproblems from ruining your data but you should be able to quickly record the necessarydata for the parts A and B below. Look at how much time is remaining. Cut, paste, andsave the data before leaving. For this lab, if necessary, the analysis can be done outsidethe lab. Your instructor will tell you how to turn in the lab (electronic or hardcopy) andthe due date.

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Case A - Constant Accelerating Force

You will now proceed to study how the acceleration changes as the mass of the cartchanges. Analyze each measurement by fitting the velocity versus time and by averagingthe acceleration data (i.e. taking the mean, SD, and SDM).

Repeat the experiment in 50 g increments by adding 50 g to the cart for each newtrial. For all measurements, there are 50 grams on the string - providing the constantaccelerating force.

Do the following mass values:

• m1=mc = mass of cart

• m1=mc + 50 g (attach 50 grams to the cart)

• m1=mc + 100 g (cart plus 100 grams attached to the cart)



At this stage, you will need to summarize this part of the experiment. Put yourresults into a table. You might wish to use different worksheets to organize your data.

m2 50 grams uncertainty = 1 gramFa=m2 g

m1 uncertainty acceleration uncertainty acceleration uncertaintyin mass (velocity data) (acceleration data)

Table 6.2: Example Case A Table - Remember Units!

Once you have obtained an acceleration value for each mass, plot a graph of theacceleration a versus the reciprocal of the total mass ([1/(m1 + m2)]. Include commentswith your result.

Compare the slope of the graph with the theoretical value of Fa = m2g.Your answer to this part requires a table, a plot, and comments.Continue on to the next section once you have your data - do the analysis later.

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Case B - Constant Total Mass

Start with 200 grams attached to the cart and m2 = 50 g. The constant total mass forthis part is m1 + m2 = mc + 250g.

Remove 20 grams from the cart and add the same 20 grams to the accelerating massm2, and find the acceleration for this system as before. Repeat this step until you havemeasured the acceleration for the following values of m2: 70g, 90g, 110g, and 130g.

Graph the acceleration a versus the accelerating force Fa = m2g. Determine the slopeof the graph. Comment on the result.

Compare the slope of the graph with the theoretical value of ([1/(m1 + m2)].Your answer to this part requires a table, a plot, and comments

Summary of Required Write-ups

Item Instructor’s EvaluationSummary: Basic measurement====physicsSummary: Equipment operationMeasurement of MassEvaluation of the Experiment:Analyze first measurement, repeat, summarizeTable of data recorded for mass added to cartPlot of a vs ([1/(m1 + m2)],comments on validity of resultComparison of expected vs observed valuesfor the applied forceTable of data recorded when moving mass to hangingpositionPlot of a vs applied force,comments on validity of resultComparison of expected vs observed valuesfor the total mass

Table 6.3: List of Required Items

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Chapter 7

Sliding along(At Home activity)

Introduction

Any good shoe store (be it brick–and–mortar or on-line) sells a bewildering array, perhapsone hundred or more different sports shoes. Besides the all–important “coolness” factor,one of the more important characteristics that differentiates between these shoes is theamount of friction (“grip”, “traction”, etc.) that their soles provide. While friction forcesare often view as unwanted, performance limiting side effects (overheating of motors,bearings, tires, speed–limiting air drag forces, etc.), they are beneficial in numeroushuman activities: imagine a car stuck in mud, with its wheels spinning helplessly, or aperson with smooth-soled shoes trying to walk across ice.

A convenient quantity that gauges the amount of friction between two surfaces iscalled the coefficient of friction, µ (Greek letter “mu”, pronounce myoo or moo). In thisexperiment you will investigate the effect of different surfaces, and different weights onthe coefficient of friction.

Formulas

The coefficient of friction1 between two surfaces is defined as:

µs/k =Ff

N(7.1)

Where Ff is the magnitude of the friction force between the surfaces and N is the magni-tude of the normal force on the surface. The index s or k denotes the “static”/”kinetic”coefficient of friction. Note that µs can be larger than one while µk is always smallerthan one. Because it is a ratio of like quantities, the friction coefficient is dimensionless.Note that eq. 7.1 merely states that the friction force is proportional with the normal

1If the two surfaces move with respect to each other the coefficient of friction is called “kinetic

coefficient of friction”. If the two surfaces do not move with respect to each other the coefficient of

friction is called static.

63

force. If, as shown in Fig. 7.1 the surface is horizontal and one has a (relatively) smallobject moving across a massive, immovable object (i.e. floor), the normal force will beequal to the weight of the object, N = Weight.

Weight

Ff

Fapplied

vN

Figure 7.1: Friction and Normal Forces

While one cannot directly measure the friction force, we can measure the force neededto overcome the friction force, as shown in Fig. 7.1. If the applied force Fapplied equalsthe friction force Ff then the object will be in equilibrium, therefore it will experienceuniform motion (the small velocity “v” shown).

Equipment/Materials


• Spring Scale (included in your PHYS140L kit)

• String

• A shoe (anything except very low profile shoes or flip–flops will do)

• A way to attach the string to the shoe (tying to shoelace, using a small bit of tape,a bent paper clip, etc)

• Access to both “smooth” and “rough” horizontal surfaces to test the shoe on.Smooth surface examples: most tiled floors, hardwood floors, floor in the JMUPhysics and Chemistry building (and in most other JMU buildings). Rough surfaceexamples: carpet, concrete (not painted), asphalt.


0. Calibrate the spring scale. Without any weight attached to it, hold the spring scalevertically and make sure that the spring scale reads 0 (zero). If it does not move

64

the metal ruler (the one that has marked pounds and newtons on it) up or downas required.

1. Attach shoe to spring scale using a ∼one foot piece of string (and tape, bent paperclip, etc).

2. Weigh the shoe using the spring scale (Gently lift the shoe off and having it hangon the spring scale). Record this weight in Table 7.1. Pay attention to units.

3. Place the shoe on a horizontal surface of your choosing (either “rough” or “smooth”).

4. Record in the “Observations” area a brief description of the surface and of the shoe(brand, how new/worn out it is), etc.

5. using the spring scale drag, very gently, the shoe across your test surface. Be sureto pull parallel to the ground. Your laboratory partner, positioned a distance awaymight be able to better asses if you are keeping the pulling force parallel to theground. Adjust as necessary.

6. As you pull the shoe across the surface note the value of the pulling force as mea-sured by the spring scale. If the shoe is moving with uniform motion (i.e. veryslowly, in a straight line, not speeding up, not slowing down) then the force recordedby the spring scale will be equal in magnitude with the friction force between theshoe and the surface. Record this force in the data table.

7. Repeat steps 5–6 at least four more times (you should consider switching placeswith your lab. partner mid-way through this process.

8. Repeat steps 5–7 for at least another surface. Record your results in Table 7.1.

9. Repeat steps 5–8 for a different shoe model. Record your results in Table 7.1.


1. Determine the average, standard deviation, and standard deviation of the mean ofthe weight for each shoe that you measured. Record these values in Table 7.2.

2. Determine the average, standard deviation, and standard deviation of the mean ofthe pulling force (needed to achieve uniform motion) for each shoe–surface combi-nation that you measured. Record these values in Table 7.2.

3. Using Eq. 7.1 compute the kinetic coefficient of friction µ for each shoe–surfacecombination. Record these values in Table 7.2.

4. Comment (in no more than 3-5 paragraphs) your coefficient of friction results? Dothe number make sense for each shoe–surface pair? Ordering the pair of surfacesaccording to their µs do they line up as you would expect or there were somesurprising results?

65

5. Comment (in no more than 3-5 paragraphs and using formulas and estimates asappropriate) on the precision of your µ measurement, given the SD, SDMs youcomputed for both the weight and the pulling force averages.

6. As the head(s) of the advertising department of the company that markets thisshoe (pick one of those tested) design a one–page add that will help market thisproduct, incorporating (some) of the results of your measurement. Print/send inelectronic format to your instructor this add.

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• Do not forget that an add “marketing” one of the shoestested must be produced/turned–in!

• Do the µ results make sense? Comment below.

• How precise are your µ results? Comment below.

67

No. Weight Pulling Force Description[N] [N]

Table 7.1: Experiment: Sliding Along. Data table. If needed, feel free to make copies ofthis table.

68

Shoe Average SD SDM Average SD SDM Friction–Surface Weight Weight Weight Pull Pull Pull Coefficient

[N] [N] [N] [N] [N] [N] µ

Table 7.2: Experiment: Sliding Along. Results table.

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Chapter 8

Crashing Carts(In Class activity)

Introduction

In this lab, you will investigate the conservation of momentum in three different cases:

1. inelastic event with both carts initially at rest

2. inelastic collision with both carts having the same final velocity

3. elastic collision

In each case, with no net force acting on the system, the total momentum of bothcarts is conserved. This means that the total momentum after the event (e.g., a collision)is the same as before the event, that is, initial momentum equals final momentum1.

In this process you will also learn how to:

• set up & become familiar with using two sensors connected to the LabPro interface.

• make predictions of experiment and then test

• verify conservation of momentum laws by observation and analysis

• see the effects of error propagation on the final results

Formulas

Momentum conservation can be written as:

~pi = ~pf (8.1)

1It would be helpful to remember that momentum, like velocity and acceleration is a vector quantity.

71

Here the indexes i and f denote the initial and final value of the momentum of thesystem. Recall that the momentum of an object is simply defined as its mass times itsvelocity:

~p = m~v (8.2)

For the particular case of a system of two objects (as you will study in this lab.) eq. 8.1becomes:

m1~v1i + m2~v2i = m1~v1f + m2~v2f (8.3)

Where the indexes 1 & 2 denote the two objects/carts.

Equipment/Materials


• a cart track

• two carts (one with spring plunger)

• a cart launcher

• two photogates

• LabPro interface

• two 500g masses

• detection sails for carts

• balance

• meter stick


This experiment has many similarities to the Atwood’s Machine lab you did previously.Recalling the techniques you used in that lab, how you set up your spreadsheet and youruncertainty analysis (See section on uncertainties below) will greatly assist in this lab.It might be a good idea to start a spreadsheet now.

Measure the mass of each cart with the detection sails installed. Record these valuesin your spreadsheet. Include uncertainty estimates. Have your instructor check this off.

As in the Atwood’s Machine lab, it is critical that the track be level and stable. Thereshould be a level tool available for your use to check this. As a check you can place onecart somewhere near the middle of the track so that it is not moving. Be sure the cartdoes not have a tendency to roll one way more than the other.

In this experiment cart velocities are required in addition to the cart masses. The ve-locities are measured using photogate sensors, attached to the LabPro interface units and

72

utilizing the LoggerPro software. The velocity is found when the “sail” passes throughthe photogate and blocks the infrared beam. It is determined by LoggerPro dividing thelength of the sail by the time that the photogate beam is blocked. This of course is anaverage velocity over the time interval.

This experiment requires the use of two sensors connected to the LabPro interface.Use the following procedure to set them up for reading:

1 Open LoggerPro. (sensor operation can be checked at this point by blocking theinfrared beam of the photogate and observing if the red LED on top of the sensorilluminates)

2 Under Experiment, click on “Set Up Sensors”, then “LabPro: 1”, or click on thesmall LabPro icon in the upper left-hand corner of the screen.

3 Click on the photogate icon in the “DIG/SONIC1” window. If this icon does notappear, select “Photogate” from the drop-down menu in the “DIG/SONIC1” win-dow. block the photogate beam now and verify the “Gate State” where indicatedabove the table in the LoggerPro screen)

4 Click on the photogate icon in the “DIG/SONIC1” window, select “Gate Timing”under Current Calibration.

5 Again click on the photogate icon, click on “Set Distance or Length”

6 Measure the length of the sail on the cart that will be passing through the photogateconnected to the “DIG/SONIC1” port and enter that value in meters. Rememberto also enter this value (and its uncertainty) in your spreadsheet.

7 Repeat steps 3–6 for the second photogate connected to “DIG/SONIC2”. Thenclose the LabPro pop-up window.

8 Under Experiments, click on “Data Collection”.

9 Under Mode, select “Time Based”, enter 10 s for length of data collection and forsampling rate enter 2000 samples/second. Click Done.

You may want to adjust the width of the table in the LoggerPro screen so that thevelocity columns for both photogates are shown.

For all of your data collection, one of your team members should be designated tocatch the cart(s) after passing through the photogate. By “softly” catching the carts, itwill keep the carts from being involved in secondary collisions, either with track bumpersor the floor, and therefore preventing damage to the carts. This is more critical in thehigher velocity runs.

Set up the photogates so that the sail on the carts pass through the beam. Positionthe photogate stands along the track about 40 to 50 cm apart. Click Collect and pushone of the carts slowly so that it passes through both photogates to make sure you set upthe data collection correctly and so that you can get a feel for what velocity is produced

73

by a given force (push). Try different levels of “push”. Have your instructor witness thisafter the data collection works correctly.

Case 1 (Carts initially at rest):In this experiment, both carts will start initially at rest between the photogates.

Momentum is imparted to the carts by the release of a spring plunger in one of the carts.Because the carts are initially at rest, both the initial momentum and the final mo-

mentum of the system is zero:

m1~v1f + m2~v2f = ~0 (8.4)

From eq. 8.4 it is clear that regardless on how you choose (left–to–right or right–to–left)your coordinate system one of the final velocities will point in the positive direction ofthe axis and one final velocity will point in the opposite direction of the axis.

You will run experiments for three cart/mass arrangements:

• Case 1A no extra masses on either cart

• Case 1B a 500g mass on Cart 2

• Case 1C two 500g masses on Cart 2 (note, measure the masses of these weightsand record with their uncertainties)

For these three sub-cases; assuming that the final velocities can be written as: ~v1f = k~v2f ,with k a scalar constant, predict the value of k for these three sub-cases based on eq.8.4. Record these three values in your spreadsheet including their uncertainty. Have yourinstructor check this off.

Set up the two carts between the photogates as in Fig. 8.1 below:

Cart Launcher

Photogate 1 Photogate 2Cart 1 Cart 2

plunger

Figure 8.1: Case 1: Both carts at rest initially (Note: your setup may be the mirrorimage of this figure)

The cart labeled Cart 1 has a built-in spring plunger with 3 set positions. To set thespring plunger, push the plunger in, and then push the plunger upward slightly to allow

74

one of the notches on the plunger bar to “catch” on the edge of the small metal bar atthe top of the hole. After setting the plunger, it is released by tapping the trigger buttonon top of the front end cap. To ensure that you do not give the cart an initial velocity,other than that supplied by the spring plunger, release the trigger by tapping it with arod or stick using a flat edge. Practice this until you are confident that your releasingtechnique is not affecting the cart velocity.

Set the plunger to the middle position. Position the carts (Case 1A, no extra masseson either cart) so that the end of the spring plunger is touching Cart 2, click Collect inLoggerPro, and then release the plunger to propel the two carts through the photogates.

• Does the initial position of the carts relative to the photogates affect the results?

• What is the optimum position of the photogates with respect to the starting positionof the carts?

• Adjust the positions of the photogates as necessary.

Record the measured velocities in your spreadsheet. Repeat for at least three trialsso that you obtain velocities that are about the same. Calculate k for each trial andfind its average value. Compare this to your predicted value for Case 1A consideringuncertainties.

Repeat the above for Cases 1B and 1C. Do the results make sense? Explain in a textbox in your spreadsheet. Review with your instructor.

Case 2 (Inelastic collision):In the inelastic collision, the carts will stick together. This is accomplished with

velcro pads on the ends of the carts. For this experiment, Cart 2 will be initially at restand since the carts stick together, the final velocity of both carts will be the same (note,this will simplify eq. 8.3).

To start this experiment, position Cart 2 initially at rest between the photogateswhile the other cart will start outside the photogates. Cart 1 will be propelled to collidewith Cart 2 in an inelastic collision. The cart launcher attachment will be used to propelCart 1.

• Before performing the collision experiment, you should understand the operationof the cart launcher.

• Loosen the thumbscrew on the adjustable latching clamp on the plunger and moveit into a position so that when the trigger lever is set on the latching clamp, theindicator is at about 3.5 cm.

• Set the launcher by compressing the spring and hooking the trigger lever on thelatching clamp. Place Cart 1 up against the rubber tip on the end of the plunger.

• Position the first photogate about 10 cm away from the cart and the second pho-togate another 40 cm away.

• Launch the cart by releasing the trigger lever while collecting data in LoggerPro.

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• You may find it helpful to have one of the members your team hold down the trackwhen the cart is launched in order to keep the relative position of the track andphotogates the same.

• Repeat this so you are comfortable with this operation.

• Record the velocity obtained at this plunger setting.

Repeat the steps above for spring compression levels of about 2.5 cm and 1.5 cm. Haveyour instructor check your velocity measurements.

Now for this set of experiments with inelastic collisions, you will run the followingthree sub-cases:

• Case 2A: no extra masses on either cart

• Case 2B: a 500g mass on Cart 2, no extra mass on Cart 1

• Case 2C: a 500g mass on Cart 1, no extra mass on Cart 2

Adjust the cart launcher spring compression to propel Cart 1 (without added mass)at a velocity of about 0.5 m/s. This may require some trial and error; adjust as necessary.

Set up the two carts relative to the photogates and cart launcher as in Fig. 8.2 below.Make sure the spring plunger in Cart 1 is well secured in its fully compressed position;you want it to stay there.

Cart Launcher

Cart 1 Cart 2

Figure 8.2: Case 2: Inelastic collision with objects moving with the same final velocity.)

Launch Cart 1, recording the velocities with LoggerPro. When the “attached” cartspass through the second photogate, is it better to use the velocity value from Cart 1 orCart 2? When you are satisfied with this operation, show your instructor. Again, considerhow the location of the photogates may improve your data and adjust accordingly.

Do this for sub-cases 2A, 2B and 2C, recording all your data. You may want torepeat each trial to be sure your velocities are generally repeatable and you are notgetting spurious data.

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For all trials, calculate the initial and final system momentum values with uncertain-ties. Evaluate your trials by comparing the initial and final momentum. Is the absolutevalue of the difference between the initial and final momenta less than the sum of theuncertainties of the initial and final momenta? If not, how does it compare to two timesthe sum of the uncertainties of the initial and final momenta? Summarize in a text boxin your spreadsheet. Review your results with your instructor.

Case 3 (Elastic collision):An elastic collision is one in which the two carts bounce off each other and in which

both momentum and kinetic energy are conserved.In this experiment, the setup will be similar to Case 2 with Cart 1 being propelled by

the cart launcher and Cart 2 initially at rest. However, since it is an elastic collision, thecarts will be turned around so that their magnet ends face one another. Use the samecart launcher setting so that Cart 1 has an initial velocity of about 0.5 m/s.

For this set of experiments with elastic collisions, you will run the following threesub-cases:

• Case 3A: no extra masses on either cart

• Case 3B: a 500g mass on Cart 2, no extra mass on Cart 1

• Case 3C: a 500g mass on Cart 1, no extra mass on Cart 2

Set up the two carts relative to the photogates and cart launcher as in Figure 3 below.

Figure 8.3: Case 3: Elastic collision.

Before you start collecting data, predict and sketch the positions of the carts afterthe collision on the track in Fig. 8.4 through Fig. 8.6 below. Use arrows to indicate thecart directions and indicate the positive velocity direction. Show your instructor.

After your instructor has checked your sketches and setup, do a trial run of case 3Ato check the data collection, again considering how the location of the photogates mayimprove your data.

Do this for sub-cases 3A, 3B and 3C, recording all your data. Record data as you didin Case 2.

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Figure 8.4: Cart positions after elastic collision, Case 3A.

Figure 8.5: Cart positions after elastic collision, Case 3B.

For all runs, calculate the initial and final system momentum values with uncertain-ties. Compare the initial and final momentum, as you did in Case 2.

Did your results match your cart position predictions? If not, what part was different?Describe in a text box on your spreadsheet. Review these results with your instructor.


(Computing the uncertainty in velocity)

Recall how the velocity is calculated with the photogate sensor. The length of whatblocks the infrared beam (the sail in this lab) is divided by the time the beam is blocked.Hence, the photogate actually measures a time differential, ∆t, vs. directly measuring avelocity. Now, you have measured the length of the sail and estimated the uncertainty forthat measurement, but how can you determine an uncertainty for the time measurement?

During the sensor setup you told LoggerPro to collect data at a rate of 2000 samples/s.That means the photogate is checking every 1/2000 of a second, that is, every 0.0005 s,to see if the infrared beam is blocked or unblocked. Now consider a situation in a run

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Figure 8.6: Cart positions after elastic collision, Case 3C.

where, for instance, the sensor determines at t = 0.1000 s the beam is unblocked andthen at the next sampling, t = 0.1005 s, it is blocked. Furthermore, lets assume somehowwe know that the beam actually becomes blocked at t = 0.1001 s. This would mean thatthe time LoggerPro records as when the beam becomes blocked is in error by 0.0004 s.How much could be the worst case error here? Since the photogate is measuring a ∆t,what will happen to the error if a similar sort of event happens when the beam becomesunblocked? Use the sum of these potential, worst case errors, as your uncertainty.

Since there are numerous runs and velocity measurements, and each velocity has aseparate uncertainty calculation, the result would be that you are spending a lot of timecalculating velocity uncertainties. To save time, it is suggested that you calculate theuncertainty for one velocity. This may be a typical value for velocity, or the minimum ormaximum velocity observed (consider which would give you the worst case uncertainty).In addition, for this uncertainty, and the velocity it is based on, calculate a relativeuncertainty. Use this relative uncertainty throughout this lab for all velocities. State onyour spreadsheet how you determined your uncertainty value.

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No. Topic Instructor Evaluation

1 Mass of carts & Uncertainties2 Sensor & data collection

“push” vs velocity3 Case1: k predictions4 Case1: k results5 Cart Launcher

velocity measurements6 Case2: Setup & operation7 Case2: Inelastic collision

Results8 Case3: Elastic collision

prediction sketches9 Elastic collision results

Table 8.1: Instructor check off table for “Crashing Carts” experiment.

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Chapter 9

Happy and Sad Balls(At Home activity)

Introduction

In this experiment we will explore the properties of the “sad/happy” balls found in yourPHYS140L take home kit. To begin with, locate these two objects: the two seeminglyidentical black balls, about an inch in diameter. To convince youseleves that these twoare only apparently identical do the following quick test: drop (do not throw) each ballin turn from a height of a couple of feet on a hard surface (tile, hardwood floor, etc.).This little test should be enough to convince you that the two balls are not identical.

Formulas

For this experiment you will need the following formulas/concepts:Density is a scalar quantity that measures how compact an object is:

ρ =m

V(9.1)

With m the mass and V the volume of the object.Archimedes’ Principle states that for every object imersed in a fluid there is an upward

force equal to the weight of the fluid displaced.In order to “float” an object’s density needs to be lower than that of the fluid in

which it is imersed.In a collission between two objects (let’s label them “1” and “2”) the restitution

coefficient is defined as:

CR =v2f − v1f

v1i − v2i(9.2)

Here the index i/f denotes the initial/final state (i.e. before and after collission). v is ofcourse the magnitude of the velocity for objects “1” and “2”. Note that as it is a ratio oflike quantities, CR is dimensionless. For an elastic collission CR would be equal to 1; fora perfectly inelastic collission CR would be equal to 0 (zero); most/all collission betweenreal objects will have CRs somewhere in between zero and one.

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For the particular case in which one of the objects is massive and static (like in thecollission between a ball and a fixed floor) eq. 9.2 becomes:

CR =v1f

v1i

(9.3)

Due to limitations in the PHYS140L take home kit you will not be able to measure(reliably) either of the velocities in eq. 9.3. Neglecting air drag one can, however, equatethe kinetic energy of the object before/after the collission with the potential energy ofthat object: K.E. = P.E.. Given the definitions for kinetic and potential energy it isstraightforward to show that:

CR =

√

h1f

h2i

(9.4)

Here hi denote the height from which the object is dropped while hf is the height towhich the object will bounce.

Equipment/Materials


• Happy and Sad balls

• Meter stick/tape

• Access to different types of floors (a hard surface floor and a carpeted floor)

• Plastic cup/container and table salt.


Determination of the density of the “happy” and “sad” balls.The formula (eq. 9.1) for density calls for the measurement of both the mass and the

volume of an object. The spring scale in your PHYS140L kit is not sensitive enough toprovide a good measurement for the masses of either the “happy” or the “sad” balls.In principle one could weigh a bunch of identical “happy” balls using the spring scaleprovided, divide by the number of balls to get the mass of a single ball, then repeat forthe “sad” ball. That is a good procedure when large numbers of identical objects areavailable. Unfortunately that is not the case in this experiment.

Alternatively one can estimate directly the density of the two balls using the followingprocedure:

• Take a container (plastic cup, drinking glass) large enough to contain the two balls(do not use too big a vessel!) and fill it to a height of ∼1.5–2 inches with water(you need the liquid level to exceed the diameter of the balls but not by much).

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• Drop the two balls in your container. What do you observe? Do the balls sink orfloat?

• Start adding regular table salt 1, one teaspoon at the time to the water. Stir alittle to allow the salt to disolve. Are the balls floating or sinking?

• After a few teaspoons of salt one of the balls will start floating. Make a note ofwhich of the two balls is the first one to float (if in doubt dry it out on a papertowel and drop it against a hard floor)

• This simple experiment should allow you to order the two balls according to theirdensity. In principle one could get an actual measurement of the density by keepingtrack of the amount of salt added. For this assignment just being able to tell whichball is more/less dense is enough.

Determination of the coefficient of restitution CR

1 Drop (from a previously measured height) one of the balls against a hard floor(tile, hardwood floor, cement, etc.) and measure the height to which the ball willbounce.

2 Record both the starting height and the bounce height in Table 9.1.

3 Repeat the procedure at least five more times (as long as you record it appropriatelythe starting height need not be exactly the same from step to step), record theresults in Table 9.1

4 Repeat steps 1–3 for the other ball. Record the results.

5 Repeat steps 1–4 this time dropping the balls against a carpeted floor (if you haveaccess to one, a Persian rug would be an acceptable substitute). Do you noticeanything different with respect to the previous set of throws?

6 For all trials above compute CR using eq. 9.4.

7 Average your CR values for each ball–type – floor combination. Record these aver-ages and the spread (SD) of your CR measurements in Table 9.2.


The ball that floats first is the ball. That means that this ball has (chose oneof higher/lower) density than the other ball.

Questions:

• Is CR a property of the ball?

1Table salt will cloud the water somewhat; that is OK. If you have it, you can substitute pickling salt

or Kosher salt - the resulting solution might be less cloudy.

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No. hi hf CR Ball & Surface[m] [m]

Table 9.1: CR Measurements using the happy and sad balls.

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No. Ball & Surface CR SD for CR

Table 9.2: CR Averages and SD.

• If one would freeze the two balls, will their respective CRs be larger or smaller?Explain.

• Same question if one would be heat (by putting them in boiling water for instance).Do not try either of these!

• Imagine that the “happy” and “sad” balls, having the same initial velocity - forinstance by rolling each one down the same incline (an open book, propped at oneend, your PHYS140/240 book comes to mind, would make a suitable incline; justroll the ball along the binding), collide (in turn) with a third ball, initially at reston a table/floor. Which would send this third ball farthest? Would it be the happyball? Or the sad ball? Explain.

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Chapter 10

Poe’s Pendulum(In Class activity)

Period of a Pendulum

A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for manyyears the pendulum was the heart of clocks used in astronomical measurements at theGreenwich Observatory. What determines how quickly the pendulum moves back andforth?

-1

-0.5

0

0.5

1

0 10 20 30 40 50 60 70 80

Time (s)

An

gle

(d

eg

rees)

1 cycle

Figure 10.1: Pendulum cycle.

Each back-and-forth motion is called a cycle. The time (usually measured in seconds)that it takes for one cycle to occur is called the period, and is given the symbol T . Thefrequency, or the number of cycles per second, is inversely related to the period: f = 1/T .

A simple pendulum consists of a mass hanging from a pivot on a string. In an idealpendulum, we imagine that the mass is concentrated at a point, the string has no mass,

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and there is no friction or air resistance.There are at least three things you could change about a simple pendulum that might

affect its period:

• the mass of the pendulum bob

• the length of the pendulum, measured from the center of the bob to the point ofsupport

• the amplitude of the pendulum swing (or how far from vertical, in degrees, thependulum is initially pulled back when it’s released)

To investigate the pendulum, you need to do a controlled experiment; that is, youneed to make measurements, changing one variable at a time, while keeping all othervariables constant. Conducting controlled experiments is a basic principle of scientificinvestigation.

In this experiment, you will use a Photogate capable of microsecond precision to mea-sure the period of one complete swing of a pendulum. By conducting a series of controlledexperiments with the pendulum, you can determine how each of these quantities affectsthe period.

In this lab, you will plot your data and draw relationships from the plot. One wordof caution: Examine the plots in Fig. 10.2. What do you think might be the functionf(x) plotted on the left? It is apparently linear. However, the plot in the center showsexactly the same data with the vertical scale starting at zero. The variation is only acouple percent, so maybe the function is a constant. The plot on the right shows thesame points along with three additional data points. In this case, there seems to be aclear (and non-linear) trend. Maybe we didn’t have enough data to see how f(x) dependson x.

9.4

9.6

9.8

10

10.2

0 1 2 3 4

x

f(x)

0

2

4

6

8

10

12

0 1 2 3 4

x

0

2

4

6

8

10

12

0 1 2 3 4 5

x

Figure 10.2: A warning about plots – pay attention to the scales on your axes!

When determining whether a parameter is relevant in affecting the period of thependulum, you must both vary the parameter(x-axis) by a sufficient amount toin order to potentially observe a measurable change and make sure that thescales on your plots are not so small that a slight change appears significant.There is no “correct scale” for your plots, but you should be conscious of this as youdraw conclusions from your data.

88

Equipment/Materials


• Vernier photogate

• Protractor

• Strings with 4 different masses

• Ring stands (2) and pendulum clamp

• Meter stick


1. Attach the string and mass to the pendulum clamp. Attach the Photogate to thesecond ring stand. Position it so that the center of the mass blocks the Photogatewhile hanging straight down, as shown in Fig. 10.3. Use care when releasingthe mass that it doesn’t strike the Photogate. The length of the pendulumis the distance from the pivot point (bottom of the clamp) to the center of massof the pendulum bob. Connect the Photogate to the DIG 1 port on the LabProInterface.

2. Prepare the computer for data collection by opening “Exp 14” from the Intro toPhysics folder. A graph of period vs. time is displayed.

3. Temporarily move the mass out of the center of the Photogate. Notice the readingin the status bar of Logger Pro at the top of the screen, which shows when thePhotogate is blocked. Block the Photogate with your hand; note that the Photo-gate is shown as “blocked.” Remove your hand, and the display should change tounblocked. Click “Collect” and move your hand through the Photogate repeatedly.After the first blocking, Logger Pro reports the time interval between every otherblock as the period, since the mass blocks the photogate twice during one completecycle. Verify that this is so.

4. Design and conduct an experiment to determine how the period depends on themass of the pendulum bob (m).

5. Design and conduct an experiment to determine how the period depends on lengthof the pendulum (L).

6. Design and conduct an experiment to determine how the period depends on theinitial amplitude (θ0).

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• For each of your three experiments, construct a data table and a graph (or graphs)that represents your data.

• Determine the empirical formula for the relationships that exist between the vari-ables tested and the period of a pendulum. That is, try to write a relationshipT ∝ ... which shows how the period depends on the three experimental variables(“∝” means “is proportional to”).

As an example, from Newton’s 2nd Law (F = ma), we know that given a fixedmass, a ∝ F and given a fixed force, a ∝ 1

m.

Ask questions as necessary, but also use experience and techniques from the previouslabs.

• Use appropriate estimates of uncertainty to qualify how well you know these re-lationships and formulas. In the lab next week, we will focus more on how todetermine experimental relationships, including determining the errors in the pa-rameters.

• Briefly write up your findings and be prepared to justify your conclusions.

Your lab report should include:

• Clearly labeled data and plots for each of the three cases.

• Uncertainties for all measured quantities listed and explained.

• Analysis tabulated and clarifying comments included.

• Results clearly stated and demonstrated.

A final measurement and a look ahead

As mentioned above, the lab next week will enable us to quantitatively determine afunctional dependence, including the errors of the fitting parameters. In order to havereliable data for this at-home activity and to save time next week, we will collect thedata now. It should take about 15 minutes to go through the following steps. Briefly,here is the goal of the analysis that you will perform next week:

When you solve physics problems involving free fall, often you are told to ignoreair resistance and to assume the acceleration is constant and unending. In the realworld, because of air resistance, objects do not fall indefinitely with constant acceleration.Instead, their acceleration is decreased by air resistance – if you drop a feather, forinstance, it will quickly reach an almost constant velocity referred to as terminal velocity,vT . (Even objects that aren’t as clearly affected by air drag, like a baseball or a skydiver,will reach terminal velocity if allowed to fall far enough!)

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Air resistance is sometimes referred to as a drag force. Experiments have been donewith a variety of objects falling in air. These sometimes show that the drag force isproportional to the velocity and sometimes that the drag force is proportional to thesquare of the velocity. In either case, the direction of the drag force is opposite to thedirection of motion.

Mathematically, the drag force can be described using Fdrag = −cv or Fdrag = −cv2.The constant c is called the drag coefficient and depends on the size and shape of theobject. When falling, there are two forces acting on an object: the weight, mg, and airresistance, −cv or −cv2. At terminal velocity, the downward force is equal to the upwardforce, so mg = −cv or mg = −cv2, depending on whether the drag force follows the firstor second relationship. That is, mg = −cvn where n is equal to 1 for the linear dragforce and 2 for the squared drag force.

How can we tell if the right drag rule for our coffee filters is the linear or the squaredtype? Mathematically, we can see if we take the log of both sides:

ln(mg) = ln(kvnT ) (10.1)

ln(m) + ln(g) = ln(k) + n ln(vT ) (10.2)

ln(m) = n ln(vT ) + ln(k/g) (10.3)

You might notice that if we call ln(m) “y” and ln(vT ) “x”, this is in the form ofy = mx + b. In order to determine which power is more appropriate, you will take yourdata for mass and velocity and make a plot of ln m vs. ln vT . In fitting this plot to astraight line, you will find that the slope n will be equal to the power.

For today, we just need to measure the terminal velocity for a few different masses.

• Disconnect the photogate used for the pendulum and connect the motion detector.Start Logger Pro by opening ”‘Exp 10, Air Resistance”’ in the Intro Physics LabFolder to take data with the motion detector.

• Position the motion detector approximately 2 meters off the ground, facing down.Place a coffee filter in the palm of your hand and hold it about 0.5 m under theMotion Detector. Do not hold the filter closer than 0.4 m to the motion detector.

• Click collect to begin data collection. When the Motion Detector begins to click,release the coffee filter directly below the Motion Detector so that it falls towardthe floor. Move your hand out of the beam of the Motion Detector as quickly aspossible so that only the motion of the filter is recorded on the graph.

• If the motion of the filter was too erratic to get a smooth graph, repeat the measure-ment. With practice, the filter will fall almost straight down with little sidewaysmotion.

• The velocity of the coffee filter can be determined from the slope of the distance vs.time graph. At the start of the graph, there should be a region of increasing slope(increasing velocity), and then it should become linear. Since the slope of this lineis velocity, the linear portion indicates that the filter was falling with a constant

91

or terminal velocity (vT ) during that time. Drag your mouse pointer to select theportion of the graph that appears the most linear. Use Logger Pro to find the slopeof the straight line.

• Record the slope in a data table (a velocity in m/s). Repeat the measurement twicemore to verify that the results are typical.

• Repeat these steps for two, three, four, and five coffee filters. Record the data tobe used at home next week.

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L

m

photogate

clamp

Figure 10.3: Experimental setup for the pendulum experiment.

93

Chapter 11

Functions/Air Drag(At Home activity)

Introduction

This lab is designed to introduce and review concepts important to the analysis of data.The delivery of this lab is through a special web based tool called LonCapa. This issimilar to Blackboard but with features more inline with solving scientific problems.

Step–by–step Guide

First connect to the web application:

http://lc.cit.jmu.edu/adm/login?domain=jmu

Second login using you standard JMU username and password.Third choose the course PHYS140L. You should have only a few options. Any one

of your courses can deliver material via the LonCapa system and you will automaticallybe given access to those courses but no others.

Fourth be sure you are on the Navigation page (Button near top of the window)Click on Navigate Contents

Fifth there is a folder called OnlineLab.sequence. Click on this folder to start thelab.

The material provided consists of

1 basic information,

2 data to be analyzed in Excel,

3 short quizzes to test your knowledge as you read the material.

You must

1 Complete the quizzes

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2 Hand in an Excel spread sheet based on the assignments requested in the lab.

Other information:

• quizzes will allow multiple tries so you can change your answer if you realize laterin the lab that you made a mistake in your previous attempt(s)

• you can navigate to different sections by returning to the navigation page clickingon the folder, followed by clicking on the part of interest.

There are 9 parts to the lab and the student should visit each part at least once:

• 4 of the parts require answers either based on material you should master or resultsfrom the requested analysis

• 5 informational pages with assignments

• One Excel spread sheet must be built as you work through the material. It willcontain the following work sheets:

– straight line

– in fit

– non linear

– non linear fit

– logarithms

Lab parts

The lab is designed to introduce concepts briefly. Some might be straightforward othersmight be complex. Students are encouraged to supplement the material with othersources. There is web a page where some additional material can be found:

http://csma31.csm.jmu.edu/physics/Courses/P140L/index.htm

[The link “SUMMARY and GLOSSARY for Fitting lab” is an outline of the lab]and there are very good references on the web for all topics covered. You can keep

multiple windows open, cut and paste ideas from these sources and from LonCapa intoa file with your accumulated notes. You can share reference material with colleagues.(Students are expected to submit their work for the assignments and maynot simply copy another student’s work.)

Students are often confused as to what data analysis means and entails. This labexplores how a set of data can be examined to learn or test an idea. The first step is toreview functional relationships (straight line, polynomials, exponentials). There is a shortdiscussion as to the ways that the data to be analyzed can be measured. Then we seewhat we mean by comparing data to a model (expected behavior). Usually the studenthas some basic idea as to how to analyze data that follows a straight line. This is reviewed

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No Description Assignments

1 Introduction info. & Assignments2 EXCEL line plot info. & Assignments3 Find slope and intercept Answers to be

(2 parts) submitted4 Fit line Answers to be

(2 parts) submitted5 Fit line info. & Assignments6 non linear info. & Assignments

plot7 Which function Answers to be

submitted8 non linear fit Answers to be

submitted9 logarithms info. & Assignments

Table 11.1: Description of activities and assignments for functions/air drag

with some emphasis on thinking about how to judge when a line best matches the data.A broader option is to compare data to more complicated functions. In making thesecomparisons the student is asked to consider how the uncertainty should enter in thisjudgment. Also there is freedom to allow the some aspects of what is usually consideredto be the function type to be changed and therefore investigated. On can ask does thedata follow a linear function, quadratic or cubic function? Finally t his question can beasked using only the straight line analysis tools if one uses logarithms to first transformthe data. Hopefully, the final exercise highlights this powerful technique.

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Chapter 12

Comedy of Errors(Final Lab Part I)

This lab is the first part of a two part lab. The data obtained in this lab will be used byyou to estimate physical quantities. The actual estimates and write-up of the results arethe focus of the second lab. For this portion of the lab, you will investigate new physicalphenomena that we have not covered yet in class. We will use the techniques we havebeen developing to investigate the new area.

Purpose

1. To become familiar with the temperature probe as means of measuring temperature

2. To measure errors carefully

3. To obtain all the data necessary to determine the latent heat values of water forthe processes of fusion and vaporization

4. To investigate new physical phenomena in the laboratory

Materials


1. Calorimeter with outside insulating container and stirrer

2. Boiler with hot plate

3. Balance

4. Warm water

5. Ice

6. Thermometer or Thermometer sensor (Lab Pro, Logger Pro-software)

99

Figure 12.1: Calorimeter Setup

100

Background Theory

When a substance changes state, a certain amount of heat is exchanged between a sub-stance and its surroundings. The amount of heat needed to melt a substance (or to beremoved to freeze the substance) depends both on how much stuff there is (mass) andas well as what the substance is. Heat of fusion (Lf) is the term applied to the ratio ofexchanged heat (Qf) per unit mass m, when a specific substance melts or freezes, or:

Lf =Qf

m(12.1)

Lf has units of calories/gram. If melting occurs, heat is absorbed by the substancefrom the surroundings. Freezing on the other hand, implies a reverse process where heatflows from substance to surroundings.

A similar expression, Lv, heat of vaporization is associated with the liquid to vaporprocess, or its opposite; again, the same rationale applies:

Lv =Qv

m(12.2)

Qv is the exchanged heat.

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Experimental Determination of Lf

To experimentally determine the heat of fusion of water, one uses a calorimeter containerof a given mass, mc, and specific heat, C. Water of mass mw is poured into an insulatingcontainer. The initial Temperature To of water and container is then measured. Ice withmass mi is then added to the container where melting takes place. You may assume thatthe ice temperature during the melting process remains at 0o C.

The law of conservation of energy may be applied during the above mixing/meltingprocess. The result is:

Heat gained by mi = Heat lost by mw and mc (12.3)

This can be expressed as:

mi(Lf + Cw(Tf − 0)) = mwCw(T0 − Tf) + mcCc(To − Tf ) (12.4)

The two terms on the left represent the heat gained by (1) the ice in melting and(2) the melted ice water going from 0o C to Tf . The two terms on the right side are,respectively, heat loss by (1) water originally in the calorimeter and (2) the aluminumcalorimeter. The symbols Cw and Cc stand for the specific heats of water and thealuminum calorimeter, respectively, for which the numerical values are 1.00 and 0.22cal / (oC-g).

Equation 12.4 readily yields a value for Lf when all other quantities appearing in theexpression have been measured or are given.

Derive an equation for Lf and show your instructor the result before proceeding:

Formula for Lf Instructor’s checkoff

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Experimental Determination of Lv

A similar approach to that above is followed in determining the heat of vaporization. Inthis instance steam of mass ms from a boiler is directed into the calorimeter where itmixes with the water already in the container and brings the temperature of the systemfrom an initial temperature To to a final temperature of Tf . A parallel statement to 12.3reads:

Heat lost by ms = Heat gained by mw and mc (12.5)

This can be expressed as:

ms(Lv + Cw(Tbp − Tf )) = mwCw(Tf − To) + mcCc(Tf − To) (12.6)

The two left hand terms represent, respectively, heat loss by (1) steam in condesationand (2) condensed steam (as liquid) in changing temperature from Tbp to Tf . It will benoted that Tbp denotes the boiling point temperature. The above equation yields a valueof Lv where all other quantities are measured or given.

One complication is that the boiling point of water (Tbp) is actually a function of thebarometric pressure. As the atmospheric pressure varies, then the boiling point of waterwill change as well. Table 12.2 gives values of barometric pressures and the correspondingTbp.

Derive an equation for Lv and show your instructor the result before proceeding:

Formula for Lv Instructor’s checkoff

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Experimental Measurements

Proper Setup of Formulas

As shown above several measurements must be plugged into a complicated formula inorder to compute the final result. There will be a more complete discussion of thesecalculations in the Appendices and the next lab. To properly estimate your final error,you will be required to combine uncertainties. In this lab, we will take the data you willneed to estimate values for Lf and Lv properly including errors.

Take time to setup the entry of your data into a spreadsheet so that you will be ableto calculate uncertainties. The best way is to proceed in steps, not to try to obtain thefinal values in a single calculation.

For example, as can be see above, one of the bits of information you will need isthe temperature difference between the final temperature and the original temperature(Tf - To) (often written as ∆T). Perform this single calculation and also calculate theuncertainty in ∆T. Each temperature has an uncertainty associated with it, so how wouldyou figure out the error in ∆T? Record how you would solve for ∆T below.

In a similar fashion, 12.4 requires you to calculate the error in the product mcCc∆T.Each of the terms (mc, Cc, and ∆T) has an uncertainty associated with them. However,since they are combined in a product rather than a sum, you use a different rule tocombine them. Record what formula you would use to solve for the error in mcCc∆Tbelow and have your instructor check it. Your instructor will tell you what value to assignto Cc and what uncertainty you can assign to it.

Because some of the rules for combining uncertainties require fractional uncertaintiesand some require absolute uncertainties, it will be convenient to provide columns for bothtypes of uncertainties. It will also be useful to name the cells so that you can enter theformulas easily and understand which terms belong to which types of error.

Show your instructor your formulas for ∆T and mcCc∆T before proceeding.

Formula to calculate uncertainty in ∆T Instructor’s checkoff

Formula to calculate uncertainty in mcCc∆T Instructor’s checkoff

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Let us consider how you might set up the cells for the mass of the water you use inyour experiment - and the error in that mass estimate.

We might set up a table as follows. We have named some of the cells. We also haveprovided the excel column name to make them easier to see.

Table 12.1: Example - Mass of Water

excel col A B C D E F G

Comment name value units abs. unc. fractional unc. commentTotal Mass MWC 55 g 2 =E2/MWC unc based on measure error

Mass of Calorimeter MC 4 g 2 =E3/MC unc based on measure errorMass of Water MW MWC - MC g ? =? unc calculated by formula

The “?” in column E and F above would be the formulas that you would use toestimate the uncertainty in the mass of the water. In column E, it would be the termthat you would associate with the sum rule for uncertainties. In column F, you might usethe product rule. This is similar to the exercise you did for the ∆T and mcCc∆Tabove.Check with your instructor if you have questions.

The key point - take some time to set up your tables to make sure that you recordthe appropriate data and uncertainties!

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Proper Setup of Experiment

There are two good calibration points for checking and calibrating thermometers fortoday’s lab.

• 0 oC - freezing point, or the ice-water equilibrium temperature at atmosphericpressure of 760 mm of Hg.

• 100 oC - boiling point, or the water-steam equilibrium temperature at atmosphericpressure of 760 mm of Hg. Try not to let the bottom of the thermometertouch the bottom of the boiler. Be careful - you can easily burn yourhand on the boiler or other hot pieces of equipment.

To calibrate the temperature sensor, you will need to do the following:

• Startup Logger Pro

• Load setup heat file

• Check the setup to ensure the configuration is sensible (rate, duration)

• Calibrate

To calibrate the thermometer, you will need to do the following:

• Use care as thermometers break easily

• Examine the scales and make sure that you can read them properly

• Check that the thermometer reads correctly at 0 oC and 100 oC.

In addition to the thermometer, we will use a device called a calorimeter. A calorime-ter is an insulated container designed to minimize thermal transfer between the experi-ment and the outside world. A diagram of a calorimeter and its components is given in12.1.

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Watching water boil

Check with your instructor about these two steps. It is often a good idea to start thewater in the boiler going so that the experiment will be ready to go.

Step 1 - Look at 12.2 and make sure that you know where the tube from the boiler willplug into the calorimeter. What will happen is that you will heat the boiler, generatingsteam. The steam will be conducted by the hose into the calorimeter, condensing intowater, and heating the water existing in the container. To make the experiment assimple as possible, we will not attach the hose until the boiler is producing steam. Beforeproceeding, check with your instructor if you have questions.

Step 2 - Heat the water in the boiler, allowing it to come to a boil. While waitingfor the water to boil, place the open end of the tube into a beaker (so that no one is hitby steam). Make sure that the tube is positioned as in Figure 2 so that steam does notcondense inside the tube. We are starting this step early to make sure that the waterin the boiler is ready when you start your heat of vaporization experiment. While thewater is heating up, you can proceed to the Heat of Fusion experiment below.

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Figure 12.2: Calorimeter with boiler

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Heat of Fusion Procedure

You may find it useful to take a picture or two of the equipment (or make a sketch) ofyour experiments, which will be helpful for next week’s lab.

Step 1- Measure the mass of the empty inner calorimeter and stirrer. Record thisvalue on the data table as mc. Estimate the uncertainty in this measurement.

Step 2 - Record the room temperature as Tr. Add water at about 10 o C above roomtemperature to your inner calorimeter container so that the container is filled to about60% of capacity. You can easily decrease or increase the water temperature by adding alittle bit of cold or hot water. Be sure to stir well so that all the water is at the sametemperature.

Step 3 - Measure the mass of the inner container (with stirrer) with the water. Recordthis as mc+w. Record an uncertainty with that measurement. This can be used with theresults of step 1 to estimate the mass of the water you are using.

Step 4 - Place the inner container within the outer insulated container as shown inFigure 1. Place the lid on the container so that the stirrer handle is sticking out andplace the stopper and temperature probe in the large hole in the top of the container.The probe should be positioned so that the tip of the probe (thermometer) is between 1and 2 cm below the surface of the water surface. Record the temperature of the water(and of course an error estimate). Wait until you get a few stable readings.

Step 5 - Remove an ice cube from the insulated container on the side table, holdingit in a paper towel. Place the cube in the inner calorimeter container, replace the lid ofthe calorimeter and gently start stirring the water until the ice has completely melted.Monitor the temperature of the water until it reaches its lowest value. Record this asTf (along with an error estimate).

Step 6 - Measure the mass of inner container (including the stirrer) with the water.Record this as mc+w+i. Record an uncertainty with this measurement. You can use thisvalue of the total mass and the values from previous steps to find the mass of the (nowmelted) ice, mi.

At this point, you have all the experimental data and uncertainties that you will needto estimate Lf . We will do that in the next lab, where you will do a complete lab write-upof this experiment.

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Heat of Vaporization

Note - it is assumed that for this part you will record all the uncertainties as you aregoing along. Again, a sketch or photo of your setup will be useful as you proceed.

Step 1 - Refill your container with cool water and record the mass as in the previousexperiment.

Step 2 - As before, measure the starting temperature of the water (and record as To).It should be at most 10 o C above room temperature. It can be cooler.

Step 3 - If you have not already done so, look at Figure 2 and make sure that youknow where the tube from the boiler will plug into the calorimeter. What will happenis that you will heat the boiler, generating steam. The steam will be conducted by thehose into the calorimeter, condensing into water, and heating the water existing in thecontainer. To make the experiment as simple as possible, we will not attach the hoseuntil the boiler is producing steam. Before proceeding, check with your instructor if youhave questions.

Step 4 - If you have not already done so, heat the water in the boiler, allowing it tocome to a boil. While waiting for the water to boil, place the open end of the tube into abeaker (so that no one is hit by steam). Wait at least one minute after the steam startscoming out of the tube before plugging it into the 1 cm hole in the calorimeter. Makesure that the tube is positioned as in Figure 2 so that steam does not condense insidethe tube.

Step 5 - As the steam is entering the calorimeter, gently stir the water and monitor thetemperature. When the temperature reads about 15 o C above room temperature (andseveral o C above your starting temperature, remove the tube and continue to monitorthe temperature until it peaks. Record this value as Tf .

Step 6 - Remove and weigh the inner container. Record this value as mc+w+s. Thisvalue, and your previous measurements can be used to calcuate the mass of the (now-condensed) steam.

Step 7 - Read the current barometric pressure from the room barometer and recordthis in your data table. With this value use Table 12.2 to find the boiling point of waterin the lab and record this as Tbp.

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Table 12.2: Barometric Pressure vs Boiling Temperature of Water

P Temp P Tempmm of Hg o C mm of Hg o C

682 97.0 687 97.2692 97.6 702 97.8707 98.0 712 98.2718 98.4 723 98.6728 98.8 733 99.0739 99.2 744 99.4749 99.6 755 99.8760 100.0 765 100.2771 100.4 776 100.6782 100.8 788 101.0793 101.2 799 101.4805 101.6 810 101.8

Before Leaving

• Make sure that you show your formulas for Lf and Lv to the instructor.

• Make sure that you show your uncertainty formulas to the instructor. You will beexpanding those formulas in the next lab, so any questions you have, ask now!

• Make sure that you show your lab data tables to the instructor.

• Make sure that you and your lab partner EACH have a copy of the lab data tables.

• Make sure that you each have a copy of any photos or sketches that you might havemade.

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Chapter 13

Tale of Woe(Final Lab Part II)

This lab is the second part of a two part lab. The data obtained in the first part ofthe lab will be used by you to estimate physical quantities. The actual estimates andwrite-up of the results are the focus of this lab. In this lab, you will draw together theerror analysis techniques you have practiced to estimate the latent heat of vaporizationand fusion for water. Each student will write up their own lab report to hand in.

Purpose

1. To use our experimental data to estimate latent heat of fusion and vaporization

2. To estimate our uncertainty in the various physical quantities

3. To write up a report describing our experiment and analysis

4. To investigate new physical phenomena in the laboratory

Materials


1. Copy of our data tables from 13.

Review from Last Lab

The term latent heat refers to the energy exchanged between a unit mass and its sur-roundings as it changes state (solid to liquid or gas to liquid). Different substanceswill have different latent heats, and the amount of latent heat will be different when asubstance melts/freezes compared to when it vaporizes/condenses.

In the last lab, you explored what happens as material is melted (ice turning to water)or condensed (steam turning to water) though the use of a calorimeter. You carefully

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measured a number of quantities (and estimated associated uncertainties) for the twoexperiments you did. One experiment investigated the latent heat of fusion (ice andwater) and the other explored the latent heat of vaporization (steam and water).

At the end of the last lab, we had all the information we needed to actually estimatethe latent heat of fusion Lf and latent heat of vaporization Lv.

Estimating Lf and Lv and estimating their Uncertain-

ties

The expressions for Lf and Lv trial are given in chapter 12. As part of that lab, yousolved for Lf and Lv. Use those formulas to set up expressions and solve for Lf and Lv.Remember that you had two or more trials, so solve for Lf and Lv for each trial you did.

In addition to the value of Lf and Lv, you also need to evaluate the uncertainty inthe estimate. As you realized in the previous lab, the uncertainty is not quite as straight-forward as some earlier labs. The expression for Lf (and Lv) involves both sums andproducts. To properly evaluate the uncertainties, you will need to break your expressiondown into smaller parts.

For example, suppose we were interested in the errors associated with an expressionsuch as

F =GM1M2

r2+

GM1M3

d2(13.1)

where we knew G, M1, M2, M3, r and d, as well as the uncertainties associated with eachof the terms. To find the uncertainty in F, we would have to find the uncertainties withterm1 and term 2 where they are defined as:

term1 =GM1M2

r2(13.2)

and

term2 =GM1M3

d2(13.3)

Finally there is the total uncertainty attributed to summing the two term’s uncer-tainties together. Only then do you have an estimate of the uncertainty in F.

So, you will need to write out your expressions for the uncertainties in Lf and Lv

broken down into smaller terms, and then sum those terms together if you are to estimatethe uncertainty in Lf and Lv properly. If you need an example of how to get started,look at the example for Lv in Appendix 3. Note that for your error analysis, you shoulduse the techniques discussed by your instructor.

Record in your spreadsheet the final uncertainty estimate for each trial of Lv and Lf .

Comparing Lf and Lv with previously measured values

This laboratory experiment is tricky, as there are a lot of ways that your data mighthave been affected. It is common for many experimenters to find values of Lf and Lv

significantly different from the ”accepted” values of Lf = 80 cal/g and Lv = 539 cal/g.

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As experimenters we are not interested in “matching” the “accepted” result. Instead, weare interested in understanding if our results are consistent within our estimates of theuncertainties.

So, take your data and the corresponding uncertainties and start comparing...

Step One

Are your values for Lf (remember you had at least two trials in the last lab) consistentwith each other within the errors? What about Lv?

For example, suppose I had measured g twice in an experiment. The first time I got9.01 m/s2 with an uncertainty of ±0.62 m/s2. The second trial gave a result of 8.72 m/s2

with an uncertainty of ±0.53 m/s2. The two trials are consistent within the estimateduncertainties.

Step Two

Are your values for Lf and Lv consistent with the established values?In the case above, my two values of g (9.01±0.62 m/s2 and 8.72±0.53m/s2 are more

than one ”sigma” away from the accepted value of 9.80 m/s2. This does not mean thatmy trials were wrong! It probably means that there were sources of uncertainty in myexperiment that I did not account for properly or completely. If one can identify those,one can improve the experiment for the next time.

A more detailed discussion of how you can interpret “sigma” values is presented inAppendix 3 in the section comparing theoretical and experimental values. As that sectionillustrates, a useful way to present your data when comparing with other experimentalvalues or a theoretical estimate is to plot your data (with 1 sigma error bars) against theother estimates. That allows you and your reader to quickly assess how well the valuesare in agreement.

Under no circumstances should you ”fix” the numbers so that your values of Lf andLv match the established values. Your results are valid for your experiment. Nor shouldyou ”bump” up the uncertainties to make your values consistent within the uncertainties.Your estimates of the uncertainties in your experiment were what you estimated at thetime to be reasonable. In the future, you can try and think of ways to improve yourtechniques or run tests of your assumptions.

Step Three

Rather than focus on how well your experimental result matches (or misses), take a lookat your various error terms. Identify the three largest sources of uncertainties in yourexperiment. The uncertainty might be in terms of absolute

For example, in my gravity case, maybe the measurement of time had a large uncer-tainty percentage wise (i.e. I had errors of 6% on my time estimates). Or perhaps I hadan uncertainty of 0.1m on my distance measurements. In some cases, a large uncertaintyin absolute terms will contribute a lot. In other cases, the uncertainty will be dominatedby the relative amount of uncertainty (i.e. fractional or percentage uncertainty). Since

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the estimates of Lf and Lv involve several different quantities, it is important to look andsee if you can identify which terms contribute the most to your overall uncertainty.

The Lab Write-up - what makes up a lab report?

This lab, in addition, to providing you a chance to work through a more complex caseof error analysis than you have done before, will also serve as way to communicate yourresults to other people. Each lab partner will write their own version of the lab report.

Your audience should be another PHYS 140L student who has not seen the labbefore. Therefore you will have to describe things clearly, so that the student could goand reproduce your lab if they needed to.

A good lab report has the following parts:

• Title Page - Includes a title, your name, names of your lab partner, sectionnumber, and your instructor’s name. It should also include a brief (one or twosentence) statement that describes the purpose of the experiment. You can considerthe statement an abstract of what your lab (done in 12).

• Introduction - Include a discussion of the physics and the formulas that you arestudying in the lab. Be sure you state what aspects of physics you were investigatingand how this physics will clarify your overall purpose. (Maximum 1 page of text).

• Procedure - Use a bullet or list style to write this section. There must be a diagramfor all your apparatus. This can be a simple block pencil sketch or somethingmore elaborate. If you took a photo of your setup during the lab, you could usethat. Label important pieces of equipment in your diagram. If you carefully showimportant aspects of the setup, you can avoid extensive discussion in the report.You can not just copy and paste the lab description from the previous lab. It hasto be in your own words - because it was your experiment.

Remember to include a description of how you calibrated your instruments. De-scribe briefly how you checked that your measurements were being done correctly.(Maximum 2 pages of text).

• Data - You should present a table containing some of the measured quantities andthe associated uncertainties in the report. You should also make available the excelspreadsheet with the complete data set so that the instructor can examine the data.(No more than one page of text).

• Analysis and Results - Here you will provide a brief description of your analysis.The discussion should include sample calculations. (No more than two pages). Inaddition, you will present a summary of your results of the analysis. You mightinclude a table containing your final values for Lf and Lv. This is where youcan discuss how various uncertainties impacted your overall result for Lf and Lv.Remember to keep proper track of units and of significant figures. (No more thattwo pages of text). Note that you can split the analysis and results into two sub-sections if you wish.

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• Conclusion - You present a brief summary that informs the reader of what youinvestigated and how well the experiment succeeded in performing the investiga-tion. Note that does not mean how well you matched the “established values” butwhether you identified and kept track of uncertainties. You can compare your re-sults with the established values and suggest areas that would be worth exploringin future experiments. This section should be short and to the point - the readershould be clear on what you accomplished. (No more than one page of text).

You can review with your lab partner the details of your experiment and procedures.However, your write-up and analysis should be your own. This is why you should havecopies of all the sketches, photos, and data that you and your lab partner took theprevious week.

You will write up a lab report that will include proper accounting for uncertaintiesand will include all of the sections listed above. It will be due on your laboratory meetingtime. Your instructor will let you know if they prefer hardcopy or electronic submission.

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General Advice

When writing, remember that what might be crystal clear to you at the conclusion of anexperiment is not clear to your reader. In general, it should be written so that a goodstudent or an instructor from any of the other lab sections could read and understand yourreport. A classmate would make a good proofreader. Be sure that you draw attention inyour write up to all the important points and link all the pieces of the lab.

Some general comments:

• Don’t get things wrong - proofread any formulas, read what you are saying. Forexample, if you start talking about boiling water and adding the resultant ice tothe calorimeter, you can expect to lose points.

• If you include tables, graphs, calculations, etc., then explain why you are includingthem. Also explain what conclusions the reader should take away from the includedmaterial.

• Plots and diagrams are very effective ways to communicate information but if youexpect your plot to illustrate a certain relationship, don’t assume the reader willmake the connection. Tell the reader what conclusion (or information) they shoulddraw from the plot.

• Don’t include all the raw data - just a sample is fine. But you will need to makeyour overall data available to the instructor.

• Don’t make complicated arguments you don’t understand. I.e. Re-read the argu-ment and if it is confusing to you, it will be confusing for your reader as well!

• Avoid hand-waving arguments.

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Appendices

1. Curve Fitting 2. Excel Spreadsheet 3. Establishing Uncertainty 4. Suggestions for Data Handling

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APPENDIX 1: FITTING DATA There are several methods that one can use to find a function that passes through a set of data points thereby revealing a mathematical relationship. To perform a fit the experimenter must choose a functional form. Functional form defines the mathematical relationship between the dependent variable (y) and independent variable (x). The form usually contains parameters whose values must be chosen to fix the relationship. Examples of some functional forms follow. A

computer program usually changes the parameters of interest in some pattern that is designed to find the best values for the parameters in an efficient way. The y-values calculated with the fit function are compared to the data y-values. The quality of the fit is judged by the difference. The method employed to search for the best parameters is unimportant as long as a good fit is found. Consider the following function

DtCeAtF tB ++= −)( There are four parameters, A, B, C, and D. Choosing different values for these parameters results in a different lines as shown in the graph below. Both lines represent the same functional form F(t) but with different values for the parameters.

F(t)=Aexp(Bt) +Ct + D

0

100

200

300

400

500

600

700

0 20 40 60 80 100 120

time t

F(t)

Line: bmxy += Parameters: m, b Exponential: x

o eAy λ= Parameters: Ao, λ Polynomial dcxbxaxy +++= 23 Parameters: a, b, c, d

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Neither function passes through the experimental values, shown as triangles with error bars. A fitting program keeps changing the parameter values and testing if the new line passes through the data points. When the line passes sufficiently close to the data (y-values from fit are close to the y-values of the data) the fitting program returns the values of the parameters. A good fitting routine can vary the parameters again to see how much a parameter can change while still passing through the error bars. This allows the routine to establish an uncertainty (range of possible values) for each parameter. In the laboratory, data analysis almost always requires both a value and an uncertainty. The fitting routines DATAFIT, Logger Pro and GRAPHICAL ANALYSIS provide values and uncertainties. Student therefore need to be able to pass data to one of these fitting programs, run the fit and retrieve parameters and uncertainties. If one uses a routine that doesn’t provide parameter uncertainties then an alternativemethod to determine these uncertainties is required. Trendline: Excel provides trend lines for charts. These lines are made to pass through the data. The parameters can be viewed by displaying the trend line function on the chart. The disadvantage of this method is that it doesn't indicate the uncertainty in the parameters. Finding Uncertainty (repeated trials method): An uncertainty can be determined (for a trend line analysis) by measuring more than one data set. Trend lines can be placed on each of the different data sets and the parameter values from each dataset (e.g. slope of a straight line) can be put into a table and compared using the SD to estimate the uncertainty in the fitted parameter (e.g. slope). This method requires the experimenter to repeat the experiment so that independent datasets are compared. This method can be used to estimate an uncertainty for any fit method. As mentioned above, routines such as DataFit provide uncertainties based on one data set. The two methods should agree. DataFit: A separate program, DataFit, is one of the best tools for general fitting. Start program using the DataFit icon. Enter the number of independent variables (usually 1). Decide if you want to have a column for y uncertainties (standard deviation column) or no

column (usually no column). Hit OK. Paste the data to the data window. Choose regression under the solve menu. Choose nonlinear. Choose the functional form from among the options or provide a custom function. If the fit is successful the results can be obtained by choosing - detailed…- in the results

menu. Scroll down until you find the table Regression Variable Results use Value and Standard Error for each parameter.

The results contain the parameters and their uncertainties (standard error) as well as a host of plots and other indicators. Ask your instructor to show you the procedure. While your instructor is demonstrating, add your own comments to the above procedure so that you can perform a fit on your own.

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Graphical Analysis: This package is supplied as part of the data collection and analysis tools from Vernier Software. This package allows the experimenter to enter or import data, to plot, calculate, graph and fit data. It has a complete set of tools so that a full analysis can be performed. It provides text boxes for comments, and graphs with sophisticated display options. Graphical analysis is a fairly complete, additional spread sheet which is available for student use. Copies of the software are available for installation on your home computer. Ask your instructor. Start Graphical Analysis by double clicking on the GA icon. Open a new analysis by choosing new under the file menu. Import or cut & paste data to the table window. Use the toolbar “Curve Fit” button or choose curve fit from the analyze menu. Choose the data to fit in the window that appears (y-column). Choose the functional form and click the “try fit” button. Complete the process by using the “OK” button. A window should appear on the graph showing the parameters and the associated uncertainties. The root mean square error, RMSE, is also given. Vernier Tech Info Library TIL # 1014 (from website)

MSE: Mean Square Error, for every data point, you take the distance vertically from the point to the corresponding point on the curve fit and square the value. Then you add up all those values for all data points, and divide by the number of points. The squaring is done so negative values do not cancel positive values. The smaller the Mean Squared Error, the closer the fit is to the data. RMSE: Root Mean Squared Error is just the square root of the mean square error. That is probably the most easily interpreted statistic, since it has the same units as the quantity plotted on the y axis. The RMSE is thus the distance, on average, of a data point from the fitted line, measured along a vertical line.

LoggerPro: Logger Pro, also provided by Vernier, has fitting functions available. These come in handy when recorded data needs to be fit quickly. Logger Pro does provide an estimate of the uncertainty for the fitted parameters.

• Highlight a section of data with the mouse. • From “Analyze” menu choose “Curve fit”. • Highlight a function. • Hit “Try Fit” button. • Hit “OK” • The results appear on the graph.

If you do not see the parameter uncertainties in the fitting summary dialog box then right click on the dialog box to obtain get the options and check the appropriate boxes. There are several interesting options available. You can vary the parameters and see how the function changes. You can define additional functions. Also see Logger Pro help files.

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The following functions are useful for the restricted case of a straight-line relationship between the dependent and the independent variable. Linest: The function LINEST returns the slope and intercept data from a straight line LSQ fit. Since there are several values returned you must:

• Enter the function LINEST(y range, x range, 1, 1) into a cell. • Select a range of cells (2 cells across, 5 cells down) that include this formula in the

upper left corner. • Type F2 function key followed by “Cntl-Shift-Enter”. (this is Excel’s array entry) • The slope and intercept values are in the first row of this 2 x 5 array • The uncertainties for the slope and intercept are in the second row of this array.

Regression Analysis: This is a just a fancy name for straight-line fitting. It assumes that the relationship between the variables is linear. It therefore can be used to find the best straight line that passes through as set of (x,y) pairs. The regression function returns a full set of quantities that can be used to describe the quality of the fit. It also provides estimates of the uncertainty in the slope and intercept. To perform a regression in excel: Use Data Analysis item in the tools menu. Choose regression. Enter the y and x values. Hit OK. View the sheet with the results. The intercept and uncertainty are tabulated. A value for the slope (x variable 1) and an associated uncertainty for this value are also

tabulated.

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Appendix 2: Working with Excel This appendix will cover ways that Excel can be used to display and analyze data. Since this is a major component of the lab, the student is encouraged to take notes and document in his/her own words how a method or tool can be used. Example spreadsheets can also be stored on the network for future reference. This is a brief guide. Excel has many features and a host of methods to accomplish the same result. If you know a method that differs from the one contained in this guide then you may use it and share it with your colleagues. As with most applications the student should explore beyond the specific lab instructions until the method is well understood. Your instructor should be able to help clarify. Once you understand the concept or method add details and summarize in your own words for future reference. WHAT IS A SPREADSHEET Spreadsheets store data in tables. Excel refers to each table as a WORKSHEET. One can change to a different worksheet using the tabs along the bottom (sheet1, sheet2). A specific location in a table (worksheet) is called a CELL. A letter and a number, for example A3, identify a cell. “A” identifies the column. “3” identifies the row. To choose a cell, move the cursor to the cell location and click the mouse. The contents of the cell are shown in the cell and in more detail in a space at the top (part of a toolbar). The cell may be empty. The cell ID (e.g. A3) also appears at top in the tool bar. Cells contain data of all types: numbers, dates, labels, formulas, and functions. The data in a cell can be displayed in many formats: date formats, percent, dollars, integer, and others. ENTERING DATA Choose a cell. You may then enter numbers or text from the keyboard. You can edit the cell’s data either in the cell or in the location provided as part of the toolbars. Use your mouse to choose the cell and the data entry or edit point for the cell chosen. MOUSE The mouse is a powerful tool in the Excel environment. The normal left-click is used to choose cells and locations (data entry windows). The left-click normally is also used to hit buttons (cancel, ok). Holding the left button down allows you to select a range of cells. The right button often reveals advanced features in a pull-down menu format. This is very useful when working with plots and graphs. Often the original plot needs updates and the mouse can be used to select a region of a plot (e.g. title or data) and then pull-up options for that region (format title, change the data source). SAVING DATA If the work will be important for future experiments you should save the file to the network or a floppy disk. If you want to safeguard current work you can save the file on the local computer (Data erased every Sunday). Your instructor will help students create a temporary area to save files and show you how to save the spreadsheet in this area using the "Save As" menu item under the "File" menu. Once you have saved the work in a file with a unique name you can periodically use the "Save" item in the "File" menu to update your work. It is good practice to periodically store your work. This enables you to recover from mistakes made while using the spreadsheet and computer problems.

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NAMING CELLS Names will simplify the use of Excel equations. Choose "Name" under the "Insert" menu. Choose "Define..." in the list. A dialog box appears. Enter "chosen name" at the top of the dialog box. (Note: If there is a name in an adjacent cell Excel will use this name by default.) The cell that is being named appears at the bottom of the dialog box. The cell name will include the worksheet name and have $ characters added. If this is the correct cell simply click the OK-button. (Note: Excel defaults to the current cell location.) If you want to chose a different cell then click the button in the bottom right-hand corner (RED arrow button) of the dialog box. This lets you select which cell or range of cells will be named. You will need to complete the cell selection with the ENTER key. The dialog box will disappear while you are choosing the cells and reappear when ENTER is hit. Test the process. Name a cell. Choose a new cell. Choose "Name" under the "Insert" menu. Now choose "Paste". A dialog box appears with the names of all the named cells. Choose one of your named cells and then hit OK. The new cell now has a formula that refers to the named cell. Hit enter. The cell contents should now be the same as the named cell. Change the value in the named cell and the new value appears in both cells. ENTERING FORMULAS A cell’s contents are interpreted as a formula if the first character is an equal sign. A formula can refer to another cell by call name or by naming the cell as described above.

=B3 set the cell’s contents to whatever is in cell B3 =vo set the cell’s content to the cell named vo. (If no cell has been named vo the error

message #NAME? appears.)

Common math operations can be used in formulas * multiplication - subtraction ^ raise to the power / divide () sets order of operations =36*B3+B4+7 multiply the contents of cell B3 by 36 and add the contents of B4

and the value 7.

There are a many special functions that can be used within a formula. Choose a cell and “insert” (on toolbar) “function” (on this menu). Choose a familiar function from the dialog box. When you have chosen your function hit OK. A new dialog box appears. This will aid in getting the arguments needed. The dialog box is similar to the one used in naming cells. The RED arrow button returns you to the spreadsheet so you can choose cells. The ENTER key completes the selection. If you chose the function =sum( ) then you need to supply a list of cells to sum. If you chose =sin( ) then you need to supply an angle. The arguments of functions can be other cells.

=sin(B3) takes the sine of the value in cell B3. =sum(B4:B8) sums the range of cells from B4->B8.

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Functions can be typed directly from the keyboard and the arguments for formulas and functions can be supplied by choosing cells with the mouse. PLOTTING Graphs or plots are powerful ways to visualize and analyze data. These tools will be used frequently in the lab. To plot data decide which columns should be plotted. 1. Choose "Insert" in the menu and then "Chart". 2. Choose "XY (Scatter)" on the first dialog window. Click "Next >". 3. Choose the data to plot by switching from "Data Range" to "Series" using the folder tab near

the top of the dialog window. a. Click "Add" to get your first data series. b. Click the button at the right of the "X Values:" entry window. The dialog box disappears

and you highlight the cells in the time column. Hit Enter after the box shows that all the desired times are selected.

4. Now use the "Y Values:" entry window and choose your position data. 5. Click finish. There are a number of refinements available for improving the graph. Labels, colors, or additional data can be added or changed. Experiment by clicking with either the left or right mouse button on various portions of the graph and seeing what you can change. You will need to explore the various options to become proficient at plotting data. A sample plot is shown below. The excel datasheet that generated this plot can be found in the desktop folder == Intro Physics Lab/excel worksheets/ 2CurvesOn1Chart.xls The worksheet describes some methods for creating charts (plots). Students may open up this folder and experiment with plotting.

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-2

-1

-1

0

1

1

2

time (s)x

(m)

-40

-20

020

40

0 0.5 1 1.5 2

v (m

/s)

position (m)velocity (m/s)

TRENDLINE To add a trendline you click on the graph on a data point. Right click to bring up a menu. (Choosing different sections of the graph will cause different menus to appear.) Choose "add trendline". Put the equation on the graph by setting the appropriate option on the trendline options page.

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APPENDIX 3: Establishing Uncertainty Every number used in the laboratory must be recorded with an uncertainty. This appendix will discuss methods for obtaining that uncertainty. This section should be used in conjunction with the Error Analysis section, which contains the definitions, formulas and some additional examples. Error and uncertainty have different meanings.

• Error is the difference between a value and its correct value or true value. The true value, of course, is not known.

• Uncertainty is an estimate of the difference between a calculated or measured value and the true value.

An instrument measures values that are in error by a certain amount. Since the exact true value of a quantity is unknown, instrumental error is also unknown. The experimenter is forced to estimate and to judge the estimation. The measured value is an estimate for the true value and the uncertainty is an estimate for the error. It will be important to understand what is an acceptable difference between two results. How much does one allow results to differ before the experiment is judged to have a serious flaw? The uncertainty is used to compare two values. It provides insight, when comparing two consecutive measurements, when comparing results with the theory, and when comparing two independent experiments. Experimenters know that their uncertainty is a safety net. Because you have an uncertainty, your results cover a range of possible values. Large uncertainties make an experiment very defensible. The result cannot be called into question if the uncertainty is so large that all reasonable results are included. On the other hand, an extremely small uncertainty is a sign of a high quality experiment, the smaller the uncertainty the better the experiment. This means that the optimal uncertainty has to balance these two opposing goals. The uncertainty should not be so small as to guarantee failure, nor so large that the experiment has no merit. It is, of course, unethical to arbitrarily increase or decrease an uncertainty without justification. Knowing that the uncertainty you claim determines the correctness and quality of the result under all future scrutiny, many experimenters spend considerable effort searching for unknown sources of error (increasing uncertainty) and pushing the limits of a technique (minimizing uncertainty ). COMMON SENSE UNCERTAINTY There are different approaches for establishing uncertainty but in this lab the focus will be on common sense rather than rigorous mathematical analysis. With this laboratory course focusing on different aspects of the experimental process, some of the rigor that is required for a real experiment will be relaxed. Often a student can use a simple straightforward method for guessing an uncertainty. Be sure to check with you lab instructor if you are not sure of your method.

• The scale on the ruler can be read to about 0.5mm. The uncertainty can be estimated based on this limitation to be 0.5 mm.

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• A few independent measurements (3 trials) can be used to calculate SD and SDM (see below). One of these two values could serve as the uncertainty. The discussion below should help you decide which one to use in any given situation.

• Some instruments may show no indication of error in their measurement. In an actual experiment a separate measurement could be used to determine the uncertainty or the experimenter could consult the instrument manual. In this lab the student may conclude that the instrument contributes a negligible error. This is true, for example for some voltage measurements.

• Instrument uncertainty is sometimes given in a manual. • Tex book constants are typically good to 3 significant figures. • Your instructor may prefer to provide the uncertainty for some quantities.

Human Error, Hand waving arguments– Student is not allowed to introduce an uncertainty unless the student has developed a method to measure that uncertainty. There will be no hand waving arguments permitted. If you believe that your experiment may be subject to errors that have not been included then you either develop a method to measure the uncertainty or you ignore it. Absolutely no error can be introduced into any discussion unless some quantitative estimate can be made for its size and all estimates need to be justified. The correct approach is to find a way to estimate these additional uncertainties, include them and reevaluate or to simply state that the experiment does not agree with theory within the uncertainty. Not allowed:

• The results are in agreement with predictions because in addition to the uncertainty measured there were some effects due to wind resistance.

• We suspect that human error and an uneven table are the source of the difference between our result and the theoretical result.

Allowed: • By measuring for a longer time period we were able to see a loss in energy over many

oscillations due to friction. As shown in the figure this resulted in a 2% change in the energy for one period. …

• Comparing the measurements of different students we were able to see an average deviation of 3 mm, which we attribute to a differences in reaction times. We therefore are factoring in a 3mm uncertainty in our analysis. …

• This experiment includes all of the measurable uncertainties that we found. The result for g, however, differs significantly from the accepted value (4 times the uncertainty). Re-examining possible pitfalls and carefully re-measuring g did not change our result.

You will be expected to include all the important sources of errors but you cannot merely state that something might be a source of error. You need to provide a justifiable guess as to how large it is. If you decide wind resistance might have influenced your measurement then, in order to mention it, you must think of a way to figure out how large an influence it is. If you cannot

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find all the errors in your experiment, you may have to conclude that your experiment failed to demonstrate the principle the lab was exploring. It is okay to have a failed experiment as long as another experimenter using the same equipment would get the same result. There are some sources of error that are to complex or subtle to be discovered in an introductory lab. There may be instruments that are not calibrated and cannot be tested by the student. Materials and components may be flawed in ways that are undetectable. It is advisable to do a dry run and perform calculations immediately to see that things are going as planned but sometimes even well designed experiments fail. If your experiment is unsuccessful and there is no obvious flaw then you should receive a good grade. Naturally your instructor will try and see why you failed. Your report may therefore require a more complete description and may be more difficult to write. Unknown – There will be times when estimating the uncertainty is not critical to the particular laboratory procedure. The student should still include the uncertainty as unknown. You should check with your instructor to see if this appropriate. Constants – Values from the textbook are usually given to 3 significant figures. You can use this rule for most of the constants used in the lab. The value of g, 9.80 m/s2, should be assigned an uncertainty of 0.01 m/s2. MEAN - STANDARD DEVIATION -STANDARD DEVIATION OF THE MEAN These are called statistics. They are functions of the data points that can be used by the experimentalist as an estimate for a quantity of interest. Usually the mean is a good estimate of the quantity measured. Most students already understand that averages can be superior to a single measurement. The underlying assumption is that the measurements are random. Some data points are greater and some less than the true value. Averaging tends to cancel these fluctuations. Be sure that you are comfortable with the notion that the mean of these data qualifies as a good estimator for the result. The standard deviation will be used as one estimate of uncertainty. The interpretation of the SD is a more subtle point. Since the SD is the average deviation of the measurements from their central value, one expects the SD could be used to estimate the uncertainty in a typical measurement. If 10 measurements of the same mass are on average 6 gm from their central value then assigning an uncertainty of 6 gm to each measurement is reasonable. The SD is then assigned as the error for each of the 10 measurements. Those close and those far from the mean are given the same value for their uncertainty (6 gm). To summarize: MEAN is an estimate for the true value of the quantity being measured. STANDARD DEVIATION is an estimate for the error in any ONE of the measurements averaged. The averaging process that provides the mean value often reduces the actual error. As mentioned above the mean is superior to a single measurement because of the cancellation due

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to the averaging. One can go further and state that averages are better when more values are used. Let N be the number of individual measurements. As N increases the average value improves as an estimate of the true value. If this point doesn’t seem obvious accept it for now and we will explore it later. This leads to the conclusion that the mean is closer to the true value than the standard deviation may suggest and that the uncertainty of the mean should depend on N. In fact another statistic, the standard deviation of the mean, SDM, is usually used to estimate the error if the mean is used as the estimate of the true value rather than SD the uncertainty in one of the individual measurements. STANDARD DEVIATION OF THE MEAN is an estimate of the error associated with using the mean as an estimate for the true value. A discussion of the mean, SD and SDM must include the limitations of these statistics as estimators. It is probably apparent that one cannot improve a measurement by simply recording and averaging more and more data. The limits arise due to a second type of error. These errors are called systematic errors. They are different from random errors because they influence each measurement in the same manner. A ruler that is too short is an example of a systematic error and such a ruler will measure all values to be short. Averaging cannot correct for this error. SYSTEMATIC errors cannot be reduced by averaging and they limit the extent to which averaging data can be used to reduce experimental error. When an experimenter judges, based on an evaluation of the experiment, that systematic errors could be a significant then the SD should be used so that one doesn’t underestimate the error. SD can be used as an overall estimate of the error (uncertainty) when the student suspects that there are systematic limits. An actual experiment must explore the extent of both types of error and develop methods to evaluate both of these errors. The introductory physics labs do not always require this level of thoroughness. More on Measurement Differences and Uncertainty An expression can be constructed for the likelihood of obtaining a certain result given the true value TV and an uncertainty σ. This expression is often a gaussian function. For illustration let us assume that you know the length of a field is exactly 50m (TV) and when measuring the length of the field the average uncertainty is 10m (σ). The gaussian function would be

2

21

21Pr

⎟⎠⎞

⎜⎝⎛ −

−

⎟⎠

⎞⎜⎝

⎛= σ

πσ

TVx

e

A plot of this function that describes this situation is shown below. The lines show those x values that are one σ (40, 60) away from TV (50).

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The likelihood of a measurement falling somewhere in this region is 68.3% (1 σ, 40 to 60). If we increase the range of values (2 σ, 30 to 70) there is a 95.4% probability that a measurement will fall somewhere within this range. If we increase the range of values (3 σ, 20 to 80) there is a 99.7% probability that a measurement will fall somewhere within this range. The probability of getting a measured value outside these ranges is 31.7%, 4.6% and 0.3%, respectively. You can conclude that finding two measured values of the same quantity that are 1 σ apart in not that unlikely but finding two measured values 3 σ apart is very unlikely and probably indicates one of the measurements is bad. When do my measurements agree with another experiment or with a theoretical value ? The graph above shows 6 measurements that were performed and compared to a theoretical prediction of g (circles, 9.8 m/s-s). The first thing to note is that both the theoretical (circles) and the experimental (squares) results have an associated uncertainty. Also note that measurement A, B, and C have a theoretical uncertainty that is very small compared to the experimental uncertainty. For measurements E, F, and G the experimenter is using a prediction for g that has a comparable uncertainty. A rigorous evaluation of the agreement or disagreement involves probability statements. However, our first goal is to get a sense of what the measurements mean and in this lab this will be the only requirement. Here are three rules of thumb one may use. The experiments are labeled as 1, 2 and 3 sigma. Sigma denotes the uncertainty chosen by the experimenter and is reflected in the size of error bars drawn.

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9

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

0 2 4 6 8 10 12

1SIGMA

2SIGMA

3SIGMA

1SIGMA

2SIGMA

3SIGMA

AFED

CB

1. Measurements and/or theoretical results that agree to within one sigma are in agreement. a. Shown as case A and D.

2. Measurements and/or theoretical results that agree to within 2 sigma are in agreement but suggest there may be problems. The experimenter needs to review his/her data and methods. The experimenter might duplicate the experiment.

a. Shown as case B and E. 3. Measurements that agree only at the 3 sigma level are in disagreement.

a. Shown as case C and F.

SIGNIFICANT FIGURES

WHEN YOU OBTAIN YOUR FINAL RESULT AND YOUR FINAL UNCERTAINTY BE SURE TO STATE THESE RESULTS WITH THE CORRECT NUMBER OF SIGNIFICANT FIGURES.

The Error Analysis Section of this lab (front of manual) has a detailed description of significant figures with numerous examples. In general, numbers presented in spreadsheet tables do not need to given with the correct number of significant figures. On the other hand, showing an excessive number of digits can clutter a spreadsheet and make it difficult to read. Intermediate results do not need to be given with the correct number of significant figures. Summary tables that are providing results for a lab section or a final result should be listed with the correct number of significant figures. If you are unsure ask your instructor.

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COMPOUNDING UNCERTAINTIES.

When a number of measured quantities are combined to get a result, the uncertainty associated with the result is a function of the uncertainties of measured quantities used to calculate the result. Examples:

• distances and times to find g, • temperatures and masses to measure the heat capacity

This type of analysis can be quite complicated. The correct way to add independent uncertainties is to add uncertainties in quadrature. The formulas in Section 3 of Appendix 4 in this Manual follow this rule and there one sees that many of the formulas sum squares and then take the square root. The following analysis will use a simpler, less rigorous, approximation. This approximation will overestimate the uncertainties. (always double check with your instructor as to the level of rigor expected in an analysis): Absolute uncertainty refers to actual uncertainty. Relative uncertainty involves the ratio of the uncertainty to another quantity. The relative

uncertainty can be expressed as: a) fractional uncertainty - the ratio of the absolute uncertainty to the measurement b) percent uncertainty – the fractional uncertainty times 100 Since dealing with percent uncertainties involves multiplying by 100 and then later dividing by 100 to get back to an absolute uncertainty, it is suggested to use fractional uncertainty to avoid this step, as is used in the discussion below. When quantities add or subtract, add absolute uncertainties. When quantities are multiplied or divided, the fractional uncertainty in the result is the sum

of the fractional uncertainties in the quantities used in the calculation. When a quantity A is raised to the power j, B=Aj. The fractional uncertainty is j times the

fractional uncertainty in A. (fractional uncertainty B) = j (fractional uncertainty A) For formulas that consist of several different operations, combine the uncertainties as you

perform the calculation. A spreadsheet is ideal for this type of calculation. Uncertainties (absolute & relative) are always positive. If when calculating uncertainties,

the measurement or calculated value is negative, then use the absolute value. One can find uncertainties by plugging in values +/- the uncertainty into a formula and see

how the result changes. This can be misleading when some of your values should be combined as smaller values with others as larger values to get the largest fluctuation. Students can typically ignore this effect.

More complicated formulas will require more complicated relationship. Discuss these with your instructor.

Example of Calculating Uncertainties Equations 4 and 6 shown below are extracted from a lab designed to measure Lf and Lv. mi(Lf + Cw(Tf-0)) = mwCw(To - Tf) + mcCc(To - Tf) (4)

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msLv + msCw(Tbp-Tf) = mwCw(Tf - To) + mcCc(Tf - To) (6) How do we find the uncertainty, for example, in equation 6 for L assuming that we measure the following quantities: quantity uncertainty fractional uncert. mass of water mw 413.6 gm Δmw 0.14 Δmw/ mw 0.0003 final temperature Tf 44.7 oC Δ Tf 0.3 Δ Tf / Tf 0.0067 initial temperature To 22.6 oC Δ To 0.3 Δ To / To 0.0133 boiling point of water Tbp 99.1 oC Δ Tbp 0.2 Δ Tbp / Tbp 0.0020 mass of the steam added ms 15.5 gm Δms 0.14 Δ ms /ms 0.0090 mass of the container mc 60.3 gm Δmc 0.07 Δmc/mc 0.0012 known sp. heat of water Cw 1.000 cal/gm oC ΔCw 0.0 ΔCw/Cw 0.0000

known sp. heat of copper

Cc 0.0924 cal/gm oC ΔCc 0.0 ΔCc/Cc 0.0000

The quantities Cw and Cc are assumed to be known exactly. The experimenter measures several quantities and determines the uncertainties in each quantity measured. There are various techniques for finding the uncertainty including statistical analysis and consulting instrument specifications. Let us solve equation 6 for Lv. msLv = mwCw(Tf - To) + mcCc(Tf - To) - msCw(Tbp-Tf)

s

fbpwsofccofwwv m

TTCmTTCmTTCmL

)()()( −−−+−=

Substituting the values above we find Lv = 543. There are three terms added together

s

ofww

mTTCm

term)(

1−

=

136

To determine the uncertainty we first note that each term includes a difference in temperatures. The uncertainty associated with these differences (add absolute uncertainties) is Term Value uncertainty fractional uncertainty Tf - To 22.1 0.3 + 0.3 = 0.6 0.0271 Tbf - Tf 54.4 0.3 + 0.2 = 0.5 0.0092 Once the temperature difference and its uncertainty have been determined, each term becomes a product (or quotient) of numbers. Therefore we add fractional uncertainties to get the uncertainty in each term. For example, to determine the uncertainty for term 1 we sum the fractional uncertainty in the mass of water, the fractional uncertainty for specific heat, the fractional uncertainty for the temperature difference (above table) and the fractional uncertainty for the mass of steam. Term Value fractional uncertainty uncertainty 1 589.7135 0.0003 + 0 + 0.0271 + 0.0090 = 0.0365 21.536 2 7.944194 0.0012 + 0 + 0.0271 + 0.0090 = 0.0373 0.297 3 54.4 0 + 0.0092 = 0.0092 0.500 Since you add the terms together we must sum up absolute uncertainty of each term in the sum. ΔLv = 21.536 +0.297 + 0.5 = 22.3 ΔLv /Lv =.041 or 4.1% Final result Lv = 543 ± 22 cal/gm

s

ofcc

mTTCm

term)(

2−

=

)(3 fbpw TTCterm −−=

137

ERROR ANALYSIS CHECK SHEET List of all directly measured quantities: Name of quantity Value Uncertainty Method for estimating error Some measured quantities are indirect and must be calculated from a set of direct measurements. Indirect measurements Quantity: Formula or relationship used to calculate this quantity: List of Direct quantities absolute uncertainty fractional uncertainty TOTAL ERROR

FINAL RESULTS - USE THE CORRECT NUMBER OF SIGNIFICANT FIGURES.

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Appendix 4 SUGGESTIONS FOR HANDLING DATA

Significant Figures

Experimental Errors (uncertainty)

Statistical Treatment of Errors (uncertainty)

Error (uncertainty) Analysis Cookbook

by

Dr. D. Chodrow Definition of terms: Error is the difference between a value and its correct value or true value. Uncertainty is an estimate of the difference between a calculated or measured value and the true value.

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1. SIGNIFICANT FIGURES Concepts and Definitions No measurement is exact. Consequently, whenever we measure any quantity it is necessary to state both the measured value and some estimate of the precision. The number of SIGNIFICANT FIGURES used in stating a measured value indicates the precision. The number of significant figures in a number is defined as follows:

1) The leftmost nonzero digit is the most significant digit.

2) If there is no decimal point, the rightmost nonzero digit is the least significant digit.

3) If there is a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

4) The number of significant figures is the number of digits from the least significant digit to the most significant digit, inclusive.

Examples:

1) 4630 has 3 significant figures. 4630. has 4 significant figures 4630.000 has 7 significant figures. 0.000 20 has 2 significant figures. 0.000 200 0 has 4 significant figures.

2) Those zeroes whose only function is to locate the decimal point in a decimal fraction such as 0.000 456 or a large integer such as 6 789 000 000 are not significant. Such numbers are best expressed in scientific notation with only the significant figures given. The numbers in this example would be given as

4.56 x 10-4 which has 3 significant figures, and 6.789 x 109 which has 4 significant figures.

When a measured value is written down, the POSITION OF THE LEAST SIGNIFICANT FIGURE indicates the magnitude of the precision. Examples:

3) If you state that the length of a rod is 34.76 cm, you are implying that the leftmost three figures are certain and that the least significant figure is uncertain to some degree. In other words, you are stating that the length of the rod is probably not less than 34.7 cm and not more than 34.6 cm, and that you are reasonably confident that the length is 34.76 cm.

140

4) You use a balance which is known to be accurate only to within 0.2 gm to make a single measurement of the mass of a machine screw. Even if the balance pointer indicates a value of 2.637 gm, you may only state the mass of the screw as 2.6 gm. This is because the second and third decimal places are meaningless here since the first place is already uncertain.

Arithmetic with Significant Figures SUMS AND DIFFERENCES: Suppose that we use three different methods to measure the lengths of the sides of a triangle. The resulting lengths, each given with the proper number of significant figures, are 27.113 cm, 8.63 cm and 19.2 cm. We wish to determine the perimeter (the sum of the lengths of the sides of the triangle. Proceeding without regard to significant figures, we add the lengths of the sides to get 27.113 cm 8.63 cm + 19.2 cm 54.943 cm. We must now interpret this result. In any number obtained by measurement, all digits following the least significant digit are UNKNOWN. Therefore, the lengths of each side, to the nearest thousandth of a cm, are 27.113 cm, 8.63X cm and 19.2YZ cm, where X,Y and Z stand for COMPLETELY UNKNOWN digits. The correct result for the perimeter is 27.113 cm 8.63X cm + 19.2Y2 cm

______________ 54.9AB cm where A and B are also unknown digits. The sum therefore has only three significant figures and is correctly given as 54.9 cm. From this example, we see that the rule for determining the number of significant figures is a sum or a difference is:

THE LEAST SIGNIFICANT DIGIT OF THE RESULT IS IN THE SAME COLUMN RELATIVE TO THE DECIMAL POINT AS THE LEAST SIGNIFICANT DIGIT OF THE NUMBER ENTERING INTO THE SUM OR DIFFERENCE WHICH HAS ITS LEAST SIGNIFICANT DIGIT FARTHEST TO THE LEFT.

Example 5: Let x = 4.231, y = 32.6, z = 29, and w = 31.7

A) If p = x+y, y is the number whose least significant digit, 6, is farthest to the left, so

141

p = 4.231 + 32.6 = 36.831 = 36.8 Thus p = 36.8 to the proper number of significant figures

B) If q = x + y - z, z is the number whose least significant digit, 9, is farthest to the left, so a = 4.231 + 32.6 - 29 = 7.831 = 8 Thus q = 8 to the proper number of significant figures.

C) If r = z - w, then r = -3 to the proper number of significant figures.

PRODUCTS AND QUOTIENTS: Suppose that we use different methods to measure the lengths of the adjacent sides of a rectangle, and that the results given to the proper number of significant figures are a = 3.24 cm and b = 4.112 cm. We wish to determine the area of the rectangle. Proceeding without regard to significant figures, we get A = ab = 3.24 cm x 4.112 cm = 13.32288 cm2

We must now interpret this result. The least significant digits of a and b are uncertain to some degree. Let us assume an uncertainty of 2 in the least significant digits. Then a could have any value between 3.22 cm and 3.26 cm, while b could have any value between 4.110 cm and 4.114 cm. Therefore A = ab could have any value between Amin = 3.22 cm x 4.110 cm = 13.2342 cm2 and Amax = 3.26 cm x 4.114 cm = 13.41164 cm2 Therefore the first decimal place is the first uncertain figure, and the area is properly reported as A = 13.3 cm2 which has THREE significant figures.

If a had been determined to two significant figures, a = 3. 2 cm, while b = 4.112 cm, we would have found A = 3.2 cm x 4.112 cm = 13.1584 cm2 Now, since a could have any value in the range from 3.0 cm to 3.4 cm while b could have any value in the range from 4.110 cm to 4.114 cm, we would have Amin = 3.0 cm x 4.110 cm = 12.33 cm2 and Amax = 3.4 cm x 4.114 cm = 13.9876 cm2 Now the first digit to the left of the decimal point is uncertain, so the area is properly reported as

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A = 13 cm2 which has TWO significant figures.

From these examples we see that the rule for determining the number of significant figures in a product (or a quotient) is:

THE NUMBER OF SIGNIFICANT FIGURES IN A PRODUCT OR QUOTIENT IS THE SAME AS THE NUMBER OF SIGNIFICANT FIGURES IN THE FACTOR WITH THE FEWEST SIGNIFICANT FIGURES.

Example 6: Let x = 6.63 x 1.0-4, y = 9.0346, z = 47320 and t = 4.2 Then

A) f = xy/t = 6.63 x 10-4, y = 9.83A6/ 4.2 = 1.552 x 10-3. This must be rounded off to two significant figures, so

f = 1.6 x 10-3

B) g = 3x2z/y = 3x(6.63 x 10-4)2 x 47320 / 9.8346 = 6.345068 x 10-3 or, to three significant figures,

g = 6.35 x 10-3

C) h = z/y2 = 47320 / (9.8346)2 = 489.2505636, or to four significant figures (z has four significant figures), h = 489.3

ROUNDOFF ERRORS, A WARNING EXAMPLE: It is often advisable to ignore the preceding rules for arithmetic with significant figures during intermediate stages in a calculation, although the final result must be given with the correct number of significant figures. This is because rounding off intermediate values in a long calculation may lead to arithmetic errors. The following example demonstrates this point. We wish to find the value of q = x4y3z where x = 1.36, y = 1.26 and z = 5.2. According to the rule for products and Quotients the value of q should have two significant figures. Before the days of hand-held electronic calculators, a common labor-saving technique was to round all data and the results of all intermediate steps to the lowest number of significant figures. In this case, x would be rounded to x1 = 1.4 while y would be rounded to y1 = 1.3. Then, to two significant figures, x1

4 = 1.44 = 3.8416 = 3.8

143

y13 = 1.33 = 2.197 = 2.2

and q1 = x1

4y13z = 3.8 x 2.2 x 5.2 = 43.472 = 43

The result a is incorrect because both x4 and y3 have been overestimated. We now calculate a, keeping all the figures provided by a ten-digit calculator: x4 = 1.364 = 3.42102016 y3 = 1.263 = 2.000376 and q = 1.364 x 1.263 x 5.2 = 3.42102016 x 2.000376 x 5.2 = 35.58529844 To two significant figures, q = 36. The value q1 = 43 is 19% too large. In order to avoid arithmetic errors arising from premature rounding off:

DO NOT ROUND OFF THE INTERMEDIATE STAGES OF A LONG CALCULATION. INSTEAD, DO THE ARITHMETIC AS IF ALL THE DATA CONSISTED OF EXACTLY KNOWN VALUES, USING YOUR CALCULATORS MEMORY TO STORE ANY INTERMEDIATE RESULTS. THE FINAL RESULT SHOULD THEN BE ROUNDED OFF TO THE CORRECT NUMBER OF SIGNIFICANT FIGURES.

144

2. EXPERIMENTAL ERRORS (uncertainty)-- AN INTRODUCTION

All measured quantities contain inaccuracies. These inaccuracies complicate the problem of determining the "true" value of a quantity. Therefore, the object of experimental work must be to determine the best estimate of the "true" value of the quantity being measured, together with an indication of the reliability of the measurement.

There are two main sources of experimental error systematic errors and statistical errors. SYSTEMATIC ERRORS are associated with the particular instruments or technique used. They can result when an improperly calibrated instrument is used or when some unrealized influence perturbs the system in some definite way, thereby biasing the result of the measurement. An example of such an influence is the small amount of friction due to air resistance, which acts on a dropped object in such a way as to reduce its acceleration by a small unknown amount.

Sometimes it is possible to correct for systematic errors. If, for example, we know that a voltmeter is calibrated so that it always reads 10% too low, it is a simple matter to compute the correct voltage by multiplying the meter reading by 10/9. Most of the time, however, the task of discovering and compensating for systematic errors is very difficult, requiring great familiarity with the experimental techniques and equipment used. There are no general methods for dealing with systematic errors.

No matter how carefully a measurement is made, it will possess some degree of variability. The errors which result from the lack of precise repeatability of a measurement are called STATISTICAL ERRORS or RANDOM ERRORS. It is often possible to minimize statistical errors by judicious choice of measuring equipment and technique, but they can never be eliminated completely. We must therefore learn how to determine the statistical error associated both with a single directly measured quantity and with a result which is calculated from several measured quantities.

The terms ACCURACY and PRECISION are often used to describe the reliability of a measurement. Although these terms are commonly used interchangeably, they have very different meanings in scientific work. A quantity is determined with great ACCURACY if the result of the measurement is close to the "true" value. In other words, great accuracy is equivalent to small systematic errors. A quantity is determined with great PRECISION if the measurements are closely repeatable. In other words, great precision is equivalent to small statistical errors.

It is possible for a measurement to be precise without being accurate or vice versa. The aim of scientific work is to achieve both accuracy and precision.

There is a third type of error, which is due entirely to poor experimental technique and carelessness. These errors are called BLUNDERS or MISTAKES and are totally unacceptable in scientific work. They can be eliminated completely with a reasonable amount of care. Be sure that you understand what you are supposed to do in the

145

laboratory before you start any experiment. Read the instructions. If you do not understand how to use a piece of equipment or how to analyze your data, reread the instructions. If you are still confused, ask your laboratory instructor for help. LABORATORY REPORTS CONTAINING BLUNDERS WILL SUFFER A SEVERE GRADE PENALTY. 3. STATISTICAL TREATMENT OF EXPERIMENTAL DATA Introduction

The variability inherent in any repeated measurement makes it impossible to determine with absolute certainty the "true" value of a physical quantity. However, it is possible to make several measurements of such a quantity and to use them to estimate both the value of the quantity and the statistical uncertainty of the estimate. (Uncertainty will be used to signify an estimate of the error.) It is also possible to estimate the value and uncertainty of a result which is calculated from other quantities whose values and uncertainties have been determined.

In the rest of this section we will assume that all sources of systematic error have been eliminated or compensated for so that only the statistical uncertainties remain to be dealt with. Estimating the Best Value and Uncertainty of a Measured Quantity

Let us assume that we have made N INDEPENDENT measurements of a quantity x, with the resulting values

x1, x2, x3, .......... xn. The problem before us is to make the best guess of the "true" value of x and of the statistical uncertainty or error in x.

We first define the MEAN or AVERAGE of the measurements to be

or

In general, a bar over a quantity indicates the mean of that quantity.

X =( x + x + x +...x )

N1 2 3 n

x =1N x

k=1

N

k∑

146

If systematic errors have been eliminated and each of the N measurements is equally reliable, then the best estimate of the "true" value of x is the mean x : Best value = Mean value

Now we must determine the reliability of this estimate. To do this, we must find the UNCERTAINTY (estimate of error) in x, Δx, which is defined by saying that it is very likely that the "true" value of x lies in the range from x - Δx to x + Δx.

The precise meaning of "very likely" depends on the particular method used to compute Δx. We will use a method for which the probability of the "true" value of x lying in the range from is about 2/3. We then present the result as

The uncertainty Δx depends both on the number N of measured values and on the

dispersion, or scatter of the individual measurements about their mean. A useful measure of the dispersion is called the STANDARD DEVIATION. It is approximately the same as the r.m.s. (root-mean-square) deviation. The RMS and SD are defined by the following expressions

Roughly 2/3 of the measured values of x should lie in the range from x - σ to x + σ.

The equations above define the standard deviation σ but do not provide an easy way to compute it. Many hand-held calculators have pre-programmed algorithms for calculating x and σ for a set of data. If such a calculator is not available and N is large so that N-1 can be replaced by N, the following equation provides an easier calculation for σ:

x = x x± Δ

)x(-)x( 22≅σ

N)x-x(+...+)x-x(+)x-x(

)x-(x=rms2

N2

22

12 =

1−N)x-x(+...+)x-x(+)x-x(=

2N

22

21σ

)x-x(N

1= 2k

N

=1k∑−1

σ

147

The standard deviation σ is usually a property of the measuring technique or equipment, and can often be thought of as a measure of the "free play" in the equipment. It seems intuitively reasonable that the reliability of a measurement should increase as the number N of data values increases. We should therefore expect that the experimental error Δx should decrease as N increases. This involves a statistical quantity called the standard deviations of the mean, σx . σx is the statistical uncertainty in x. It can be shown that the STANDARD DEVIATION OF THE MEAN is

Therefore 1) The best estimate of x is x . 2) The statistical uncertainty of x is Δx = σx.

This means that the probability that the "true" value of x lies between x - Δx and x + Δx is roughly 2/3. We then write

Notice that the denominator in the equation for x vanishes when N = 1. This is because the concept of a statistical uncertainty based on the dispersion of the data becomes meaningless when there is only a single data value. Occasionally it is necessary or convenient to make only one measurement of a quantity. In that case, the statistical uncertainty in that quantity should be taken to be the RESOLUTION of the measuring device, which is the smallest increment in the quantity which can be distinguished. For example, if the thickness of a rod is measured once using a caliper which can be read to within 0.02 cm, then the uncertainty in t is Δt = 0.02 cm. Example 1: Ten measurements of the length of a stretched spring yield the values (all in cm) 7.2, 6.9, 7.1, 7.0, 7.1, 7.2, 6.9, 7.0, 6.9, 7.1 Solution -- In this case, N = 10. The best estimate of x is the mean

N=x

σσ

x = x x± Δ

x =1N x

k=1

N

m∑

148

= (7.2 + 6.9 + 7.1 + 7.0 + 7.1 + 7.2 + 6.9 + 7.0 + 6.9 + 7.1) / 10

Note that we do not round x off to 2 significant figures. In fact, we do not round anything off until all the calculations are finished. To find the standard deviation use a calculator which is pre-programmed or make the following calculations

= (7.22 + 6.92 + 7.12 + 7.02 + 7.12 + 7.22 + 6.92 + 7.02 + 6.92 + 7.12) /10

Then (using the approximation that N-1=9 is about the same as 10).

which we have rounded to three significant figures. The standard deviation of the mean is

x = 7.04cm

( x ) =1N

x2

k=1

N2

m∑

( x )= 49.574 cm2 2

σ = ( x ) - ( x ) = 49.574 - (7.04 ) = 0.111cm2 2 2

cm0.03=0.111cm=N

=x 510

σσ

149

This is the uncertainty in x, Δx = 0.035 cm. Then x = (7.04 ± 0.035) cm. We can now determine the correct number of significant figures in x. It is highly probable that the "true" value of x lies between x - Δx = 7.005 cm and x + Δx = 7.075 cm. We see that the tenth' place is certain but the hundredths' place is not. Therefore, x should be stated with three significant figures. Since it does not make any sense to give a numerical value for the uncertainty in a figure which is completely uncertain, we round the uncertainty off to one significant figure (or two at the most) and quote the value of x, with its uncertainty, as x = (7.04 ± 0.04) cm. Avoid serious blunders by rounding off only at the end of the calculation. Example 2: In the previous example we saw how the number of significant figures quoted in an experimental result is determined by the uncertainty. Usually we only give the uncertainty to one figure. For example, if our calculations yield v = 43.2684 m/s and Δv = 0.02162 m/s we should quote the result as v = (43.27 ± 0.02) m/s Sometimes however, we will quote the uncertainty to two significant figures and keep an extra figure in the mean. This is done only when it is necessary to prevent rounding off in such a way as to affect two figures in the result. For example, if T = 8.9631 s and ΔT = 0.3421 s, we may quote T as either T = (9.0 ± 0.3) s or T = (8.96 ± 0.34) s Strictly speaking, the first equation is correct, but the second equation is more convenient. However, it would be totally incorrect to state that T = (8.9631 ± 0.3421) s. Sometimes instead of stating the ABSOLUTE uncertainty Δx, one states the RELATIVE uncertainty in a quantity x. There are two ways to do this. The FRACTIONAL uncertainty in x is

while the PERCENT UNCERTAINTY in x is simply 100 times the fractional uncertainty

FractionalError =x

xΔ

150

Example 3: If M = 4.79 kg and ΔM = 0.08 kg then the fractional uncertainty in M is

and the percent uncertainty is 1.7%, which could equally well have been rounded off to 2 %. We may quote M as either M = (4.79 ± 0.08) kg or M = 4.79 kg ± 2 % Note that we never give a percent uncertainty to more than two significant figures. Propagation of Uncertainties (errors): Estimating the Best Value and Uncertainty of a Result Calculated from Several Independently Measured Quantities

Let the quantity q depend on the quantities x, y, z, .... through the equation q = f(x, y, z, ...) If x, y, z, ... are measured independently with the results

we must find the best value for q together with its uncertainty.

We will assume that the relative uncertainties in x, y, z, ... are small. Then the best estimate for q is found by substituting the best estimates for x, y, z, ... into the equation which defines q:

PercentError = 100x

xΔ

ΔMM

=0.08kg4.79kg

= 0.017

x = x x, y = y Y,z = z z± ± ±Δ Δ Δ

q = f( x, y,z,... )

151

There are two very important special cases for which the statistical uncertainty in q can be calculated from easy to remember formulas: Special Case 1 -- q is a SUM q = Ax + By + Cz + .... where A, B, C, ... are constants. In this case, the uncertainty in q is

Special Case 2 -- q is a PRODUCT

where K, A, B, C, . . . . . are constants. In this case, the fractional uncertainty in q is

and the uncertainty is

Examples: 4) If q = 7y2 and y = 26.3 + 0.8, find q and Δq. Solution: Here y = 26.3 and Δy = 0.8. Then q = 7( y )2 = 4841.83. We find Δq by first finding the fractional uncertainty

Δ Δ Δ Δq = A ( x ) + B ( y ) + C ( z ) +...2 2 2 2 2 2

q = K x y zA B C

Δ Δ Δ Δqq

= Ax

x+ B

yy

+ Cz

z+...2

22

2

22

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

ΔΔ

q =q

qq

⎛

⎝⎜

⎞

⎠⎟

152

Then the uncertainty is Δq = 0.0608 q = 0.0608 x 4841.83 = 294. Then, since q = 4.8 x 103 and Δq = 0.3 x 103, q = (4.8 ± 0.3) x 103. 5) If w = 18z -1/3 and z = (7.24 ± 0.06), find w and its uncertainty. Solution: Here z = 7.24 so w = 18(7.24)-1/3 = 18 x 0.5169 = 9.304. Now

so Δw = 0.00276 x 9.304 = 0.0257 = 0.03. Then w = 9.30 ± 0.03. 6) A rectangle of sides x and y has perimeter L = 2x + 2y and area A = xy. x and y are measured and found to be x = (3.0 ± 0.1) m and y = (2.65 ± 0.02) m. Find the perimeter and area of the rectangle together with their uncertainties. Solution: Here x = 3.0 m, y = 2.65 m, Δx = 0.1 m and Δy = 0.02 m. Perimeter: L = 2 x + 2 y = 2(3.0) + 2(2.65) = 11.3 m. Since L is a sum,

Then L = (11.3 ± 0.2) m. Area: A = xy = (3.0) (2.65) = 7.95 m2 . Since A is a product,

Δ Δqq

= 2y

y=

2x0.826

.3 = 0.0608

Δ Δww

= -13

zz

= 0.00276

Δ Δ ΔL = 2 ( x ) + 2 ( y ) = 0.2042 2 2 2 m

0.034=yy

2+xx

1=AA

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ⎟⎠⎞

⎜⎝⎛ ΔΔ

153

Then ΔA = 0.034 A = 0.034 x 7.95 m2 = 0.27 m2, and A = (7.95 ± 0.27) m2 or A = (8.0 ± 0.3) m2 7) A cylinder of radius r and height h has volume V = πr2 h and surface area S = π r2 + 2 πrh. If the radius and height of the cylinder have been measured with the results r = (4.60 ± 0.05) cm and h = (6.0 ± 0.1) cm, find the volume and surface area of the cylinder together with their uncertainties. Solution: Here r = 4.60 cm, h = 6.0 cm, Δr = 0.05 cm and Δh = 0.1 cm. Volume:

Since V is a product,

then V = 0.027 V = 0.027 x 398.9 cm3 = 10.8 cm3 = 11 cm3, and V = (399±11) cm3 or V = (4.0 ± 0.1) x 102 cm3. 8) Great care is needed when two experimentally determined quantities whose values are close are to be subtracted. For example, if x = (123 ± 4) cm and y = (129 ± 3) cm and if the quantity of interest is w = y - x, then

while

so w = (7 ± 5) cm. Even though x and y are determined with precision of 3.3% and 2.3%, w has a percent uncertainty of 71%.

V = ( r ) ( h )= (4.60 ) (6.0)= 398.9 cm2 2 3π π

Δ Δ ΔVV

= 2r

r+1

hh

= 0.02722

22

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

w = y - x = 129 -123 = 7cm

Δ Δ Δ Δ Δw = (-1 ) ( x ) +(1 ) ( y ) = ( x ) +( y ) = 5cm2 2 2 2 2 2

154

4. ERROR (uncertainty) ANALYSIS COOKBOOK Measurements Suppose N values are recorded x1, x2, x3, ...xN Mean or Average: (Best estimate = Mean value)

Nx+x+x+x=x N321 K

Standard Deviation: )x(-)x()x-x(N

1= 222k

N

=1k

≅− ∑1

σ

Standard Deviation of the Mean: (Statistical Uncertainty = Standard Deviation of the

Mean)

N=x

σσ

Propagation of Uncertainties Suppose q = f(x,y,z,....) Best Estimate:

Statistical Uncertainty: (General Case)

Statistical Uncertainty: (Special Case where q is a sum q = Ax + By + Cz ...)

Statistical Uncertainty: (Special Case where q is a product q = KxAyBzC....)

q = f( x, y,z, )K

Δ Δ Δ Δq = (qx

) ( x ) +(qy

) ( y ) +(qz

) ( z ) +2 2 2 2 2 2∂∂

∂∂

∂∂

K

Δ Δ Δ Δq = A ( x ) + B ( y ) + C ( z ) +2 2 2 2 2 2K

|q

q|= A (

xx

) + B (y

y) + C (

zz

) +2 2 2 2 2 2Δ Δ Δ ΔK

physics 140l laboratory manual -...

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