GENERAL PHYSICS 1 LABORATORY - hcmiu.edu.vn guide.pdf · ... 6 4. EXPERIMENT ... International...

VIETNAM NATIONAL UNIVERSITY-HCMC

INTERNATIONAL UNIVERSITY

GENERAL PHYSICS 1

LABORATORY

Instructor: Trinh Thanh Thuy

HCMC, September, 2017

International University-VNU-HCMC General Physics 1 Laboratory

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CONTENTS

LAB 0: ERRORS AND UNCERTAINTIES ...................................................................................................... 1

1. EQUIPMENT ................................................................................................................................................. 1

2. INTRODUCTION .......................................................................................................................................... 1

3. THEORY ........................................................................................................................................................ 1

3.1 Accuracy and Precision ............................................................................................................................. 2

3.2 Types and Sources of Experimental Errors ............................................................................................... 2

3.2.1 Systematic Errors ............................................................................................................................... 3

3.2.2 Random Errors ................................................................................................................................... 3

3.3 Calculating Experimental Error ................................................................................................................ 4

3.3.1 Significant Figures ............................................................................................................................. 4

3.3.2 Percent Error ..................................................................................................................................... 4

3.3.3 Percent Difference ............................................................................................................................. 4

3.3.4 Mean and Standard Deviation ........................................................................................................... 5

3.4 Reporting the Results of an Experimental Measurement .......................................................................... 6

4. EXPERIMENT ............................................................................................................................................... 8

4.1 Apparatus .................................................................................................................................................. 8

4.2 Experiment set up ...................................................................................................................................... 9

4.3 Procedure ................................................................................................................................................ 10

LAB 1: PROJECTILE MOTION ...................................................................................................................... 12

1. EQUIPMENT ............................................................................................................................................... 12

2. INTRODUCTION ........................................................................................................................................ 12

3. THEORY ...................................................................................................................................................... 12

4. EXPERIMENT ............................................................................................................................................. 14

4.1 Horizontal launching ............................................................................................................................... 14

4.1.1 Experiment set up (for muzzle velocity and muzzle velocity vs. time of flight) ................................. 14

4.1.2 Procedure ......................................................................................................................................... 15

4.1.3 Data Analysis ................................................................................................................................... 16

4.1.4 Experiment set up and procedure (for horizontal range) ................................................................. 16

4.2 Launching at an Angle ............................................................................................................................ 17

4.2.1 Experiment set up ............................................................................................................................. 17

4.2.2 Procedure ......................................................................................................................................... 17

4.2.3 Analysis ............................................................................................................................................ 18

4.3 Launching at an Angle from a Height ..................................................................................................... 19

4.3.1 Experiment Set up ............................................................................................................................ 19

4.3.2 Procedure ......................................................................................................................................... 19

4.3.3 Analysis ............................................................................................................................................ 20

LAB 2: NEWTON’S LAW ............................................................................................................................... 22

1. EQUIPMENT ............................................................................................................................................... 22

2. INTRODUCTION ........................................................................................................................................ 22

3. THEORY ...................................................................................................................................................... 22


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4. EXPERIMENT ............................................................................................................................................. 25

4.1 Newton’s first law ................................................................................................................................... 25


4.1.2 Procedure ......................................................................................................................................... 25

4.2 Newton’s second law .............................................................................................................................. 26


4.2.2 Procedure ......................................................................................................................................... 26

4.2.3 Calculations ..................................................................................................................................... 27

4.3 Newton’s third law .................................................................................................................................. 28


4.3.2 Procedure ......................................................................................................................................... 29

LAB 3: CONSERVATION OF MOMENTUM ............................................................................................... 30

1. EQUIPMENT ............................................................................................................................................... 30

2. INTRODUCTION ........................................................................................................................................ 30

3. THEORY ...................................................................................................................................................... 30

4. EXPERIMENT ............................................................................................................................................. 32

4.1 Experiment set up .................................................................................................................................... 32

4.2 Procedure ................................................................................................................................................ 32

4.2.1 Forces between Interacting Objects ................................................................................................. 32

4.2.2 Newton’s Law and Momentum Conservation .................................................................................. 37

LAB 4: CONSERVATION OF ANGULAR MOMENTUM ........................................................................... 41

1. EQUIPMENT ............................................................................................................................................... 41

2. INTRODUCTION ........................................................................................................................................ 41

3. THEORY ...................................................................................................................................................... 41

4. EXPERIMENT ............................................................................................................................................. 43

4.1 Setup apparatus ....................................................................................................................................... 43

4.2 Procedure ................................................................................................................................................ 43

4.3 Analysis ................................................................................................................................................... 44

4.4 Questions ................................................................................................................................................. 45

LAB 5: ROTATIONAL INERTIA ................................................................................................................... 46

1. EQUIPMENT ............................................................................................................................................... 46

2. INTRODUCTION ........................................................................................................................................ 46

3. THEORY ...................................................................................................................................................... 46

4. EXPERIMENT ............................................................................................................................................. 49

4.1 Set up apparatus ...................................................................................................................................... 49

4.2 Procedure ................................................................................................................................................ 49

4.2.1 Measurements for the theoretical rotational inertia ........................................................................ 49

4.2.2 Measurements for the experimental method..................................................................................... 50

4.3 Calculations ............................................................................................................................................. 52

4.4 Extra experiment ..................................................................................................................................... 52

5. APPENDIX ................................................................................................................................................... 53


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LAB 6: SLIDING FRICTION .......................................................................................................................... 57

1. EQUIPMENT ............................................................................................................................................... 57

2. INTRODUCTION ........................................................................................................................................ 57

3. THEORY ...................................................................................................................................................... 57

4. EXPERIMENT ............................................................................................................................................. 61

4.1 Set-up experiment ................................................................................................................................... 61

4.2 Procedure ................................................................................................................................................ 61

4.3 Analysis ................................................................................................................................................... 62

4.4 Conclusions ............................................................................................................................................. 64

LAB 7: VARIABLE-G PENDULUM .............................................................................................................. 65

1. EQUIPMENT ............................................................................................................................................... 65

2. INTRODUCTION ........................................................................................................................................ 65

3. THEORY ...................................................................................................................................................... 65

3.1 Simple pendulum .................................................................................................................................... 65

3.2 Physical pendulum .................................................................................................................................. 68

4. EXPERIMENT ............................................................................................................................................. 70


4.2 Procedure ................................................................................................................................................ 71

4.3 Analysis ................................................................................................................................................... 72

4.4 Questions ................................................................................................................................................. 72

LAB 8: VIBRATING STRINGS ...................................................................................................................... 74

1. EQUIPMENT ............................................................................................................................................... 74

2. INTRODUCTION ........................................................................................................................................ 74

3. THEORY ...................................................................................................................................................... 74

4. EXPERIMENT ............................................................................................................................................. 77


4.2 Procedure ................................................................................................................................................ 78

LAB 9: GYROSCOPE ...................................................................................................................................... 80

1. EQUIPMENT ............................................................................................................................................... 80

2. INTRODUCTION ........................................................................................................................................ 80

3. THEORY ...................................................................................................................................................... 80

3.1 Torque and rotational inertia ................................................................................................................... 80

3.2 Precession................................................................................................................................................ 82

3.3 Gyroscope ............................................................................................................................................... 85

4. EXPERIMENT ............................................................................................................................................. 88

4.1 Measuring the precession rate ................................................................................................................. 88


4.1.2 Procedure ......................................................................................................................................... 88

4.2 Measuring quantities for the theoretical value ........................................................................................ 89


4.2.2 Procedure ......................................................................................................................................... 90


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4.2.3 Analysis ............................................................................................................................................ 90

LAB 10: BERNOULLI’S PRINCIPLE ............................................................................................................ 92

1. EQUIPMENT ............................................................................................................................................... 92

2. INTRODUCTION ........................................................................................................................................ 92

3. THEORY ...................................................................................................................................................... 92

4. EXPERIMENT ............................................................................................................................................. 94

4.1 Pre-Set up Preparations ........................................................................................................................... 94

4.1.1 Connect apparatus ........................................................................................................................... 94

4.1.2 Water Supply .................................................................................................................................... 95

4.1.3 Water-flow measurement by a stop watch ........................................................................................ 96

4.2 Clean-up .................................................................................................................................................. 97

4.3 Storage .................................................................................................................................................... 97

LAB 11: IDEAL GAS LAW............................................................................................................................. 98

1. EQUIPMENT ............................................................................................................................................... 98

2. INTRODUCTION ........................................................................................................................................ 98

3. THEORY ...................................................................................................................................................... 98

4. EXPERIMENT ........................................................................................................................................... 101

4.1 Description of apparatus (Ideal Gas Law Syringe) ............................................................................... 101

4.2 Ideal Gas Law Syringe .......................................................................................................................... 101

4.2.1 Procedure ....................................................................................................................................... 101

4.2.2 Analysis .......................................................................................................................................... 102

4.3 Constant Temperature ........................................................................................................................... 103

4.3.1 Procedure ....................................................................................................................................... 103

4.3.2 Analysis .......................................................................................................................................... 103

4.4 Further Investigations............................................................................................................................ 104

4.5 Adiabatic Compression ......................................................................................................................... 104

4.5.1 Procedure ....................................................................................................................................... 104

4.5.2 Analysis .......................................................................................................................................... 104

LAB 12: GAY-LUSSAC’S LAW ................................................................................................................... 105

1. EQUIPMENT ............................................................................................................................................. 105

2. INTRODUCTION ...................................................................................................................................... 105

3. THEORY .................................................................................................................................................... 105

4. EXPERIMENT ........................................................................................................................................... 107

4.1 Constant volume (Gay-Lussac’s law) ................................................................................................... 107

4.1.1 Description of apparatus ................................................................................................................ 107

4.1.2 Procedure ....................................................................................................................................... 107

4.1.3 Analysis .......................................................................................................................................... 107

4.2 Further Investigations............................................................................................................................ 108

4.3 Conclusion ............................................................................................................................................ 108

4.4 Determining Absolute Zero while Keeping the Number of Gas Moles (n) Constant ........................... 108

LAB 13: HEAT ENGINE CYCLES ............................................................................................................... 110


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1. EQUIPMENT ............................................................................................................................................. 110

2. INTRODUCTION ...................................................................................................................................... 110

3. THEORY .................................................................................................................................................... 111

4. EXPERIMENT ........................................................................................................................................... 113

4.1 Set up .................................................................................................................................................... 113

4.2 Software Set Up .................................................................................................................................... 114

4.3 Procedure .............................................................................................................................................. 114

4.4 Analysis ................................................................................................................................................. 115

LAB 14: BLACKBODY RADIATION .......................................................................................................... 117

1. EQUIPMENT ............................................................................................................................................. 117

2. INTRODUCTION ...................................................................................................................................... 117

3. THEORY .................................................................................................................................................... 118

4. EXPERIMENT ........................................................................................................................................... 120

4.1 Experiment set up .................................................................................................................................. 120

4.2 Procedure .............................................................................................................................................. 123

4.3 Analysis ................................................................................................................................................. 124

5. APPENDIX ................................................................................................................................................. 125


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LAB 0: ERRORS AND UNCERTAINTIES

1. EQUIPMENT

INCLUDED Scienceworkshop

1 Photogate Head ME-9498A

1 Large Rod Stand ME-8735

1 45 cm Long Steel Rod ME-8736

NOT INCLUDED BUT REQUIRED

1 ScienceWorkshop® 750 Interface CI-6450 or CI-7599

1 DataStudio Software CI-6870

2. INTRODUCTION

Reliability estimates of measurements greatly enhance their value. Thus, saying that the

average diameter of a cylinder is 10.00 0.02 mm tells much more than the statement that

the cylinder is a centimeter in diameter.

3. THEORY

No physical quantity can be measured with perfect certainty; there are always errors in

any measurement. This means that if we measure some quantity and, then, repeat the

measurement, we will almost certainly measure a different value the second time. How can

we know the “true” value of a physical quantity? The short answer is that we can’t.

However, as we take greater care in our measurements and apply ever more refined

experimental methods, we can reduce the errors and gain greater confidence that our

measurements approximate ever more closely the true value.

“Error analysis” is the study of uncertainties in physical measurements, and a complete

description of error analysis would require much more time and space than we have in this

course. However, by taking the time to learn some basic principles of error analysis, we

can:

1) Understand how to measure experimental error,

2) Understand the types and sources of experimental errors,

3) Clearly and correctly report measurements and the uncertainties in those

measurements, and


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4) Design experimental methods and techniques and improve our measurement skills to

reduce experimental errors.

Two excellent references on error analysis are:

John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in

Physical Measurements, 2d Edition, University Science Books, 1997

Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis

for the Physical Sciences, 2d Edition, WCB/McGraw-Hill, 1992

3.1 Accuracy and Precision

Experimental error is the difference between a measurement and the true value or

between two measured values. Experimental error is measured by its accuracy and

precision.

Accuracy measures how close a measured value is to the true value or accepted value.

Since a true or accepted value for a physical quantity may be unknown, it is sometimes not

possible to determine the accuracy of a measurement.

Precision measures how closely two or more measurements agree with other. Precision

is sometimes referred to as “repeatability” or “reproducibility”. A measurement which is

highly reproducible tends to give values which are very close to each other. Figure 0.1

defines accuracy and precision by analogy to the grouping of arrows in a target.

Figure 0.1 Accuracy and Precision

3.2 Types and Sources of Experimental Errors

When scientists refer to experimental errors, they are not referring to what are

commonly called mistakes, blunders, or miscalculations. Sometimes also referred to as

“illegitimate”, “human”, or “personal” errors, these types of errors can result from

measuring a width when the length should have been measured, or measuring the voltage

across the wrong portion of an electrical circuit, or misreading the scale on an instrument,


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or forgetting to divide the diameter by 2 before calculating the area of a circle with the

formula A = πr2 . Such errors are surely significant, but they can be eliminated by

performing the experiment again correctly the next time.

Experimental errors, on the other hand, are inherent in the measurement process and

cannot be eliminated simply by repeating the experiment no matter how carefully. There

are two types of experimental errors: systematic errors and random errors.

3.2.1 Systematic Errors

Systematic errors are errors that affect the accuracy of a measurement. Systematic

errors are “one-sided” errors, because, in the absence of other types of errors, repeated

measurements yield results that differ from the true or accepted value by the same amount.

The accuracy of measurements subject to systematic errors cannot be improved by

repeating those measurements. Systematic errors cannot easily be analyzed by statistical

analysis. Systematic errors can be difficult to detect, but once detected can be reduced only

by refining the measurement method or technique.

Common sources of systematic errors are faulty calibration of measuring instruments,

poorly maintained instruments, or faulty reading of instruments by the user. A common

form of this last source of systematic error is called “parallax error”, which results from the

user reading an instrument at an angle resulting in a reading which is consistently high or

consistently low.

3.2.2 Random Errors

Random errors are errors that affect the precision of a measurement. Random errors are

“two-sided” errors, because, in the absence of other types of errors, repeated measurements

yield results that fluctuate above and below the true or accepted value. Measurements

subject to random errors differ from each other due to random, unpredictable variations in

the measurement process. The precision of measurements subject to random errors can be

improved by repeating those measurements. Random errors are easily analyzed by

statistical analysis. Random errors can be easily detected, but can be reduced by repeating

the measurement or by refining the measurement method or technique.

Common sources of random errors are problems estimating a quantity that lies between

the graduations (the lines) on an instrument and the inability to read an instrument because

the reading fluctuates during the measurement.


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3.3 Calculating Experimental Error

When a scientist reports the results of an experiment, the report must describe the

accuracy and precision of the experimental measurements. Some common ways to

describe accuracy and precision are described below.

3.3.1 Significant Figures

The least significant digit in a measurement depends on the smallest unit which can be

measured using the measuring instrument. The precision of a measurement can then be

estimated by the number of significant digits with which the measurement is reported. In

general, any measurement is reported to a precision equal to 1/10 of the smallest graduation

on the measuring instrument, and the precision of the measurement is said to be 1/10 of the

smallest graduation.

For example, a measurement of length using a meter stick with 1-mm graduations will

be reported with a precision of ±0.1 mm. A measurement of volume using a graduated

cylinder with 1-ml graduations will be reported with a precision of ± 0.1 ml.

Digital instruments are treated differently. Unless the instrument manufacturer indicates

otherwise, the precision of measurement made with digital instruments are reported with a

precision of ± ½ of the smallest unit of the instrument.

For example, a digital voltmeter reads 1.493 volts; the precision of the voltage

measurement is ± ½ of 0.001 volts or ± 0.0005 volt.

3.3.2 Percent Error

Percent error (sometimes referred to as fractional difference) measures the accuracy of a

measurement by the difference between a measured or experimental value E and a true or

accepted value A.

The percent error is calculated from the following equation:

%Error 100E A

A

(0.1)

3.3.3 Percent Difference

Percent difference measures precision of two measurements by the difference between

the measured or experimental values E1 and E2 expressed as a fraction the average of the

two values. The equation to use to calculate the percent difference is:

1 2

1 2

%Difference 100

2

E E

E E

(0.2)


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3.3.4 Mean and Standard Deviation

When a measurement is repeated several times, we see the measured values are grouped

around some central value. This grouping or distribution can be described with two

numbers: the mean, which measures the central value, and the standard deviation which

describes the spread or deviation of the measured values about the mean

For a set of N measured values for some quantity x, the mean of x is represented by the

symbol <x> and is calculated by the following formula:

N

i 1 2 3 N-1 N

i=1

1 1x = x = x +x +x +...+x +x

N N (0.3)

where xi is the ith

measured value of x. The mean is simply the sum of the measured

values divided by the number of measured values. The standard deviation of the measured

values is represented by the symbol x and is given by the formula:

x

1

1

1

N2

i

i=

σ = x - xN -

(0.4)

The standard deviation is

sometimes referred to as the “mean

square deviation” and measures how

widely spread the measured values

are on either side of the mean. The

meaning of the standard deviation

can be seen from Figure 0.2, which

is a plot of data with a mean of 0.5.

SD represents the standard

deviation. As seen in Figure 0.2, the

larger the standard deviation, the

more widely spread the data is about

the mean. For measurements which have only random errors, the standard deviation means

that 68% of the measured values are within x from the mean, 95% are within 2 x from

mean, and 99% are within 3 x from the mean.

3.4 Reporting the Results of an Experimental Measurement

When a scientist reports the result of an experimental measurement of a quantity x, that

result is reported with two parts. First, the best estimate of the measurement is reported.

Figure 0.2: Measured Values of x


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The best estimate of a set of measurement is usually reported as the mean x of the

measurements. Second, the variation of the measurements is reported. The variation in the

measurements is usually reported by the standard deviation x of the measurements.

The measured quantity is then known have a best estimate equal to the average, but it

may also vary from xx to xx . Any experimental measurement should then be

reported in the following form:

xx x (0.5)

Example

Consider Table 0.1, which lists 30 measurements of the mass m of a sample of some

unknown material.

Table 0.1: Measured Mass (kg) of unknown Material

1.09 1.01 1.1 1.14 1.16

1.11 1.04 1.16 1.13 1.17

1.14 1.03 1.17 1.09 1.09

1.15 1.06 1.12 1.08 1.20

1.08 1.07 1.14 1.11 1.05

1.06 1.12 1.00 1.1 1.07

We can represent this data on a type of bar chart called a histogram (Figure 0.3), which

shows the number of measured values which lie in a range of mass values with the given

midpoint.

For the 30 mass measurements, the mean mass is given by:


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133 04 kg 110 kg

30m . . (0.6)

We see from the histogram that the data does appear to be centered on a mass value of

1.10 kg. The standard deviation is given by:

30

m

1

11.10 0 05 kg

30-1

2

i

i=

σ = m - . (0.7)

We also see that from the histogram that the data does, indeed, appear to be spread

about the mean of 1.10 kg so that approximately 70% (= 20/30×100) of the values are

within m from the mean.

The measured mass of the unknown sample is then reported as:

110 0 05 kgm .. (0.8)

----- End of Theory-----

Reference from: http://www.ece.rochester.edu/courses/ECE111/error_uncertainty.pdf


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4. EXPERIMENT

The major objectives of this experiment is to develop a basic understanding of what it

means to make an experimental measurement and provide a methodology for assessing

random and systematic errors in this measurement process. In addition this lab will also

give you a minimal framework in which to introduce you to the PASCO© interface

hardware and software.

By now you should have had numerous opportunities to become familiar with time and the

concept of a time interval. The increment of one second will be used as an intuitive

reference point. In this lab you will test your ability to internalize this one second time

interval by making and recording a repetitive flicking motion with your finger. By flicking

your finger back and forth you will move it though an infrared beam sensor (i.e. the

PASCO photogate) and each full cycle (back and forth, approximately 2 seconds) will be

simultaneously recorded, plotted and tabulated by the PASCO interface software. Your goal

in this experiment is to assess the size of systematic and random errors in your data set

and learn a simple methodology for distinguishing between the two.

SYSTEMATIC ERRORS: These are errors which affect the accuracy of a measurement.

Typically they are reproducible so that they always affect the data in the same way. For

instance if a clock runs slowly you will make a time measurement which is less than the

actual reading.

RANDOM ERRORS: These are errors which affect the precision of a measurement. A

process itself may have a random component (as in radioactive decay) or the measurement

technique may introduce noise that causes the readings to fluctuate. If many measurements

are made, a statistical analysis will reduce the uncertainty from random errors by

averaging.

4.1 Apparatus

a. Computer with monitor, keyboard and mouse.

b. A PASCO photogate and stand: This device emits a narrow infrared beam in the gap

and occluding the beam prevents it from reaching a photodetector. When the beam

is interrupted the red LED should become lit. (Plugged into DIGITAL CHANNEL

#1.)

c. A PASCO Signal Interface (CI-700 or CI-750) monitors the photodetector

output vs time and can be configured to tabulate, plot and analyze this data.


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4.2 Experiment set up

To configure the experiment you should refer to Figure 0.1 below. Adjust the photogate so

that one member can easily and repetitively flick his/her finger through the gap. The phone-

jack cable from the photogate should be plugged into the DIGITAL CHANNEL #1 socket.

Ignore the other sensors which may already be plugged into other sockets. It

is important that the PASCO interface be turned on before the computer. If not the

computer will not recognize it and, therefore, it must be rebooted to properly communicate

with the PASCO module.

Figure 0.1

To initiate the PASCO interface software you will need to click the computer mouse on the

telescope icon in the “toolkit” area below. Figure 0.2 below gives a good idea of how the

display should appear. Note that, while you are able to reconfigure the display parameters,

the default values that are specified on start-up will allow you to do most of this experiment

without necessitating any major changes.

You will note that a “dummy” first data set already exists on start-up showing a typical data

run. In the table you can view all 47 data points and the statistical analysis, including mean

and standard deviation. In addition there should be a plot of this data and a histogram.


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Figure 0.2

4.3 Procedure

1. Start the preliminaries by CLICKing on the icon and practice “flicking” a

finger back and forth so that a two second interval appears in the window. CLICK

on the Stop icon when done. The same person need not perform both operations.

This will produce a second data set. (There is also a “monitor” function which can

permit adjustments and trials without storing the results in memory. To access this

type ALT-M).

2. Each run gets its own data set in the “Data” display window. (If there are any data

sets in existence you will not be able to reconfigure the interface parameters or

sensor inputs.) A data set can be deleted by moving the mouse cursor to the “Run

# 1” position, CLICKing the left mouse button and then striking the “Delete” key.

3. Once you are comfortable with the procedure then click on the icon and

cycle a finger back and forth over fifty times. The click on the “Stop” icon. DO

NOT watch the time display while you do this, since you want to find out how

accurately and precisely you can reproduce a time interval of 2 s using only your

mind.


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4. What is the mean time per cycle? What is the standard deviation? The mean t and

standard deviation are given by:

N2

iNi=1

i

i=1

t - t

t = t N and σ =N - 1

Questions to consider:

I. Is your mean suggestive of a systematic error?

II. Does your data qualitatively give the appearance of a normal distribution (i.e. a

Gaussian bell curve.)

5. For analyzing and quantifying random errors, you need to asses how a data set is

distributed about the mean. The standard deviation is one common calculation

that does this. In the case of a normal distribution approximately 68% of the data

points fall within 1 of the mean (90% within 2). Is your data consistent with

this attribute?

6. Assessing the possibility of systematic behavior is somewhat more subtle. In

general is a measure of how much a single measurement fluctuates from the

mean. In this run you have made fifty presumed identical measurements. A better

estimate of how well you have really determined the mean is to calculate

the standard deviation of the mean N

. After recording in you lab book,

can you now observe any evidence that there is a systemic error in your data?

Answer this same question with respect to the first “sample” data set.

7. OPTIONAL: Systematic errors can sometimes drift over time. In the best-case

scenario they drift up and down so that they hopefully average out to zero. (Clearly

it would be better if they could be eliminated entirely.) With respect to t and for

the first 25 and second 25 cycles do you observe any systematic trends? Use the

“Zoom Select” feature on the Graph. Simply by draging the mouse will highlight a

subset of the data. The mean, and other statistical attributes will appear at the

bottom of the table.

References

http://badger.physics.wisc.edu/lab/manual/


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LAB 1: PROJECTILE MOTION

1. EQUIPMENT

1 Mini Launcher ME-6825

1 Smart Timer ME-8930

1 Time of Flight Accessory ME-6810

2 Photogate Head ME-9498A

1 Photogate Bracket ME-6821

1 Universal Table Clamp ME-9376B

1 Carbon Paper SE-8693

1 Metric Measuring Tape SE-8712A


1 Xplorer GLX Pasport PS 2002

1 Glue

2. INTRODUCTION

The purpose of this experiment is to predict the horizontal range of a projectile shot from

various heights and angles. In addition, students will compare the time of flight for

projectiles shot horizontally at different muzzle velocities.

3. THEORY

Figure 1.1. Trajectory of the projectile motion


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The horizontal range, R, for a projectile can be found using the following equation:

xR v t (1.1)

where vx is the horizontal velocity and t is the time of flight. vx0= v0cosϴ0

To find the time of flight, t, the following kinematic equation is needed:

0

21

2y ya t v tH (1.2)

where is the height, ay is the acceleration due to gravity and vy0 is the vertical component

of the initial velocity. vy0= v0sinϴ0

A. Projectile is fired horizontally

When a projectile is fired horizontally (from a height), the time of flight can be found from

rearranging Equation (1.2). Since the initial velocity is zero, the last term drops out of the

equation yielding:

2

y

tH

a (1.3a)

B. Projectile is fired at an angle

When a projectile is fired at an angle and it lands at the same elevation from which it was

launched, the first term in Equation (1.2) is dropped. Rearranging yields:

y

y

a

vt

02 (1.3b)

C. Projectile is fired from a height at an angle

When a projectile is fired from a height, none of the terms drop out and Equation (1.2) must

be rearranged as follows:

2

0

10

2y ya t v t H (1.3c)

Equation (1.3c) must be solved quadratically to find the time of flight, t.



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4. EXPERIMENT

4.1 Horizontal launching

4.1.1 Experiment set up (for muzzle velocity and muzzle velocity vs. time of

flight)

1. Choose one corner of a table to place the projectile launcher. Make sure a distance of

about 3 meters is clear on the floor around the table.

2. Clamp the launcher to the corner of the table using the Universal Table Clamp (see

photo below).

3. Using the attached plumb bob, adjust the angle of the launcher to 0o.

4. Slide the Photogate Bracket into the groove on the bottom of the launcher and tighten

the thumbscrew.

5. Connect two photogates to the bracket (see photo below).

Safety tips

Wear Safety Goggles.

Do not place foreign objects into the

Launcher.

Do not look into the Launcher.

Do not aim the Launcher at others.


Trinh Thanh Thuy 15

6. Plug the photogate closest to the launcher into port 1 on the Smart Timer. Plug the other

photogate into port 2.

7. Turn on the Smart Timer. Using the red "Select Measurement" button, choose the

"Time" Measurement."

8. Using the blue "Select Mode" button, choose the "Two Gates Mode". This will measure

the time it takes for the projectile to travel between the two photogates.

4.1.2 Procedure

4.1.2.1 Muzzle Velocity

1. Using the cross-hairs on the side, record the height of the projectile. In addition, record

the spacing between the two photogates.

2. Place the steel ball into the launcher and use the push rod to load the ball until the “3rd

click” is heard.

3. Hold a piece of cardboard a few centimeters past the 2nd

photogate to block the ball.

4. Press the Start button on the Smart Timer.

5. Pull the launch cord on the launcher.

6. Record the time from the Smart Timer display.

7. Repeat steps 2-6 for 2 clicks and 1 click.

Data Table 1.1

Projectile Height: _________ m

Photogate Spacing: ________________ m

Number of Clicks Time Between Photogates (s)

3rd Click

2nd Click

1st Click

4.1.2.2 Muzzle Velocity vs. Time of Flight

1. Remove the photogate from port 2 of the Smart Timer and replace it with the Time of

Flight Accessory.


Trinh Thanh Thuy 16

2. Load the ball into the launcher to the 3rd click.

3. Predict where the ball will land and explain your prediction.

4. Launch the ball and note where it lands. Place the Time of Flight Accessory such that

the ball will land on it.

5. Place the steel ball into the launcher and use the loader to push the ball in until the “3rd

click” is heard.

6. Press the Start button on the Smart Timer. Note: Use the same Smart Timer setting as

Part 4.1.2.1.

7. Pull the launch cord on the launcher.

8. Record the time from the Smart Timer display into Data Analysis Table 1.2.

9. Repeat steps 2-8 for 2 clicks and 1 click.

4.1.3 Data Analysis

Muzzle Velocity vs. Time of Flight

1. Use the time between the photogates and the spacing between the photogates to find the

muzzle velocity of the projectile for each firing.

2. Record these values into Data Analysis Table 1.2.

Data Analysis Table 1.2

Number of Clicks Muzzle Velocity (m/s) Time of Flight (s)

3rd Click

2nd Click

1st Click

4.1.4 Experiment set up and procedure (for horizontal range)

Prediction: Using the initial height of the projectile and the muzzle velocity from the "3rd

click," calculate the theoretical horizontal range of the ball.

1. Tape a target to the floor in front of the projectile launcher at a distance equal to the


Trinh Thanh Thuy 17

range prediction calculated above.

2. Place carbon paper over the target.

3. Align the projectile launcher.

4. Launch the ball from the 3rd click. Repeat four more times.

5. Remove the carbon paper. Observe the locations where the ball struck the Bull's Eye.

4.2 Launching at an Angle

4.2.1 Experiment set up

1. Clamp the launcher to the edge of a table using the Universal Table Clamp so that the

ball launches from and lands at the same elevation (see photo above).

2. Adjust the angle of the launcher to 25o. Note: With the photogate bracket and

photogates attached to the launcher, the lowest angle is approximately 23o.





5. Using the blue "Select Mode" button, choose the "Two Gates Mode." This will measure

the time it takes the projectile to travel between the two photogates.

4.2.2 Procedure

1. Using the push rod, push the ball as far as possible into the Launcher. Make sure two

clicks are heard. Using the string, pull back on the trigger. Note the location on the table

where the ball lands.

2. Tape a sheet of blank paper at this location. Place carbon paper over the blank paper.

3. Load the Launcher.


5. Launch the ball.

6. Use the tape measure to find the horizontal range.

7. Record the experimental data. Enter the value of the angle in degrees, the time between


Trinh Thanh Thuy 18

photogates, and the horizontal range in meters into the “Measured Range” data table.

8. Repeat the steps 1-7 for 35, 45, 65, 85 degrees.

Data Table 1.3: Measured Range

Distance Between Photogates: __________ m

Angle

(degrees)

Time between

Photogates (s)

Horizontal

Range (m)

25

35

45

65

85

4.2.3 Analysis

1. Using the distance between the photogates and the time between the photogates (Data

Table 1.3), calculate the initial velocities of the ball. Record these values into the Initial

Velocity Analysis Table.

Analysis Table 1.4: Initial Velocity

Angle (degrees) Initial Velocity (m/s)

25

35

45

65

85

2. Using the initial velocity and the angle; calculate the horizontal range in meters. Enter

this value for each angle into the “Calculated Horizontal Range” Analysis Table (Table

1.5). Hint: Calculate the components of the initial velocities. See the “THEORY”

section.

Analysis Table 1.5: Calculated Horizontal Range

Angle (degrees) Horizontal Range (m)

25

35


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45

65

85

3. Use DataStudio to plot both the Measured Horizontal Range vs. Angle and the

Calculated Horizontal Range vs. Angle on the same graph.

4.3 Launching at an Angle from a Height

4.3.1 Experiment Set up

1. Clamp the launcher to the edge of a table using the Universal Table Clamp so that the

ball launches from a position above where it will land (see photo above).

2. Adjust the angle of the launcher to -20o.





5. Using the blue "Select Mode" button, choose the "Two Gates Mode." This will measure

the time it takes the projectile to travel between the two photogates.

4.3.2 Procedure

1. Using the cross hairs on the side of the launcher, measure and record the height of the

ball's starting point.

2. Using the plunger, push the ball as far as possible into the Launcher. Make sure three

clicks are heard. Using the string, pull back on the trigger. Keep track of the location on

the floor where the ball lands.

3. Tape a sheet of blank paper at this location. Place carbon paper over the blank paper.

4. Load the Launcher.


6. Launch the ball.

7. Use the tape measure to find the horizontal range.


Trinh Thanh Thuy 20

8. Record the experimental data. Enter the value of the angle in degrees, the time between

photogates, and the horizontal range in meters into the “Measured Horizontal Range”

data table 1.6.

9. Repeat the steps 1-8 for the angles listed below in Data Table 1.6.

Data Table 1.6: Measured Horizontal Range

Height: ________________________ m

Distance Between Photogates: ______ m

Angle

(degrees)

Time between

Photogates (s)

Horizontal

Range (m)

-20

-10

0

10

20

30

40

45

50

60

70

80

4.3.3 Analysis

1. Using the distance between the photogates and the time between the photogates (Data

Table 1.6), calculate the initial velocities of the ball. Record these values into the Initial

Velocity Analysis Table (Table 1.7).

Analysis Table 1.7: Initial Velocity

Angle (degrees) Initial Velocity (m/s)

-20

-10

0

10


Trinh Thanh Thuy 21

20

30

40

45

50

60

70

80

2. Using the initial velocity and the angle; calculate the horizontal range in meters. Enter

this value and the angle into the “Calculated Range” Analysis Table (table 1.8). Hint:

Find the vertical component of the initial velocity and the initial height of the projectile.

Use these values in equation 1.3c and solve it quadratically to find the time of flight.

Analysis Table 1.8: Calculated Range

Angle (degrees) Horizontal Range (m)

-20

-10

0

10

20

30

40

45

50

60

70

80

Using DataStudio software, plot both the Measured Horizontal Range vs. Angle and the

Calculated Horizontal Range vs. Angle on the same graph.


Trinh Thanh Thuy 22

LAB 2: NEWTON’S LAW

1. EQUIPMENT

INCLUDED: PASPORT

Newton’s first Law

1 PAScar Dynamics System ME-6955

1 Motion Sensor PS-2103

1 Discover Friction Accessory ME-8574

1 Physics String SE-8050

1 Computer Interface PS-2001


Newton’s second and third Law

2 Force Sensor PS-2104

1 Smart Pulley with Clamp ME-9448A

1 Adjustable Feet (Optional) ME-9470

2. INTRODUCTION

The purposes of these experiments are to:

a. Determine how external forces influence an object's motion. The following objects are

pushed briefly: a Cart and a Friction Tray. The resulting velocity is measured with a

Motion Sensor. An analysis of this motion yields Newton's 1st Law.

b. Determine Newton’s 2nd

Law. A modified version of Atwood’s machine is set up with

a mass tied to string that hangs over a pulley at the end of a table. The other end of the

string is tied to a Force Sensor mounted on a cart. A Motion Sensor records the velocity

of the cart.

c. Determine the relationship between interacting forces. Two Force Sensors are used to

measure the paired forces in a rubber band tug-o-war and the paired forces in a collision

of two carts.

3. THEORY

Newton's laws of motion are three physical laws that, together, laid the foundation for

classical mechanics. They describe the relationship between a body and the forces acting

upon it, and its motion in response to those forces. They have been expressed in several

different ways, over nearly three centuries and can be summarized as follows.


Trinh Thanh Thuy 23

First law: When viewed in an inertial reference frame, an object either remains at rest or

continues to move at a constant velocity, unless acted upon by an unbalanced

force.

Second law: In an inertial reference frame, the vector sum of the forces F on an object is

equal to the mass m of that object multiplied by the acceleration vector a of

the object: F = ma.

Third law: When one body exerts a force on a second body, the second body

simultaneously exerts a force equal in magnitude and opposite in direction on

the first body.

First law:

Students may be familiar with the following definition of inertia - also known as

Newton's 1st Law:

"An object at rest will remain at rest. An object in motion will remain in motion."

However, how does a scientist quantify the statement above? Specifically, what affects the

object's speed and direction? These are all questions best left for direct investigation...

Second law:

The following equation is Newton’s 2nd

Law: F = ma

F represents the forces acting upon a mass, m; and a represents the resulting

acceleration. Imagine an object in space pulled in opposite directions by two equal forces.

The sum of these forces, therefore, equals zero. According to Newton’s 2nd

Law, the mass

will not accelerate. If, however, the forces are not equal, then the object will accelerate in

the same direction as the net force.

In this lab, the acceleration must be measured from a velocity-time graph. Since

acceleration is defined as the change in the velocity per unit time, then the slope of the

velocity-time graph equals the acceleration.

Third law:

Students may be familiar with the following definition of Newton's 3rd Law:

"For every action there is an equal and opposite reaction."

The statement means that in every interaction, there is a pair of forces acting on the two

interacting objects. The size of the forces on the first object equals the size of the force on


Trinh Thanh Thuy 24

the second object. The direction of the force on the first object is opposite to the direction

of the force on the second object. Forces always come in pairs - equal and opposite action-

reaction force pairs.

However, how does the statement above manifest itself in physical interactions?

Specifically, what determines the magnitude and direction of the forces? These are all

questions best left for direct investigation...



Trinh Thanh Thuy 25

4. EXPERIMENT

4.1 Newton’s first law


1. Connect the Motion Sensor to the PASPORT Interface. Connect the PASPORT

Interface to the computer.

2. Make sure the switch on the top of the motion sensor is set to "cart."

3. Adjust the alignment bar on the side of the motion sensor so that it points slightly

downward.

4. Open the file “1st Law (PP).ds.”

4.1.2 Procedure

1. Place the Friction Tray about 1 meter away from the Motion Sensor.

2. Press the Start button in DataStudio.

3. Push and release the Friction Tray in the direction of the motion sensor.

4. Data collection will stop after several seconds.

5. To erase data, select "Experiment" from the menu bar and choose "Delete last data run".

6. Repeat the above steps, if necessary, until you have one representative data run.

7. Repeat for the Hover Puck and the Cart, making sure someone prevents collisions with

the Motion Sensor.


Trinh Thanh Thuy 26

4.2 Newton’s second law


1. Connect the Motion Sensor to a PASPORT interface. Make sure the switch on the top

of the Motion Sensor is set to "cart."

2. Connect the Force Sensor to a PASPORT interface. Connect the interface to the

computer.

3. Using the long thumbscrew, attach the Force Sensor to the cart.

4. Place the Motion Sensor on one end of the track as in the picture above. Adjust the

alignment knob on the side of the Motion Sensor so that it points parallel to the track.

5. Level the track.

5. Optional: Use adjustable feet on both ends to level the track. Attach

the Motion Sensor to the end of the track as shown at right.

6. Clamp the pulley to the other end of the track. Place this end over

the edge of the table.

7. Wrap one end of a one meter length of string around the notch of the

mass hanger.

8. Place the Cart/Force Sensor assembly on the track. Tie the other end of the string to the

hook of the Force Sensor. Hang the mass hanger over the pulley.

9. Level the string by adjusting the pulley.

10. Open the file “2nd Law (PP).ds.”

4.2.2 Procedure

1. With no tension on the string, press the "TARE" or "ZERO" button on the Force

Sensor.

2. Pull the cart back as far as possible without allowing the mass hanger to contact the


Trinh Thanh Thuy 27

pulley.

3. Simultaneously press the START button at the top of DataStudio and

release the cart. Prevent the cart from colliding with the pulley.

4. Make sure the Force Sensor’s cord does not impede the cart’s motion.

5. Data recording will stop automatically.

6. Using the cursor, highlight only the section of the velocity graph that corresponds to the

intended motion. Press the Fit button and select “Linear Fit.” Enter the value of

the acceleration into the data table 2.1.

7. Using the cursor, highlight only the section of the force graph that corresponds to the

accelerated motion. The legend displays the mean force for this highlighted section.

Enter the value of the mean force into the data table 2.1.

8. Go to the EXPERIMENT menu and select "Delete all Data Runs."

9. Repeat the previous steps until a total of 4 data runs are collected. Each time increase

the mass by 5 grams. No mass bar is added to the cart. (Run #1, #2, #3, #4)

10. Repeat more the previous steps until a total of 4 data runs are collected. Each time add

more 250g mass bar to the top of cart. Keep the hanging mass constant at run #4 in step

9.

11. Observe the Force vs. Acceleration graph. Press the Fit button and select “Linear Fit.”

Record the values of the slope and vertical intercept.

12. Find the mass in kilograms of the Cart and Force Sensor.

4.2.3 Calculations

Calculate the theoretical acceleration when the mass is constant and the net force is

changed, and record the calculations in the data table 2.1.

1. The theoretical acceleration is the ratio of the net force divided by the total mass.

hanging

cart+force sensor hanging

m .ga =

m + m

2. For runs #5, #6, #7 and #8, the total mass of the system (mass of cart + hanging mass +

mass bar) increases and the net force (hanging mass 9.8) remains constant.

3. Assuming no friction, the net force is the weight of the hanging mass (mass 9.8 N/kg).

Find the percent difference between the theoretical and experimental acceleration and

record it in the data table 2.1.


Trinh Thanh Thuy 28

theoretical - experiment%diff 100

theoretical=

Table 2.1

Trial

No

Mass of

Cart +

Force

sensor

(kg)

Mass of

hooked +

the

hanging

(kg)

Mean

force

(N)

Measured

acceleration

(m/s2)

Theoretical

acceleration

(m/s2)

%diff

Slope -

vertical

intercept of

Force vs.

Acceleration

graph

1

2

3

4

5

6

7

8

4.3 Newton’s third law


1. Connect one Force Sensor to a PASPORT interface. Connect the other Force Sensor to

a PASPORT interface. Connect the interface(s) to the computer.

2. With nothing connected to the Force Sensors, press the "ZERO" or "TARE" buttons on

the Force Sensors.

3. Attach the hooks of the Force Sensors to the ends of a long rubber band as in the picture

above.


Trinh Thanh Thuy 29

4. Open the file “3rd Law Tug-O-War (PP).ds.”

4.3.2 Procedure


2. Play a small-scale game of tug-o-war with neither Person A nor Person B winning.

3. Data Collection will end after several seconds.

4. If necessary to delete unwanted data, click the "Experiment" button and select "Delete

all data runs."

5. Record the direction and magnitude of the:

A. Force of person A on Person B (FAB)

B. Force of person B on Person A (FBA)

6. Repeat steps 1-5 above with Person A winning.

7. Repeat steps 1-5 above with Person B winning.


Trinh Thanh Thuy 30

LAB 3: CONSERVATION OF MOMENTUM

1. EQUIPMENT

INCLUDED:

1 2.2 m PAScar Dynamics System ME-6956

2 Dynamics Track Mount CI-6692

2 Pasport Motion Sensor PS-2103


1 Balance SE-8723

NOT INCLUDED, BUT REQUIRED:

1 Pasport Xplorer GLX PS-2002


2. INTRODUCTION

The purpose of this activity is to investigate the momentum of two carts before and after

they collide. Use two Motion Sensors to measure the motion of two carts (same/different

masses) before and after they collide. Compare the momentum of the carts before they

collide with the momentum after they collide.

3. THEORY

When objects collide, whether locomotives, shopping carts, or your foot and the

sidewalk, the results can be complicated. Yet even in the most chaotic of collisions, as long

as there are no net external forces acting on the colliding objects, one principle always

holds and provides an excellent tool for understanding the collision. That principle is called

the conservation of momentum. The law of momentum conservation can be stated as

follows:

“For a collision occurring between object 1 and object 2 in an isolated system, the total

momentum of the two objects before the collision is equal to the total momentum of the

two objects after the collision. That is, the momentum lost by object 1 is equal to the

momentum gained by object 2.”

The momentum of an object depends on its mass and velocity.

vmpMomentum (3.1)

The direction of the momentum is the same as the direction of the velocity. During a

http://www.physicsclassroom.com/Class/momentum/u4l2c.cfm


Trinh Thanh Thuy 31

collision, the total momentum of the system of both objects is conserved because the net

force on the system is zero if external forces (such as friction) are ignored. This means that

the total momentum just before the collision is equal to the total momentum just after the

collision. For a two-object collision, momentum conservation is stated mathematically by

the following equation: + = ′ + ′ (3.2)

If the momentum of one object decreases, the momentum of the other object increases

by the same amount. This is true regardless of the type of collision, and even in cases where

kinetic energy is not conserved. The law of conservation of momentum is stated as

Total Before Collision Total After Collisionp p (3.3)

The change in momentum for each object is its mass times its change in velocity.

1 1 2 2

' '

1 1 1 2 2 2

m v = m v

m v - v = m v - v

(3.4)

The kinetic energy of a cart also depends on its mass and velocity but kinetic energy is a

scalar.

2

2

1mvKE (3.5)

The total kinetic energy of the system of two carts is found by adding the kinetic

energies of the individual carts.



Trinh Thanh Thuy 32

4. EXPERIMENT


1. Set up the track so that it

is level. (Place a cart on

the track. If the cart rolls

one way or the other,

adjust the track to raise or

lower one end.)

2. Attach the first motion

sensor at the left end of

the track. Attach the

second sensor at the right

end of the track.

3. Remove the hooks from

the Force Sensors.

Replace them with the

rubber bumpers.

4. Using long thumbscrews, attach the Force Sensors to the carts as in the picture

above. Place the carts on the track.

5. With nothing connected to the Force Sensors, press the "ZERO" or "TARE" buttons

on the Force Sensors.

6. Measure and record the mass m of each cart.

NOTE: The procedure is easier if one person handles the carts and a second person

handles the Xplorer GLX.

4.2 Procedure

4.2.1 Forces between Interacting Objects

There are many situations where objects interact with each other, for example during

collisions. In this investigation we want to compare the forces exerted by the objects on

each other. In a collision, both objects might have the same mass and be moving at the

same speed, or one object might be much more massive, or they might be moving at very

different speeds. What factors might determine the forces the objects exert on each other? Is

there some general law that relates the forces the objects exert on each other? Is there some

Figure 3.1 Equipment set up


Trinh Thanh Thuy 33

general law that relates the forces?

4.2.1.1 COLLISION INTERACTION FORCES

What can we say about the forces two objects exert on each other during a collision?

Prediction 1: Suppose

two objects have the

same mass are moving

toward each other with

the same speed, so that: m1 = m2 and 1 2v v

Predict the relative magnitudes of the forces between object 1 and object 2 during the

collision. Place a check next to your prediction:

______ Object 1 exerts a larger force on object 2.

______ The objects exert the same size force on each other.


Prediction 2:

Suppose two

objects have the

same mass and

object 1 is moving

toward object 2, but object 2 is at rest, so that: m1 = m2 and 1 20 0v v , .






Prediction 3: Suppose the mass of object 1 is greater than that of object 2, and that it is


Trinh Thanh Thuy 34

moving toward object 2, which is at rest, so that: m1 > m2 and 1 20 0v v ,






Provide a summary of your predictions. What are the circumstances under which you

predict that one object exert a greater force on the other object?

In order to test the predictions you made, you can study gentle collisions between two force

probes attached to carts. You can strap additional masses to one of the carts to increase its

total mass so it has significantly more mass than the other. If a compression spring is

available you can set up an "explosion" between the two carts by compressing the spring

between the force probes on each cart and letting it go.

Set up the apparatus as shown in the following figure.

Figure 3.7: Experiment set up for “forces between interacting objects”

The force probes should be securely fastened to the carts. The spacing between the magnet

and the sensor should be about 1.0 cm. The hooks should be removed from the force probes

and replaced by rubber stoppers which should be carefully aligned so that they will collide

head-on with each other. If the carts have friction pads, these should be raised so that they

don't rub on the track.

1. Remove the hooks from the Force Sensors. Replace them with the rubber bumpers.

2. Using long thumbscrews, attach the Force Sensors to the carts as in the picture

above. Place the carts on the track.

3. With nothing connected to the Force Sensors, press the "ZERO" or "TARE" buttons

on the Force Sensors.


Trinh Thanh Thuy 35

4. Open the file “Collision (PP).ds.”

1. Refer to the prediction number (#) to place the appropriate mass to each cart.


3. With the rubber bumpers facing each other, briefly and gently push the other cart

into the stationary cart.

4. Data collection will end after several seconds.

5. If necessary to delete unwanted data, click on the "Experiment" menu and select

"Delete all data runs."

6. Record the direction and magnitude of the forces.

7. Remove the extra mass and repeat the procedures above.

Use the two carts to explore various situations which correspond to the predictions you

made about interaction forces. Your goal is to find out under what circumstances one

object exerts more force on another object. Try collisions (a) - (c) listed below. Be sure

to Zero the force sensors before each collision with the stoppers close together but not

touching. Also be sure that the forces during the collisions do not exceed 10 N. Sketch the

graphs for each collision on the axes above, or print them and affix them over the axes. Be

sure to label your graphs. For each collision also find the values of the impulses exerted by

each cart. Record these values, and carefully describe what you did and what you

observed.


Trinh Thanh Thuy 36

(a) (Prediction 1) Two carts of the same mass moving towards each other at about the

same speed.

(b) (Prediction 2) Two carts of the same mass, one at rest and the other moving

towards it.

(c) (Prediction 3) One cart twice or three times as massive as the other, moving toward

the other cart which is at rest.

4.2.1.2 OTHER INTERACTION FORCES

Interaction forces between two objects occur in many other situations besides

collisions. For example, suppose that a small car pushes a truck with a stalled engine, as

shown in the picture below. The mass of object 1 (the car) is much smaller than object 2

(the truck).

At first the car doesn't push hard enough to make the truck move. Then, as the driver

pushes down harder on the gas pedal, the truck begins to accelerate. Finally the car and

truck are moving along at the same constant speed.

Prediction 4: Place a check next to your predictions of the relative magnitudes of the

forces between objects 1 and 2.

Before the truck starts moving:

The car exerts a larger force on the truck

The car and truck exert the same force on each other

The truck exerts a larger force on the car

While the truck is accelerating:




After the car and truck are moving at a constant speed:





Trinh Thanh Thuy 37

Test your predictions.

1. Use the same setup as in the last activity (Fig. 3.7) with the two force probes

mounted on carts. Add masses to cart 2 (the truck) to make it much more massive

than cart 1. Zero both force probes with the stoppers close together but not

touching, just before you are ready to take measurements.

2. Your hand will be the engine for cart 1. Move the carts so that the stoppers are

touching, and then hit Start. When graphing begins, push cart 1 toward the right. At

first hold cart 2 so it cannot move, but then allow the push of cart 1 to accelerate

cart 2, so that both carts move toward the right, finally at constant velocity.

3. Sketch your graphs on the axes on the previous page, or print them and affix them

over the axes.

4.2.2 Newton’s Law and Momentum Conservation

Your previous work should have shown that interaction forces between two objects are

equal in magnitude and opposite in sign (direction) on a moment by moment basis for all

the interactions you might have studied. This is a testimonial to the seemingly universal

applicability of Newton's Third Law to interactions between objects.

As a consequence of the forces being equal and opposite at each moment, you should

have seen that the impulses of the two forces were always equal in magnitude and opposite

in direction. You may have seen in lecture or in your text a derivation of the Conservation

of Momentum Law by combining these findings with the impulse-momentum theorem

(which is really another form of Newton's Second Law since it is derived mathematically

from the Second Law). The argument is that the impulse 1F acting on cart 1 during the

collision equals the change in momentum of cart 1, and the impulse 2F acting on cart 2

during the collision equals the change in momentum of cart 2:

1 1 2 2; F p F p

But, as you have seen, if the only forces acting on the carts are the interaction forces

between them, then 1 2 F F . Thus, by simple algebra:

1 2 1 2or 0p p p p

i.e., there is no change in the total momentum p1 + p2 of the system (the two carts).

If the momenta of the two carts before (initial--subscript i) and after (final--subscript f)

the collision are represented as in the diagrams below, then:


Trinh Thanh Thuy 38

i fp p

i 1 1i 2 2i f 1 1f 2 2fwhere and p = m v m v p = m v m v

In the next activity you will examine if momentum is conserved in one simple collision-

-an inelastic collision between two carts of (un)equal mass.

1. Set up the carts, track and motion detector as shown below. Remove the force

sensors from the carts, and place the Velcro sides toward each other so that they will

stick together after the collision. Add masses to cart 1 so that it is about twice as

massive as cart 2.

Figure 3.8: Velcro Bumpers for Inelastic Collisions

2. Measure the masses of the two carts.

m1 = kg m2 = kg

3. Set up the motion detector to take velocity data at about 50 points per second. Set up

axes to graph velocity-time like those below


Trinh Thanh Thuy 39

Prediction 5: You are going to give the more massive cart 1 a push and collide it with

cart 2 which is initially at rest. The carts will stick together after the collision. Suppose that

you measure the total momentum of cart 1 and cart 2 before and after the collision. How do

you think that the total momentum after the collision will compare to the total momentum

before the collision. Explain the basis for your prediction.

4. Test your prediction. Begin with cart 1 at least 20 cm from the motion

sensor. Press Start, and when you hear the clicks of the motion sensor, give cart 1 a

brisk push toward cart 2 and release it. Be sure that the motion detector does not

see your hand.

5. Repeat until you get a good run when the carts stick, and move together after the

collision. Then sketch the graph on the axes on the previous page, or print it and

affix it over the axes.

6. Use the software to measure the velocity of cart 1 just before the collision, and the

velocity of the two carts together just after the collision. You may want to find the

average velocities over short time intervals just before and just after the

collision. Record these values with unit

v1i = ________________m/s v1f =________________m/s

v2i = ________________m/s v2f =________________m/s

7. Calculate the total momentum of carts 1 and 2 before the collision and after the

collision. Show your calculations below.

pi =________________kg-m/s pf =________________kg-m/s

8. Calculate the total momentum of the two-cart system before and after the collision

9. Compare these two by calculating the percent difference.

100before after

before

p p%difference %

p


Trinh Thanh Thuy 40

Extension: Use the same mass carts with the magnetic bumpers toward each other so

the carts will bounce off each other and the collision will be elastic (see Figure 3.9).

Repeat the collisions and record all the resuls as above.

Figure 3.9: Magnetic Bumpers for Elastic Collisions


Trinh Thanh Thuy 41

LAB 4: CONSERVATION OF ANGULAR

MOMENTUM

1. EQUIPMENT

INCLUDED:

1 Rotary Motion Sensor PS-2120

1 Rotational Accessory CI-6691




1 PASPORT Interface


2. INTRODUCTION

A non-rotating ring is dropped onto a rotating disk, and the final angular speed of the

system is compared with the value predicted using conservation of angular momentum.

3. THEORY

When the ring is dropped onto the rotating disk, there is no net torque on the system

since the torque on the ring is equal and opposite to the torque on the disk. Therefore, there

is no change in angular momentum; angular momentum (L) is conserved.

i i f f= =L I I (4.1)

where Ii is the initial rotational inertia and ωi is the initial angular speed. The initial

rotational inertia is that of a disk

𝑖 = 𝑅 (4.2)

and the final rotational inertia of the combined disk and ring is = 𝑅 + + (4.3)

where r1 and r2 are the inner and outer radii of the ring. So the final rotational speed is

given by


Trinh Thanh Thuy 42

12

1 12 2

2

1 1f i2 2 2

1 1 2 1 2

M Rω = ωM R + M r +r

(4.4)

The rotational kinetic energy of a rotating object is given by 212

KE I .



Trinh Thanh Thuy 43

4. EXPERIMENT

4.1 Setup apparatus

1. Mount the Rotary Motion Sensor to a support rod and connect it to the PASPORT

interface. Place the disk directly on the pulley and tighten the screw as shown in

Figure 4.1.

2. Open the DataStudio file called "angmom.ds".

Figure 4.1: Setup for dropping ring onto disk

4.2 Procedure

1. Hold the ring with the pins up just above the center of the disk. Give the disk a spin

using your hand and start collecting data by clicking on START. After about 25 data

points have been taken, drop the ring onto the spinning disk.

2. Stop collecting data. If the ring is not on center

(Figure 4.2), repeat the drop.

3. Select the Smart Tool and move the cursor to the

data point immediately before the collision.

Record the Angular Velocity at this point. Move

the cursor to the data point immediately after the

collision and record the Angular Velocity at this

point.

4. If a second disk is available, drop the second disk onto the

spinning disk and use the Smart Tool to determine the initial

and final velocities for this new collision.

5. Measure the mass of the disk and ring and measure the radii

record these values in the Data Table 4.1.

Figure 4.2: Offset Ring


Trinh Thanh Thuy 44

4.3 Analysis

1. Calculate the expected (theoretical) value for the final angular velocity and record

this value in the Data Table 4.2.

2. Calculate the percent difference between the experimental and the theoretical values

of the final angular velocity and record this value in the Data Table 4.2.

3. If you also dropped a second disk on the first disk, calculate the theoretical value for

the final angular velocity and compare it to the experimental value.

4. For the ring and disk, calculate the total rotational kinetic energy before and after

the collision.

Table 4.1 Data Table

Mass (kg) Radius (m) Theoretical Rotational

Inertia (kgm2)

Disk 1

Disk 2

Ring

Table 4.2 Data and Results

Run # Collision ωi

(rad/s)

ωf

(rad/s)

Theory

ωf

(rad/s)

% Diff. Total

rotational

kinetic

energy

before

collision

Total

rotational

kinetic

energy after

collision

1

Ring on

Disk

2

3

4

5

Average

6 Disk on

Disk

7

8


Trinh Thanh Thuy 45

9

10

Average

4.4 Questions

1. How does the experimental result for the final angular velocity compare with the

theoretical value for the final angular velocity?

2. What percentage of the rotational kinetic energy was “lost” during the collision?

Calculate the energy lost.

2 2

i i f f

2 2

i i

1 1

2 2%KE lost1

2

I ω I ω

I ω


Trinh Thanh Thuy 46

LAB 5: ROTATIONAL INERTIA

(Moment of inertia)

1. EQUIPMENT

INCLUDED Scienceworkshop

1 Mini-Rotational Accessory CI-6691

1 Rotary Motion Sensor

(3-step Pulley: 10; 29; 48 mm

diameters)

CI-6538

1 Base and Support Rod ME-9355

1 Mass Set (5 g resolution) ME-9348

1 Triple Beam Balance SE-8723

Paper clips (for masses <1 g)

NOT INCLUDED BUT REQUIRED

1 ScienceWorkshop® 750 Interface CI-6450 or CI-7599


2. INTRODUCTION

The purpose of this experiment is to find the rotational inertia of a point mass

experimentally and to verify that this value corresponds to the calculated theoretical value.

3. THEORY

Moment of inertia is the name given to rotational inertia, the rotational analog of mass

for linear motion. It appears in the relationships for the dynamics of rotational motion. The

moment of inertia must be specified with respect to a chosen axis of rotation. Theoretically,

for a point mass, the moment of inertia, I, is just the mass times the square of perpendicular

distance to the rotation axis 2I MR , where M is the mass, and R is the distance from the

mass to the axis of rotation.

That point mass relationship becomes the basis for all other moments of inertia since

any object can be built up from a collection of point masses.


Trinh Thanh Thuy 47

Figure 5.1: Some rotational inertias

Since this experiment uses two masses equidistant from the center of rotation, the

total rotational inertia will be

2total totalI M R

(5.1)

where Mtotal = M1 + M2, the total mass of both point masses and R is their distance from the

rotation axis. These are attached by a rod, pulley, and axle to the Rotary Motion Sensor,

which also contribute an unknown amount to the total rotational inertia of the system. We

will measure:

experiment apparatus pointI I I (5.2)

and make a separate measurement of the moment of inertia of the apparatus, so we can

subtract it and determine the rotational inertia of the “point” masses alone.

The experimental moment of inertia is found by applying a known torque to the object

and measuring the resulting angular acceleration. Since = I ,


Trinh Thanh Thuy 48

experimentI

(5.3)

where α is the angular acceleration and

= r T (5.4)

is the torque due to the tension T in the thread wrapped around the 3-step pulley of radius r.

This is a two-body problem, so the tension is not equal to mg. The mass m hanging on the

thread is also accelerated. We must solve Newton’s Second Law for the hanging mass (See

Figure 5.2)

F mg T ma (5.5)

solving for the tension in the string gives:

T = m(g-a) (5.6)

where the linear acceleration a = α.r, with r

= the radius of the pulley. Once the angular

acceleration of the rotating system is

measured, we can calculate the linear

acceleration and the tension, and use the

tension to find the torque and moment of

inertia.


Figure 5.2: Rotational Apparatus and Free-

Body Diagram


Trinh Thanh Thuy 49

4. EXPERIMENT

4.1 Set up apparatus

1. Attach a mass on each end of the rod

(part of the Mini-Rotational

Accessory) equidistant from the rod

center. You may choose any radius

you wish.

2. Tie one end of the string to the Mass

Hanger and the other end to one of

the levels of the 3-step Pulley on the

RMS.

3. Mount the thin rod to the pulley on the Rotary Motion Sensor. Please note the

orientation of the 3-step Pulley.

4. Mount the RMS to a support rod and connect it to a computer. Make sure that the

support rod does not interfere with the rotation of the accessory rod. See Figure 5.3.

5. Mount the clamp-on Super Pulley to the Rotary Motion Sensor.

6. Drape the string over the Super Pulley such that the string is in the groove of the

pulley and the Mass Hanger hangs

freely (see Figure 5.3).

Note: The clamp-on Super Pulley must be

adjusted at an angle, so that the thread runs

in a line tangent to the point where it leaves

the 3-step Pulley and straight down the

middle of the groove on the clamp-on

Super Pulley (Figure 5.4).

7. Adjust the Super Pulley height so that

the thread is level with the 3-step pulley.

4.2 Procedure

4.2.1 Measurements for the theoretical rotational inertia

1. Weigh the masses to find the total mass Mtotal and record in Table 5.1.

2. Measure the distance from the axis of rotation to the center of the masses and record

this radius in Table 5.1.

Figure 5.3: Experiment set up

Figure 5.4: Super pulley position


Trinh Thanh Thuy 50

Table 5.1: Theoretical Rotational Inertia Data

Trial No Mtotal

(kg)

Distance from the

axis of rotation to

the center of the

mass (m)

Diameter of the

pulley (m)

Radius of the

pulley (m)

Theoretical

Rotational Inertia

(…)

1

2

3

4

5

Average

4.2.2 Measurements for the experimental method

4.2.2.1 FINDING THE ACCELERATION OF THE POINT MASSES AND

APPARATUS

1. Open DataStudio and create an experiment.

2. In the Sensors list of the Experiment Setup window, click and drag the Rotary Motion

Sensor icon to the two digital ports that the RMS is plugged into on the interface.

3. In the Experiment Setup window, double click on the Rotary Motion Sensor icon to

open the Sensor Properties dialog.

4. In the Measurement tab of the Sensor Properties dialog, select "Angular Velocity

(rad/s)."

5. In the Rotary Motion Sensor tab, select 360 divisions/rotation, and choose the

appropriate pulley in the Linear Calibration menu; click OK.

6. Put the 50 g mass on the Mass Hanger and wind up the thread. Click on the Start button;

then release the 3-step Pulley, allowing the mass to fall. Click the Stop button to end the

data collection.

HINT: Click the Stop button before the mass reaches the floor or the end of the thread to

avoid erroneous data.

7. In the Graph Display window, click on the Statistics button; then select the linear curve

fit from the pop-up menu.

The slope of the linear fit represents the angular acceleration (α) and should be entered

in Table 5.2.


Trinh Thanh Thuy 51

4.2.2.2 FINDING THE RADIUS

Using calipers, measure the diameter of the pulley about which the thread is wrapped

and calculate the radius. Record in Table 5.2.

4.2.2.3 FINDING THE ACCELERATION OF THE APPARATUS ALONE

In 4.2.2.1, “Finding the Acceleration of the Point Mass and Apparatus” the apparatus is

rotating and contributing to the rotational inertia. It is necessary to determine the

acceleration and the rotational inertia of the apparatus by itself, so this rotational inertia can

be subtracted from the total, leaving only the rotational inertia of the point masses.

1. Take the point masses off the rod and repeat the procedure under “Finding the

Acceleration of the Point Mass and Apparatus” for the apparatus alone. You may need

to decrease the amount of the hanging mass, so that the apparatus does not accelerate so

fast that the computer cannot keep up with the data collection rate.

2. Record the data in Table 5.2.

Table 5.2: Experimental Rotational Inertia Data

Trial No Point mass and apparatus Apparatus alone Radius of

the pulley

(m)

Hanging

mass (kg)

Slope (the angular

acceleration (α))

(rad/s2)

Hanging

mass (kg)

Slope (the angular

acceleration (α))

(rad/s2)

1

2

3

4

5

Average


Trinh Thanh Thuy 52

Table 5.3: Result

4.3 Calculations

1. Calculate the experimental value of the rotational inertia of the point masses and

Rotary Motion Sensor together and record in Table 5.3.

2. Calculate the experimental value of the rotational inertia of the Rotary Motion

Sensor alone. Record in Table 5.3.

3. Subtract the rotational inertia of the Rotary Motion Sensor from the rotational

inertia of combination of the point masses and Rotary Motion Sensor. This will be

the rotational inertia of the point masses alone. Record in Table 5.3.

4. Calculate the theoretical values of the rotational inertia of the point masses and

record in Table 5.3.

5. Use percent differences to compare the experimental values to the theoretical

values. Record in Table 5.3.

100Experimental Theoretical

%differenceTheoretical

4.4 Extra experiment

Repeat the experiment above with changing the point mass to two different objects: a

ring and a disk.

Theoretically, the rotational inertia, I, of a ring is given by

2

2

2

12

1RRMI

(5.7)

where M is the mass of the ring, R1 is the inner radius of the ring, and R2 is the outer radius

of the ring. The rotational inertia of a disk is given by

2

2

1MRI

(5.8)

where M is the mass of the disk and R is the radius of the disk.

Component Rotational Inertia

Point Masses and Apparatus Combined

Apparatus Alone

Point Masses (experimental value)

Point Masses (theoretical value)

Percent (%) Difference


Trinh Thanh Thuy 53

5. APPENDIX

Set up the “DataStudio” program

1. Lauch PASportal.

2. The “How would you like to use it” window will appear.

Click on the “Launch DataStudio” icon.

3. A picture of the SW750 interface will appear. The

rotary motion sensor is already connected to the left-most

connectors, so you need to click on the one on the far left as

shown in the picture here.

4. The “Choose Sensor” window will appear.

There is an alphabetical list of sensors in the

left-hand column. Double click on the “Rotary

Motion Sensor”. After you do this, an icon of

the sensor will appear, showing how it is

supposed to be plugged into the SW750

interface. See picture below.

The yellow

connector should

already be plugged into the left-most socket like you see

here. We select “Angular Velocity” and “rad/s” units.

You don’t have to change any other settings but click the

sensor tab to confirm that it is collecting data at 10 Hz.

5. Click on the “Measurement” tab and verify that the

box for “Angular Velocity” for Channels 1&2 with units


Trinh Thanh Thuy 54

of (rad/s) is checked and that none of the other boxes is checked. It should look like the

example at the bottom of the previous page.

6. There is a list of display options at the bottom of the left

column of the Setup window. Double-click on the one that

says “Graph”, and a display called “Graph 1” will open up.

This is where our data will appear.

7. Resize the “Graph 1” window so it fills most of the screen. We

are now ready to take data, which should be angular velocity (y axis) as a function of

time (x axis).

Collect and analyze the data

1. Select a mass to put on the mass hanger (30 g to 50 g works well) and weigh it along

with the mass hanger on your balance. Record the result into Table 5.2.

2. Carefully wind the string onto the middle pulley, checking that the Super Pulley turns

freely and that the string goes straight over the Super Pulley. One person should hold

onto the rod until you are ready to start.

3. Release the rod and click on the “Start” button. Data

analysis is simplified if you start collecting data just after

the rod is released, like we did in the air table

experiments.

4. Click the “Stop” button to end the collection of data.

Again, data analysis will be simpler if you stop collecting data before you run out of

string and/or the mass hits the floor, but there is an easy work-around if you don’t.

5. You will now have a set of data displayed in your “Graph 1” window. The buttons

across the top of this window (see below) are used to analyze the data. If you position

the mouse cursor over a button, a window will tell you what it does. We will mainly use

the leftmost button (to select part of the data to view or fit), the menu under the “Fit”

button, and the menu under the “Data” button.

6. A sample data set is shown at right. If your data look like

this bogus set, you need to start over and try again, but


Trinh Thanh Thuy 55

these fake data make it easy to see how the options work.

7. Here we have data both before and after the acceleration took place. We can click on

the left-most button, which is a selection tool, and use the cursor to click-and-drag a box

around the portion of the data we want to fit as shown here.

8. Use the pull-down menu under “Fit” to select a “Linear

Fit”.

9. The fitted line will be shown along with a box

containing the fit parameters. The fit parameters are the

same ones we get from “Graphical Analysis”. We get

uncertainties as well as the correlation coefficient “r”.

Notice that the data being fit are highlighted in yellow.

10. Record the slope ± its standard deviation along with the

correlation coefficient “r”.

11. Remove the two masses (M1+M2) from the rod and

repeat steps 1 through 10. This is for “apparatus alone”

process. Because the moment of inertia of the rod and

the rest of the apparatus is so small, you must use less

mass this time. Usually the mass hangar alone (about 10

g) is enough. Record those data into table 5.2.

12. You can either delete the previous run or have the new

data appear over the old ones. The “Data” menu controls

which set is active and being fit.

Calculations

For both the “point mass” and “apparatus alone” cases:

1. Copy the value for the slope into Table 5.2 for α. Use the pulley radius in Table 5.1 to

calculate the linear acceleration, a, then use Equation (5.6) to find the string tension, T.

2. Use Equation (5.4) to calculate the torque, and then use Equation (5.3) to calculate the

moment of inertia. Enter the results in Table 5.3. Remember to include units.

3. Subtract the moment of inertia of the apparatus alone from the combined experimental


Trinh Thanh Thuy 56

moment of inertia of the “point” masses plus the apparatus. (See Equation (5.2).)

Record this experimental estimate of the rotational inertia of the “point” masses in

Table 5.3.

4. Calculate the theoretical value of the “point” masses from Equation (5.1) and enter the

result in Table 5.3.

5. Calculate the percent difference and record it in Table 5.3.

Other details

! The DataStudio program can produce printouts of your fits. Try one and see, but you

don’t need to include it at the end of your lab report unless your instructor tells you to.

! Shut down the computer program when you are

done. Answer “No” to the dialog box that shows up.

Remember to shut down the program and log off of the computer!


Trinh Thanh Thuy 57

LAB 6: SLIDING FRICTION

1. EQUIPMENT

INCLUDED: PASPORT

1 Discover Friction Accessory ME-8574


4 Motion Sensor


1 Hanging mass holder (5 grams) and set

of Pasco masses

1 1.2 meter dynamics track


1 Pasport PS-2100


2. INTRODUCTION

The purpose of this experiment is to find the coefficient of static friction and the

coefficient of kinetic friction for different solid objects and surfaces. As it pulls a Friction

Tray from rest to a constant velocity, the Force Sensor can measure both the static friction

and the kinetic friction. A plot of each of these forces versus their respective normal forces

yields both coefficients.

3. THEORY

We use the word friction to describe the resisting force that arises when we slide, or

attempt to slide, a solid object on a solid surface. The frictional force is always directed

parallel to the surface between the objects. Experiments have shown that the magnitude of

the frictional force depends on the materials the objects are made of, on how smooth or

rough the surfaces in contact are, and on the magnitude of the force pressing the objects

together. This “pressing together” force is called the normal force (FN) because it is always

normal to the surface.

Counter to what you might suppose, the frictional force does not depend on either the

area of contact between the objects or the relative speed between them, as long as the speed

isn’t too great. The relationship between the frictional force and the normal force is


Trinh Thanh Thuy 58

expressed by the linear equation:

f NF F (6.1)

where the Greek letter (mu) is a dimensionless constant of proportionality called the

coefficient of friction, Ff is the frictional force, and FN is the normal force.

Your everyday experience tells you that there are two different kinds of friction. A force

of static friction occurs when two objects are at rest relative to each other and you attempt

to make one of them slide over the other one. This static frictional force arises to oppose

any applied force trying to cause motion along the surface of contact. When a force is

applied to an object resting on a surface, it will not move until the force applied to it is

greater than the maximum force due to static friction. The coefficient of static friction (s)

is simply the ratio between the maximum static frictional force (Fs) and the normal force

(FN):

N

ss

F=F

(6.2)

The magnitude of this frictional force is related to the magnitude of the normal force by

the equation:

s s NF F (6.3)

The meaning of the inequality in this relationship is that the static frictional force is a

vector that is equal in magnitude but opposite in direction to the applied force. The static

force increases as the applied force increases from zero up to a maximum value given by

SFN. If the applied force exceeds this value, then the surfaces begin to move relative to one

another.

Another kind of friction occurs when the two surfaces in contact are moving relative to

each other. To keep the object moving at a constant velocity, a force must be applied to the

object equal to the kinetic frictional force. The kinetic frictional force is also characterized

by a proportionality constant, this time called the coefficient of kinetic friction. Hence, the

coefficient of kinetic friction (k) is the ratio between the kinetic frictional force (Fk) and

the normal force (FN):

N

kk

F=F

(6.4)

Note that in general k < s for any two materials. Table 6.1 below gives approximate

values for coefficients of static and kinetic friction between some common materials.


Trinh Thanh Thuy 59

Table 6.1: Coefficients of Friction

Materials s k

Aluminum on Aluminum 1.05 1.40

Glass on Glass 0.94 0.35

Rubber on Concrete (dry) 1.20 0.85

Rubber on Concrete (wet) 0.80 0.60

Steel on Aluminum 0.61 0.47

Steel on Steel (dry) 0.75 0.48

Teflon on Steel 0.04 0.04

Teflon on Teflon 0.04 0.04

Steel on Ice 0.02 0.02

Waxed Wood on Snow 0.05 0.03

Wood on Aluminum 0.30 0.20

Wood on Wood 0.58 0.40

The coefficient of friction between two surfaces is primarily related to their roughness.

Even a surface that appears to be smooth can actually look quite rough when examined

under a microscope as in Figure 6.1. When you place a flat-bottomed block of wood on a

flat aluminum track, the microscopic peaks and valleys in the bottom surface of the block

tend to interlock with those in the track below, like two sheets of sandpaper placed face to

face. When you attempt to slide the block on the track, the interlocked micro-irregularities

of the two surfaces oppose the force that you apply, giving rise to the static frictional force.

Figure 6.1 Zoom in the contact area between the two objects surface

Note that as long as at least one micro-peak remains locked in place, the block cannot

slide. Thus, to a good approximation, the static frictional force does not depend on the

contact area between the surfaces. To free the locked surfaces without breaking off the


Trinh Thanh Thuy 60

micro irregularities, you must raise the block over the bumps, overcoming the force

pressing the block against the track.

-----End of Theory-----


Trinh Thanh Thuy 61

4. EXPERIMENT

4.1 Set-up experiment

Figure 6.4. The apparatus used for the kinetic friction experiment set up and ready for

measurement

1. Connect the Force Sensor and the Motion Sensor to PASPORT GLX interface.

2. Cut about 50 cm of Physics String. Tie the string in a loop. Thread one end of the loop

through the hole in the holding bracket on the front of the Friction Tray. Pass the rest of

the loop back through this loop and tighten it on the holding bracket so it is centered.

This prevents the tray from rotating sideways as it moves. Repeat for the cork

bottomed, felt bottomed and one of the plastic bottomed trays.

3. Loop the other end of the string over the hook on the Force sensor.

4. Set the switch on the Motion Sensor to the Cart, short range setting. Set the angle to

zero degrees so the Sensor is not tipped up or down.

5. Place the Friction Tray 15 cm in front of the Motion Sensor. Try to use the same section

of the table each time since variations in the surface can affect the results.

6. Set up the 1.2 meter track as shown in Figure 6.4 with the motion sensor clipped onto

the end of the track.

4.2 Procedure

1. Use a force sensor to weigh each of the friction tray and the mass bar. Record the mass

of the friction tray and the mass bar on to the Table data 6.1.

2. With no tension on the string, press the "zero" button on the force sensor.


Trinh Thanh Thuy 62

3. Place one 250 g mass in the cork bottomed Friction Tray. Center the mass in the tray to

prevent the tray from trying to turn sideways.

4. Click START. With the force sensor tied to the tray, slowly increase the pull on the

Friction Tray horizontally until it begins to move, and then continue pulling it across the

lab station at a constant (slow) velocity. Look at the Velocity vs. Time plot above to

verify the velocity is roughly constant. It is difficult to keep it exactly constant, but you

should be able to identify regions where it is nearly constant. It is probably better to

watch the Friction Tray and try to keep its motion uniform. There is a tendency to over

correct and oscillate up and down if you try to watch the Velocity vs. Time plot.

Continue pulling at a constant velocity for several seconds. Click STOP to stop data

collection. This will record the speed and force as a function of time and graph both.

Read the approximate speed from the Velocity vs. Time graph and enter it in the Data

table 6.1.

5. Your Force vs. Time graph should look similar to that shown below. If you don’t have a

peak followed by a lower horizontal section, you may not be keeping the cart moving at

a constant speed or there may be contaminates on the track where the cart moves. Try

cleaning the table top, moving to a different section of the dynamics track, or just

practice moving the cart at a constant speed. Click the Delete Last Run button at the

bottom right of the screen to delete unwanted runs.

Figure 6.5 Maximum static friction determined

6. Repeat step 1-5 with the felt and plastic bottomed trays with 2, 3, 4 mass bars.

4.3 Analysis

1. Use the cursor to highlight the region that corresponds to the maximum static friction.

The legend box gives the value of the maximum Static Friction. Enter this value into


Trinh Thanh Thuy 63

the corresponding STATIC data table. Also, enter the value of the Friction Tray's

normal force.

Figure 6.6 Average kinetic friction determined

2. Use the cursor to highlight the region where the velocity is constant. The legend box

gives the MEAN or average Kinetic Friction. Enter this value into the corresponding

KINETIC data table. Also, enter the value of the Friction Tray's normal force.

3. Repeat with different mass bars. Record these Static ad kinetic values into the table in

datastudio.

4. Click on the graph labeled "Friction v Normal."

5. In the legend box , highlight the Data that corresponds to the static frictional

force. From the "Fit" menu at the top of the graph, select "Linear." Record the

slope and vertical intercept.

6. From the legend box, highlight the Data that corresponds to the kinetic frictional force.

From the "Fit" menu at the top of the graph, select "Linear." Record the slope and

vertical intercept.

7. Repeat the previous steps using the other Friction Trays.


Trinh Thanh Thuy 64

4.4 Conclusions

1. The graph above is representative of the force applied to an object as it is pulled across

a horizontal surface. Draw a force diagram for each of the positions labeled in the graph

above. Describe the motion of the object for the positions labeled in the graph.

2. What relationship exists between the static frictional force and the normal force on an

object?

3. What specific equation describes this relationship? (Include numbers and units for both

the slope and vertical intercept)

Your report should include your observations about the following:

1. Is the coefficient of friction constant depending only on the types of surface in

contact? (Remember it is very difficult to reproduce exactly the same value for the

coefficient of friction on each run.)

2. Does the normal force change the coefficient of friction?

3. Does the surface area change the coefficient of friction?

4. Did the coefficient of friction vary significantly with acceleration?

5. What are your sources of experimental error?


Trinh Thanh Thuy 65

LAB 7: VARIABLE-G PENDULUM

1. EQUIPMENT

INCLUDED:



1 Angle Indicator ME-9495A

1 Adjustable Angle Clamp ME-8744

1 Mini-Rotational Accessory (Need

rod and masses only)

CI-6691

1 Rotary Motion Sensor PS-2120


1 Science Workshop 750 Interface UI-5000

1 DataStudio Software UI-5400

2. INTRODUCTION

This experiment explores the dependence of the period of a simple pendulum on the

acceleration due to gravity. A simple rigid pendulum consists of a 35-cm long lightweight

(26 g) aluminum tube with a 150-g mass (2 separated masses) at the end, mounted on a

Rotary Motion Sensor. The pendulum is constrained to oscillate in a plane tilted at an angle

from the vertical. This effectively reduces the acceleration due to gravity because the

restoring force is decreased.

3. THEORY

3.1 Simple pendulum

The variable-g pendulum is a variation of the simple pendulum, so a logical place to

begin is with the theory of the simple pendulum. Figure 7.1 shows a simple pendulum in its

equilibrium position. Note that the force of gravity, Fg1 = mg, where m is the mass of the

pendulum, is acting downwards parallel to the arm of the pendulum.


Trinh Thanh Thuy 66

Figure 7.1: Diagram of a rigid pendulum composed of a mass m hanging at a length L from

the mount point. The force of gravity mg is also labelled

Figure 7.2 shows the forces acting on the pendulum while it is in motion. The motion is

due entirely to a restoring force. In the case of the simple pendulum, the restoring force is

the component of the force of gravity that acts parallel to the direction of motion. The

restoring force for the simple pendulum, Frestore1, is:

Frestore1 = −mg sin (7.1)

Figure 7.2: Diagram of the rigid pendulum in Figure 7.1 with the mass displaced by an

angle . The force of gravity and Frestore1 are also labelled.

where is the angular displacement of the pendulum from its equilibrium position. Note

that the angular displacement, and the position of the pendulum are defined to be positive

when to the right of the equilibrium point. Equation (7.1) has a negative sign because when

the angle is positive the force is acting in the negative direction, and when the angle is

negative the force is acting in the positive direction.


Trinh Thanh Thuy 67

A variable-g pendulum differs from a simple pendulum in that the support for the

pendulum can be tilted. This characteristic combined with the rigidity of the arm allows the

pendulum to rotate in a different plane. This rotation changes the restoring force, therefore

a correction must be made to Equation (7.1). Figure 7.3 shows a variable-g pendulum that

has been tilted by an angle and is still at its equilibrium position. Recall that for the

simple pendulum the force of gravity acted parallel to the pendulum’s arm. In the case of

the variable-g pendulum this force is but a component of the total force of gravity and is

referred to as Fg2 for the effective force of gravity. The effective force of gravity is given by

the expression:

Fg2 = mg cos (7.2)

Figure 7.3: Diagram of a rigid pendulum in Figure 7.1 that has been tilted by the angle .

The pendulum is allowed to rotate into and out of the page by the angle . The force of

gravity mg and Frestore2 are also labelled.

A correction can be made to the restoring force of the simple pendulum, Equation (7.1),

by replacing mg with Fg,eff, Equation (7.2). The result is an expression for the restoring

force of the variable-g pendulum, Frestore2, which is

Frestore2 = −mg cosθ sin (7.3)

By applying the small angle approximation, which states that for a small angle θ

sin ≈ , (7.4)

to Equation (7.3) it can be written as

Frestore2 = −mg cosθ (7.5)


Trinh Thanh Thuy 68

The arc length theorem states that

x = L (7.6)

where x is the arc length or displacement of the pendulum and L is the length of the

pendulum. Using this, Equation (7.5) rewritten as = − 𝜃 (7.7)

The motion of the pendulum can also be defined by the differential form of Newton’s

second law, as following: = 𝑎 ↔ − 𝜃 = (7.8)

where x is the pendulum’s displacement. Equation (7.8) has the form of a differential

equation describing simple harmonic motion. The solution is therefore periodic, with period

given by 𝑇 = √ 𝜃 (7.9)

Note that, as might be expected, when 𝜃 is set to zero the period becomes that of the simple

pendulum, and that when 𝜃 = π/2 the pendulum no longer moves.

3.2 Physical pendulum

The pendulum we use in this experiment is actually a physical pendulum (not a point

mass) so Equation (7.9) must be modified. A physical pendulum is the generalized case of

the simple pendulum. It consists of any rigid body that oscillates about a pivot point.

Figure 7.4: A physical pendulum with an object with optional shape

The motion can be described by Newton’s second law for rotation:

= I × = Frestore × Lcm (7.10)

Where the torque: = − mg Lcm sin (7.11)

https://www.boundless.com/physics/definition/physical-pendulum/

https://www.boundless.com/physics/definition/physical-pendulum/

https://www.boundless.com/physics/definition/simple-pendulum/

https://www.boundless.com/physics/definition/rigid-body/

https://www.boundless.com/physics/definition/rigid-body/


Trinh Thanh Thuy 69

From (7.10) and (7.11): ≈ − 𝑐𝑚𝐼

Lcm is the distance from the pivot point to the center of mass, is the angular acceleration.

This is of the same form as the conventional simple pendulum and this gives a period of: 𝑇 = √ 𝐼 𝑐𝑚 (7.12)

So the period of the pendulum used in this experiment (titled at an angle) can be

calculated by the equation:

2 cos

cm

IT

Mg L

(7.13)

In this experiment, I is the total moment of inertia of the system of a point mass about the

fixed axis with the rod (I = Irod + Imasses), M is the total mass (brass masses (150 g) + rod (26

g)), and Lcm is the distance from the axis to the center of mass of the rod plus masses (~31.4

cm). We may then re-write Equation (7.13) in the form of Equation (7.9):

2 21

32( ) cos

rod cm masses cm

rod masses cm

m L m

L

L

Tm m g

(7.14)

-----End of Theory-----


Trinh Thanh Thuy 70

4. EXPERIMENT


Figure 7.5: Components of g

1. Remove the black thumb screw from the clamp on the Rotary Motion Sensor. See

Figure 7.6. Remove the mobile rod clamp from the Adjustable Angle Clamp. Screw

the Adjustable Angle Clamp onto the Rotary Motion Sensor.

Figure 7.6: Attaching the Rod Clamp

2. Mount the Rotary Motion Sensor on the rod stand (see Figure 7.7).

g

θ

θ geff = g cos gperp = g sin


Trinh Thanh Thuy 71

3. Attach the Angle Indicator as in Figure 7.7.

Figure 7.7: Set up

4. Put the pulley on the Rotary Motion Sensor with the largest step outward. Attach the

rod to the Rotary Motion Sensor pulley and put the two 75 g masses on the end of

the rod.

5. Plug the Rotary Motion Sensor into Channels 1 and 2 on the Science Workshop

interface.

6. Adjust the Adjustable Angle Clamp for an initial angle of 0o.

7. Open the DataStudio file called “Variable-g Pendulum”

4.2 Procedure

1. Clamp the pendulum clamp at zero degrees. Click on START and displace the

pendulum from equilibrium (no more than 20 degrees amplitude) and let go. Read

the period on the digits display and type the value into the table on the line next to

zero degrees. Do NOT click on STOP.

2. Clamp the pendulum at 5 degrees. Displace the pendulum from equilibrium (no

more than 20 degrees amplitude) and let go. Record the new period in the table.

3. Repeat Step 2 for 10 degrees to 85 degrees, in increments of 5 degrees. Then click

on STOP.

4. Collect all data into table 7.1

# Pendulum Angle

(θ) (degrees) geff = g cosθ

Period (s)

(Experimental)

1 0

2 5

3 10

4 15


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5 20

6 25

7 30

8 35

9 40

10 45

11 50

12 55

13 60

14 65

15 70

16 75

17 80

18 85

5. Examine the graph of the period vs. geff. To determine how the period depends on g,

use the Curve Fit by clicking on the Fit button at the top of the graph. Select various

functions to try to find which function best fit the data.

4.3 Analysis

1. On the graph screen, click on the Fit button at the top of graph. Select various

functions to try to find which function fits the data.

2. Describing the graph of Period vs. geff.

3. Comparing the theoretical and experimental values in the Period Times of the

pendulum.

4. What does this show about Equation (7.14) from Theory?

4.4 Questions

1. How does the period depend on the acceleration due to gravity?

2. What do the constants in the curve fit for the Period vs. g data represent? Calculate

what they should be theoretically and compare the theoretical value to the curve fit

constants.

3. Would the pendulum be longer or shorter on the Moon?

4. What would the period be if the pendulum had been inclined to 90 degrees? What

value of g does this correspond to?


Trinh Thanh Thuy 73


Trinh Thanh Thuy 74

LAB 8: VIBRATING STRINGS

1. EQUIPMENT

INCLUDED:

1 String Vibrator WA-9857


1 GLX Power Amplifier PS-2006


1 C-clamp (small) SE-7286

1 Patch Cords SE-9750

1 Tape Measure SE-8712A


1 Xplorer GLX PS-2002


2. INTRODUCTION

The general appearance of waves can be shown by means of standing waves in a string.

This type of wave is very important because most of the vibrations of extended bodies, such

as the prongs of a tuning fork or the strings of a piano, are standing waves. The purpose of

this experiment is to study how the speed of the wave in a vibrating string is affected by the

stretching force and the frequency of the wave.

3. THEORY

Standing waves (stationary waves) are produced by the interference of two traveling

waves, both of which have the same wavelength, speed and amplitude, but travel in

opposite directions through the same medium. The necessary conditions for the production

of standing waves can be met in the case of a stretched string by having waves set up by

some vibrating body, reflected at the end of the string and then interfering with the

oncoming waves.


Trinh Thanh Thuy 75

Figure 8.1

Standing Waves in Strings

A stretched string has many natural modes of vibration (three examples are shown

above). If the string is fixed at both ends then there must be a node at each end. It may

vibrate as a single segment, in which case the length (L) of the string is equal to 1/2 the

wavelength ( ) of the wave. It may also vibrate in two segments with a node at each end

and one node in the middle; then the wavelength is equal to the length of the string. It may

also vibrate with a larger integer number of segments. In every case, the length of the string

equals some integer number of half wavelengths. If you drive a stretched string at an

arbitrary frequency, you will probably not see any particular mode; many modes will be

mixed together. But, if the tension and the string's length are correctly adjusted to the

frequency of the driving vibrator, one vibrational mode will occur at a much greater

amplitude than the other modes.

For any wave with wavelength and frequency f, the speed, v, is

(8.1)

The speed of the wave on the string is also given by


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(8.2)

Where:

L is the length of the string and n is the number of segments. (Note that n is not the

number of nodes). Since a segment is 1/2 wavelength then

(8.3)

Setting the wave speed in Equation (8.1) equal to the wave speed in Equation (8.2) and

solving for the tension gives

2 2

TF = f (8.4)

Substituting for the wavelength from Equation (8.3) yields

22

T 2

LF = 4 f

n (8.5)

The tension required to achieve 2 segments will be measured for various driving

frequencies. Equation (8.5) shows that a graph of tension versus the square of the frequency

will result in a straight line with

Slope

2

2

L4

n (8.6)



Trinh Thanh Thuy 77

4. EXPERIMENT


Set up the equipment as shown in Figure 8.2. Adjust the vibrator clamp on the side to

position it firmly in the vertical orientation. Run one end of the string from a vertical bar

past the vibrator and over the pulley. The vibrator is connected to the string with an

alligator clip. Attach a mass hanger to the other end of the string. Throughout this

experiment, you will be changing both the density of the medium (by using different

strings) and the tension (by using different weights). To measure vibration amplitudes, it is

helpful to have a meter stick clamped vertically near the string

1. Turn on the signal interface and the computer.

2. Call up Datastudio. Under “Hardware Setup”, click on the output channels of the

interface to connect the mechanical vibrator. Under “Signal Generator”, click on

“SW750 Output”.

3. Note that a sine wave has already been selected. We will use only sine waves in this

experiment. Set the amplitude and frequency of the signal generator initially to

approximately +3V and 10 Hz, respectively. Then click “On”. You should see the

string vibrate. Adjust the frequency to observe the multiple harmonics of the

standing wave. Remember that you can obtain frequency steps of various sizes by

clicking on the up and down arrows. Click the right and left arrows to adjust the size

of these steps.

Figure 8.2: Set up


Trinh Thanh Thuy 78

4.2 Procedure

A. Part 1

In this section, we will keep the tension and density of the string constant to find

experimentally the relationship between frequency and number of antinodes.

1. Adjust the frequency until you obtain a nice standing wave with two antinodes (n =

2). Record this frequency in the “Data” section.

2. Obtain and record the frequencies for consecutive n values. Take at least six

measurements, starting with the fundamental mode.

3. Calculate and record the wavelength, Eq. 2, and wave speed, Eq. 1, corresponding

to each n.

4. Plot a graph of frequency as a function of n. What is the relationship between the

two variables?

B. Part 2

In this section, we will keep n constant and change the weights to find the relationship

between frequency and tension.

1. Choose one of the three strings. If you like to see data that agrees well with theory,

choose the finest string. If you would rather see more interesting data, for which you

might need to explain the discrepancy, choose the most massive string. Measure and

record the linear mass density (µ = M/L) of the string by obtaining its total mass M

and total length L. Use the digital scale to weigh the string. Keep all units in the SI

system (kilograms and meters).

2. Using the 50-g mass hanger, measure and record the frequency for the n = 2 mode.

(Note: You may choose any integer for n, but remember to keep n constant

throughout the rest of this section.)

3. Add masses in increments of 50 g, and adjust the frequency so that the same number

of nodes is obtained. Take and record measurements for at least six different

tensions.

4. The wave speed should be related to the tension F and linear mass density µ by v =

(F/µ)1 /2 (eq (2)). Calculate and record the wave speed in each case using Eq. 2, and

plot v2 as a function of F/µ. (You have calculated v2 from the frequency and

wavelength; these are the y-axis values. You have calculated F/µ from the measured

tension and linear mass density; these are the x-axis values. Be sure to convert the

tension into units of Newton.) You now have the experimental points.


Trinh Thanh Thuy 79

5. Now plot the “theoretical” line v2 = F/µ. This is a straight line at 450 on your graph,

if you used the same scale on both axes. Do your experimental and theoretical

results agree well? If not, what might be the reasons?

C. Part 3

In this section, we will determine the relationship between frequency and the density of

a medium through which a wave propagates.

1. Measure the linear mass densities (µ = M/L) of the two other strings as described

above.

2. Keeping the tension and mode number constant at, 50 g and n = 2, measure and

record the frequencies for the three strings.

3. Calculate and record the “experimental” wave speed from the frequency and

wavelength for each string density.

4. Calculate and record the “theoretical” wave speed for each string density from v =

(F/µ)1/2, and compare these speeds with the experimental values.


Trinh Thanh Thuy 80

LAB 9: GYROSCOPE

1. EQUIPMENT

INCLUDED:

1 Gyroscope ME-8960

1 Stopwatch SE-8702

1 Super Pulley ME-9450

1 Pulley Mounting Rod SA-9242

1 Mass and Hanger Set ME-9384

1 Balance, Meter stick, Table clamp

for pulley, thread (1.5m)

2. INTRODUCTION

The purpose of this experiment is to measure the precession rate of a gyroscope and

compare it to the theoretical value.

3. THEORY

3.1 Torque and rotational inertia

We are all aware that a massive wheel has rotational inertia. In other words, it is hard to

start the wheel rotating; and, once moving, the wheel tends to continue rotating and is hard

to stop. These effects are independent of friction; it is hard to start a wheel rotating even if

its bearings are nearly frictionless. Rotational inertia is a measure of this resistance to

rotational acceleration, just as inertia is a measure of resistance to linear acceleration.

We also have an intuitive idea of torque, the tendency of a force to rotate a body. To

produce the maximum rotational acceleration, we want to push perpendicular to the rotation

axis, and at as large a distance r from the rotation axis as possible.

Consider a small mass m at a distance r from the rotation axis. If a force F acts on it, the

linear acceleration of the mass around the circle will be a = F⊥/m, where F⊥ is the

component of F perpendicular to the radius arm. The angular velocity ω = v/r of the mass

is increasing, but the angular acceleration α = a/r is constant.


Trinh Thanh Thuy 81

From the equation F⊥ = ma, we have:

rF⊥ = mr 2a/r = mr

2α, (9.1)

or

torque = = Iα. (9.2)

We recognize the torque = rF⊥ in vector form τ = r × F. The rotational inertia I is

equal to mr2 for a small particle of mass m. For an assembly of small particles, each of mass

mi, we sum to get the rotational inertia: = ∑ 𝑖 𝑖 . (9.3)

And if the mass distribution is continuous, we integrate: = ∫ . (9.4)

We can continue to define the rotational analogs to linear motion. For example, the

angular momentum L is

L = Iω, (9.5)

in analogy to

p = mv, (9.6)

and the rotational kinetic energy is

rotational KE = (1/2)Iω 2, (9.7)

in analogy to

translational KE = (1/2)mv 2. (9.8)

Linear momentum p = mv is conserved in the absence of external forces. Likewise,

angular momentum L = Iω is conserved in the absence of external torques. One interesting

difference between rotational motion and linear motion is that; since rotational inertia

depends on the positions of the masses, it is easy to change the rotational inertia “on the

fly”, so to speak. A spinning ballerina, ice skater, or star with a large rotational inertia I and

small angular velocity ω can increase the angular velocity of spin by pulling mass in to

reduce the rotational inertia: the ballerina and ice skater by pulling in their arms and legs,

and the star by collapsing smaller by gravity


Trinh Thanh Thuy 82

3.2 Precession

A common lecture demonstration of gyroscopic precession is to hang a bicycle wheel

by one end of its axle. If the bicycle wheel is not spinning, it flops down.

But if the wheel is spinning, it doesn’t fall. Instead it precesses around: its axle rotates

in a horizontal plane.

A necessary condition for precession is a torque aligned in a different direction from the

spin. In the case of the bicycle wheel, gravity acts downward on the center of mass so the

torque is in a horizontal direction. We are particularly interested in the case when the torque

is perpendicular to the spin axis.


Trinh Thanh Thuy 83

Let’s try to understand precession in general. Consider linear motion first. If an

object is at rest, and a force acts on it, the force will increase the speed of the object in the

direction of the force.

But if the object is already moving and the force acts perpendicular to the motion, the

speed will not be changed. Instead the force will curve the velocity around, producing

uniform circular motion if the force is always perpendicular to the velocity

Something similar happens with rotational motion. When the wheel is not spinning, the

torque from the weight produces an angular velocity about the torque axis, in this case the

y-axis.

But if the wheel is already spinning, it has spin (angular velocity) about the x-axis. The

torque of the weight adds some spin about the y-axis, perpendicular to the original spin. The

resulting spin axis is turned a little in the xy-plane. The torque doesn’t change the value of

the spin; instead it “curves” the spin. (Again, note that here the torque axis is perpendicular


Trinh Thanh Thuy 84

to the spin axis. This need not be true in general, but we are considering this simplified

case.)

Mathematically 𝜏 = 𝑳 (9.9)

which is the rotational analog of 𝑭 = 𝒑 (9.10)

To keep the ideas clear, we will call the angular velocity of the spinning object itself its

spin ω, and the turning around of the spin axis the precession angular velocity Ω. The

angular momentum L of the spin is L = Iω, where ω is the angular velocity of the spin, and

I is the rotational inertia of the wheel.

Thus, in a time ∆t, the torque produces a change in the angular momentum of the spin given

by

∆L = ∆t. (9.11)

But for small changes in the angle of L, ∆L = L∆.

Thus,

∆ L = L∆ = ∆t.


Trinh Thanh Thuy 85

The angular velocity of precession Ω = ∆φ/∆t = /L, the last equality following from the

equation above. Since L = Iω, we have finally Ω = 𝜏/ 𝜔 (9.13)

This then is our basic equation relating the precession angular velocity to the rotational

inertia, spin, and torque.

3.3 Gyroscope

A gyroscope is a spinning wheel or disc in which the axis of rotation is free to assume

any orientation by itself. When rotating, the orientation of this axis is unaffected by tilting

or rotation of the mounting, according to the conservation of angular momentum. Because

of this, gyroscopes are useful for measuring or maintaining orientation.

A torque is applied to the gyroscope by hanging a mass on the end of the shaft. This

torque causes the gyroscope to precess at a certain angular velocity, Ω.

Assume that the gyroscope is initially balanced in the horizontal position, θ = 90˚. The

disk is spun at an angular velocity (ω) and then a mass, m, is attached to the end of the

gyroscope shaft at a distance, d, from the axis of rotation. This causes a torque: = mgd.

But the torque is also equal to dL/dt, where L is the angular momentum of the disk. As

shown in Figure 9.1, for small changes in angle, d, dL = Ld

Figure 9.1: Torque applied to horizontal gyroscope

Substituting for dL in the torque equation gives 𝜏 = = = (9.14)

Since d/dt = Ω, the precession velocity, τ = = LΩ (9.15)


Trinh Thanh Thuy 86

and the precession rate (precession velocity) is given by Ω = 𝐼ω (9.16)

where I is the rotational inertia and ω is the angular velocity of the disk.

To find the rotational inertia of the disk experimentally, a known torque is applied to the

disk and the resulting angular acceleration is measured. Since = Iα, = 𝜏𝛼 (9.17)

where α is the angular acceleration which is equal to a/r and is the torque caused by the

weight hanging from the thread which is wrapped around the pulley on the disk.

= rF (9.18)

where r is the radius of the pulley about which the thread is wound and F is the tension in

the thread when the disk is rotating.

Applying Newton’s Second Law for the hanging mass, m, gives (See Figure 9.2)

ΣF = mg – F = ma (9.19)

Figure 9.2: Rotating disk and free-body diagram

Solving for the tension in the thread gives

F = m (g – a) (9.20)

So, once the linear acceleration of the mass (m) is determined, the torque and the

angular acceleration can be obtained for the calculation of the rotational inertia. The

acceleration is obtained by timing the fall of the hanging mass as it falls from rest a certain

distance (y). Then the acceleration is given by 𝑎 = (9.21)



Trinh Thanh Thuy 87


Trinh Thanh Thuy 88

4. EXPERIMENT

4.1 Measuring the precession rate


Level the gyroscope base.

1. Purposely make the apparatus unbalanced by moving the 900g counterweight

towards the center.

2. Adjust the leveling foot on one of the legs of the base until the gyroscope disk is

aligned over the leveling foot on the other leg of the base. See Figure 9.3.

3. Rotate the gyroscope 90 degrees so the gyroscope axle is parallel to one side of the

“A” and adjust the other leveling foot until the shaft will stay in this position. See

Figure 9.3.

4. Adjust the position of the 900g counterweight until the gyroscope is balanced

without the add-on mass. The 30g counterweight can be used to fine tune the

balance.

Figure 9.3: Leveling the base

4.1.2 Procedure

1. Weigh the add-on mass and record its mass in Table 9.1. Attach the add-on mass to

the end of the shaft. Measure the distance (d) from the axis of rotation to the center

of the add-on mass. Record this distance in Table 9.1

2. While holding the gyroscope so it can’t precess, spin the disk at about two

revolutions per second (rev/s). Time 10 revolutions of the disk to determine the

angular velocity (ω) of the disk. Record in Table 9.1.


Trinh Thanh Thuy 89

3. Let the gyroscope precess and time two revolutions to find the precession rate.

Record in Table 9.1.

4. Immediately repeat the measurement of 10 revolutions of the disk. The before-and-

after data will be used to find the average angular velocity of the disk during the

precession.

Table 9.1: Angular velocity measurements

Add-

On

mass

Distance

d

Time for

ten

revolutions

(initial)

Time for

ten

revolutions

(final)

Average

Angular

Velocity of

Disk

Time for

Precession

T

Experimental

Precession

Rate =2/T

kg m rad/s rad/s rad/s sec rad/s

4.2 Measuring quantities for the theoretical value


1. Clamp the gyroscope shaft in the horizontal position. See Figure 9.4.

Figure 9.4: Experiment set up

2. Attach a Super Pulley with rod to the table using a table clamp.


Trinh Thanh Thuy 90

3. Wind a thread around the pulley on the center shaft and pass the thread over the

Super Pulley.

4.2.2 Procedure

Accounting for Friction

Because friction is not included in the theory, it will be compensated for in this

experiment by finding out how much mass over the pulley it takes to overcome kinetic

friction. When the mass drops at a constant velocity, the weight of the mass is equal to the

kinetic friction. Then this “friction mass” will be subtracted from the mass used to

accelerate the apparatus.

To find the mass required to overcome kinetic friction, put just enough mass hanging

over the pulley so that the velocity is constant. Record this friction mass in Table 9.2.

Finding the Acceleration of the Disk

1. To find the acceleration, put about 30 g (record the exact hanging mass in Table 9.2)

over the pulley. Wind the thread up and let the mass fall from the table to the floor,

timing the fall.

2. Repeat this for a total of 5 times, always starting the hanging mass in the same

position.

3. Measure the height that the mass falls and record this height in Table 9.2.

Measure the Radius

Use calipers, measure the diameter of the pulley about which the thread is wrapped and

calculate the radius. Record the radius in Table 9.2.

Table 9.2: Rotational inertia data

Friction

Mass

Hanging

Mass

Original

Mass

Height

Mass

Falls

Radius

of

Pulley

Average

Times

Linear

Acceler

ation

Tension Torque Angular

Acceleration

Experimental

Rotational

Inertia

Kg Kg Kg m m s m/s2 N Nm rad/s2 Kgm2

4.2.3 Analysis

1. Using the average time from Table 9.2, calculate the acceleration and record the

result in Table 9.3.

2. Calculate the rotational inertia:

(a) Subtract the “friction mass” from the hanging mass used to accelerate the disk to


Trinh Thanh Thuy 91

determine the mass, m, to be used in the equations.

(b) Calculate the experimental value of the rotational inertia and record it in Table 9.3.

3. Using the times for 10 revolutions in Table 9.1, divide by 10 to find the periods.

Average these two periods and calculate the average angular velocity (ω = 2π/T).

Record the angular velocity in Table 9.3.

4. Find the experimental value for the precession rate by dividing the precession time

by two and calculating Ω = 2π/T. Record in Table 9.3.

5. Calculate the theoretical value for the precession rate and record in Table 9.3.

6. Calculate the percent difference between the experimental and theoretical values of

the precession velocity.

Table 9.3: Results

Solid

Disk

Mass

(M)

Solid

Disk

Radius

(K)

Theoretical

Rotational

Inertia

Add-

On

Mass

Distance

(d)

Gravitational

Acceleration

(g)

Average

Angular

Velocity

of Disk

()

Theoretical

Precession

Rate

Experimental

Precession

Rate

Precession

Rate

Difference

Kg m kgm2 kg m m/s

2 m/s

2 rad/s rad/s


Trinh Thanh Thuy 92

LAB 10: BERNOULLI’S PRINCIPLE

1. EQUIPMENT

1 Venturi Apparatus base and top plate ME-8598

1 Restriction Clamps (qty. 2) 640-052

1 Fluid Tubing, 6 mm ID, 6 m long 640-012

1 Water Reservoir (or similar container) ME-8594

1 Container to catch water

1 Table Clamp ME-9472

1 120 cm rod ME-8741

1 Quad Pressure Sensor PS-2164


1 Xplorer GLX Pasport PS 2002

2 Three-finger clamps SE-9445

1 Motion Sensor PS-2103

1 Stopwatch SE-8702B

2. INTRODUCTION

In the Venturi Apparatus, air or water flows through a channel of varying width. As the

cross-sectional area changes, the volumetric flow rate remains constant but the velocity

and pressure of the fluid vary. With a Quad Pressure Sensor connected to the built-in

Pitot tubes, the Venturi Apparatus allows the quantitative study and verification of the

Continuity Equation, Bernoulli’s Principle and the Venturi effect.

3. THEORY

Bernoulli's law states that if a non-viscous fluid is flowing along a pipe of varying cross

section, then the pressure is lower at constrictions where the velocity is higher, and the

pressure is higher where the pipe opens out and the fluid stagnate. Many people find this

situation paradoxical when they first encounter it (higher velocity, lower pressure). This is

expressed with the following equation:

const2ρv

+ ρgz+P2

(10.1)


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Where (in SI units):

P = fluid static pressure at the cross section in N/m2.

= density of the flowing fluid in kg/m3

g = acceleration due to gravity in m/s2 (its value is 9.81 m/s

2 = 9810 mm/s

2)

v = mean velocity of fluid flow at the cross section in m/s

z = elevation head of the center of the cross section with respect to a datum

Figure 10.1: Fluid flow through a pipe of varying diameter

An incompressible fluid of density flows through a pipe of varying diameter. As the

cross-sectional area decreases from Ao (large) to A (small), the speed of the fluid increases

from vo to v. The flow rate, R, (volume/time) of the fluid through the tube is related to the

speed of the fluid (distance/time) and the cross-sectional area of the pipe. This relationship

is known as the Continuity Equation and can be expressed as

R = Aovo = Av (10.2)

In this experimental set up, the centerline of all the cross sections are considering lie on

the same horizontal plane (which we may choose as the datum, z = 0, and thus, all the ‘z’

values are zeros. As the fluid travels from the wide part of the pipe to the constriction, the

speed increases from vo to v and the pressure decreases from Po to P. If the pressure change

is due only to the velocity change, then Bernoulli’s Equation can be simplified to:

𝑃 = 𝑃 − ( − 𝑜) (10.3)



Trinh Thanh Thuy 94

4. EXPERIMENT

4.1 Pre-Set up Preparations

Remove the top plate from the Venturi Apparatus. Measure the depth of the channel

and the widths of the wide and narrow sections. Calculate the largest cross-sectional

area (AL) and the smallest cross-sectional area (AS) by multiplying the depth by the width

for the two sections of the apparatus. Re-assemble the Venturi Apparatus. Do not over-

tighten the T-screws. Ensure that the Quad Pressure Sensor couplings are not connected to

the 4 tubes at the bottom of the Venturi. Connect the sensors to the GLX Pasport and

launch Datastudio.

4.1.1 Connect apparatus

Connect the Quad Pressure Sensor to your PASPORT interface (but do not connect

tubing to the pressure ports yet). If you are using a computer, start DataStudio. Calibrate the

Quad Pressure Sensor. The purpose of this calibration is to fine-tune all four pressure

measurements so they read the same when exposed to the atmosphere. This will allow the

small pressure differences that occur in the apparatus to be measured more accurately.

Conduct this procedure with all four pressure ports exposed to the same pressure. This

process is used for Datastudio:

Click the Setup button to open the Experiment Setup window

Click the Calibrate Sensors button to open the calibration window

At the top of the Calibrate Sensors window, select Quad Pressure Sensor

Select the Calibrate all similar measurements simultaneously option

Select the 1 Point (Adjust Offset Only) option.

Click Read From Sensor (in the Calibration Point 1 section of the window).

Click OK.

Connect each of the four pressure tubes extending from the underside of the apparatus

to the ports of the Quad Pressure Sensor as indicated in Figure 10.2. Ensure that they are

connected in the right order. The one closest to the flow IN to the apparatus is connected to

number 1 etc.

Important: Do not allow water to enter the sensor. Ensure that there is no water near the

sensor end of the pressure tubes


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Figure 10.2: Quad pressure sensor connected to apparatus

1. Place the top plate on the apparatus and secure it with eight T-knob screws. Tighten the

screws no more than necessary to prevent leaking.

2. Set up the fluid supply and flow-rate measurement as following.

4.1.2 Water Supply

1. Set up the apparatus with at least 1.5 m of vertical drop from the top surface of the

water reservoir to the bottom of the drain tube. Elevate the reservoir above your lab

bench and put the catch basin on the floor (see Figure 10.3).

Figure 10.3: Set up for water


Trinh Thanh Thuy 96

2. Cut the water tubing into two pieces of suitable length. Connect one piece of tubing to

the outflow port of the apparatus and run it over the side of the lab bench into the catch

basin. Secure the tubing so water will not spill onto the floor. Place both hose clamps on

the outflow tubing. Close one of the clams partially to regulate the flow rate. Close the

other clamp completely; you will open and close this clamp to start and stop water flow.

3. Run the other piece of tubing from the reservoir to the inflow port of the apparatus.

Note: Observe the correct direction of water flow through the apparatus indicated in

Figure 10.2.

4. Connect the Quad Pressure Sensor if it is not already connected

Important: Do not allow water to enter the sensor’s pressure ports. Connect the quad

pressure sensor to the apparatus before filling it with water. Once water is in the

apparatus, do not disconnect the sensor; otherwise water will flow through the

pressure tubes.

5. Fill the reservoir with water.

6. Open the clamp to let some water through the apparatus; then close it. Initially, there

will be air in the apparatus; tilt it so that the air moves to the outflow port. Let some

more water through to flush out the air. Repeat this process until all air has been

removed from the apparatus and inflow tubing. Do not let the reservoir run empty, or

new bubbles will enter. Close the clamp. Refill the reservoir.

4.1.3 Water-flow measurement by a stop watch

In this method, you measure a volume and elapsed time to determine the average flow.

Do this before collecting pressure data:

Start with the catch basin empty.

Start the stopwatch and open the clamp to start water flow.

After a measurable amount of water has flowed through, stop the stopwatch and close

the clamp. Be sure to read the water level by looking at the bottom of the meniscus.

Measure the volume of water that flowed out of (or into) the apparatus. When the flow is

steady note the pressures on the four sensors. Enter your data in the table 10.1. Repeat the

measurements several times more. Be sure to return all the water to the reservoir. Make

sure that the apparatus stays free of air bubbles.

Calculate the average flow rate: 𝑅 = ∆𝑉∆


Trinh Thanh Thuy 97

where ΔV is the volume of water and Δt is the elapsed time.

Typically the flow rate varies with the level of water in the reservoir. To keep the flow

rate close to constant, make the pressure measurements with the water level approximately

the same as it was for the flow rate measurement.

Appendix: Constants

Density of water: 1000 kg/m3

Wide cross-sectional area of channel: 1.99 cm2

Narrow cross-sectional area of channel: 0.452 cm2

4.2 Clean-up

1. Allow the water reservoir to run empty. Tilt the apparatus to empty water from it.

2. With the apparatus empty of water, disconnect the pressure tubes from the sensor.

(Leave the tubes connected to the underside of the apparatus.)

3. Remove the top plate from the apparatus. Allow the apparatus and tubing to dry

completely.

4.3 Storage

Store the apparatus with the top plate loose to avoid permanently deforming the seal.


Trinh Thanh Thuy 98

LAB 11: IDEAL GAS LAW

1. EQUIPMENT

REQUIRED:

1 Ideal Gas Law Syringe TD-8596

2 Pressure/Temperature Sensor PS-2146

1 Absolute Zero Apparatus TD-8595

1 Plastic Containers 740-183


Hot Water

Cold Water or Ice

Calipers

2. INTRODUCTION

The volume of a gas dep ends on the pressure as well as the temperature of the gas.

Therefore, a relation between these quantities and the mass of a gas gives valuable

information about the physical nature of the system. Such a relationship is referred to as the

equation of state. One of the most fundamental laws used in thermal physics and chemistry

is the Ideal Gas Law that deals with the relationship between pressure, volume, and

temperature of a gas.

3. THEORY

3.1 Boyle’s Law

In 1662, Robert Boyle discovered that the product of the pressure (P) and volume (V) of

a gas at a constant temperature is constant. Boyle’s Law gives the relation between the

pressure and volume of a given amount of gas at constant temperature. It states that the

volume is inversely proportional to the pressure of the gas.

1

VP

(11.1)

Where V is the volume of the gas and P is the pressure. This can also be written as

PV = k1= constant. (11.2)

The plot of pressure versus volume is shown in Fig. 11.1 below.


Trinh Thanh Thuy 99

Figure 11.1: Pressure versus volume plot

3.2 Charles’s Law

In the 1787, Jacques Charles experimentally verified that the volume and temperature

(T) of a gas at constant pressure are directly proportional.

V T (11.3)

Figure 11.2a shows the plot of volume versus temperature in Celsius. Note that the

graph is a straight line that when extrapolated back, intersects the horizontal axis at –273C.

At this temperature the volume of the gas goes to zero. This temperature is defined as the

absolute zero temperature or 0 kelvin (0 K). The same graph is plotted using the Kelvin 6

scale or absolute scale in Fig. 11.2b. Now the graph goes through the origin.

Figure 11.2: Plot of volume versus temperature

3.3 Gay-Lussac’s Law

In 1802, Joseph Gay-Lussac discovered the direct relationship between the pressure and

temperature of a gas at constant volume:

P T (11.4)

A plot of pressure versus temperature will be very similar to that of volume versus

temperature. Again, note that in Fig. 11.3a, the straight line graph intersects the temperature

axis at –273 C, and in Fig. 11.3b it passes through the origin.


Trinh Thanh Thuy 100

Figure 11.3: Plot of pressure versus temperature

3.4 The Ideal Gas Law

The three gas laws discussed above can be combined to give a more general relation

between the volume, pressure, and temperature of a gas. It relates the absolute pressure (P)

and volume (V) of a gas to the absolute temperature (T) in degrees Kelvin.

PV T (11.5)

Equation (11.5) describes the behavior of one variable when the other two variables are

changed. If the temperature is kept constant, then this reduces to Boyle’s Law. If the

pressure or volume is kept constant, Eq. (11.5) reduces to Charles’s Law or Gay-Lussac’s

Law respectively. The more general form of the gas law includes the amount of gas present

and is expressed as

PV nRT (11.6)

where R = 8.314 J/(mol·K) is the universal gas constant and n is the number of moles of the

gas. n can be calculated using the definition

mass gramsmol

molecular mass g/moln (11.7)

Note: If the pressure is in pascals, Pa, the volume is in m3, n is in mol, temperature is in

Kelvin, and R is 8.314472 J/(mol ·K).

3.5 Adiabatic Compression

Adiabatic compression happens when a gas is compressed so quickly that all the work put

into the compression goes into heating the gas, i.e. no heat escapes to the outside

environment during compression. For adiabatic compression the pressure and volume are

related by:

i i f fPV P V (11.8)

where γ is the ratio of specific heats. For air γ ≈ 1.40.




4. EXPERIMENT

In the first part of the experiment you will maintain a constant temperature and measure

the pressure for two different volumes. From this you will determine the correction to be

applied to account for the tubing portion of the apparatus. In the second part, you will

measure the pressure and temperature for two different volumes and verify the ideal gas

law. In part three you will determine the number of moles of gas in your apparatus from a

plot of volume versus pressure. Finally, in the last part you will examine the behavior of the

gas when it is subjected to adiabatic expansion.

4.1 Description of apparatus (Ideal Gas Law Syringe)

The Ideal Gas Law Syringe allows simultaneous

measurements of temperature and pressure of a gas as it

is compressed. The mini stereo jack is connected to a

low thermal mass thermistor built into the end of the

syringe to measure temperature changes inside the

syringe. The mini stereo jack plugs directly into the

temperature sensor.

The white plastic tubing coupler attaches to the port

of the pressure sensor: A slight twisting motion locks

the coupler onto the port. This white plastic connector

can be disconnected and re-connected during the

experiment to allow for different initial plunger

positions. All of the clear plastic fittings are glued in

place and cannot be removed.

The plunger is equipped with a mechanical stop that

protects the thermistor, and also allows for a quick, predetermined change in volume. Never

slam the plunger down on the table. Always grip the syringe and plunger as shown to

compress the air.

4.2 Ideal Gas Law Syringe

4.2.1 Procedure

1. In DataStudio, construct a graph of Pressure (kPa) vs. Time and Temperature (K) vs.

Time. Set the sample rate to 20 Hz.

2. Disconnect the white plastic pressure coupler from the pressure sensor. Press the



plunger of the syringe all the way in until the handle of the plunger bottoms out on the

mechanical stop. Record this minimum volume in the Data Table 1 as your final

volume. It should be close to 20 cc.

3. Set the plunger at 40 cc, and then re-connect the coupler to the sensor. Record this on

the worksheet as the initial volume of gas.

4. Click Start to record data and quickly compress the plunger all the way in and keep it

compressed. The plunger handle should be all the way down and flush against the

mechanical stop.

5. Watch the graphs of pressure and temperature, and continue to hold the plunger in until

the values are no longer changing. This should take around 10 seconds.

6. After the temperature and pressure have equalized, release the plunger. Again, watch

the graphs and wait until the values are no longer changing.

7. Click Stop to stop the data collection.

4.2.2 Analysis

1. Look at your pressure and temperature graphs. Correlate the changes in pressure and

temperature to the movement of the plunger.

2. What happened to the temperature when the air was compressed? Why?

3. What is the equilibrium temperature of the gas when it was compressed? Why? What is

the equilibrium pressure? Why does it not go back to “room pressure”?

4. What happened to the temperature during the expansion (when you released the

plunger)? Why? Does it go below room temperature? Does the pressure go below

“room pressure”? What would you have to do to make this happen?

5. Create a Data Table in DataStudio of the Pressure and Temperature data.

6. Measure the initial temperature (T1) and pressure (P1) of the gas from your data just

before you compressed it. You can highlight an area (click and drag) in the graph and

that data will appear in the data table. This data corresponds to an initial volume (V1) of

40 cc.

7. Highlight the area on the temperature graph where it peaks. Pick the place where the

temperature has peaked, not the pressure. It takes the temperature sensor about 1/2

second to respond. Record the peak temperature (T2) and the corresponding pressure

(P2) for that time. You want two values that occurred at the same time. This data

corresponds to the volume (V2) of 20 cc. Note: If the compressed volume marked on the

syringe is different than 20 cc, use that value instead.



8. Use the Ideal Gas Law to show that the ratio of volumes can be expressed as

where the subscript 1 refers to the initial state (volume = 40 cc) and the subscript 2

refers to the final state (volume = 20 cc) after compression.

9. Use your values of pressure and temperature to calculate the ratio of volumes. How

does this compare to the actual ratio? Are they about the same?

4.3 Constant Temperature

4.3.1 Procedure

1. Add a digits display of temperature in DataStudio.

2. Disconnect the white plastic coupler from the pressure sensor. Set the plunger at 45 cc,

and then re-connect the coupler to the sensor.

3. Start recording data. Compress the plunger to 40 cc and hold it at this position. Watch

the temperature on the digits display and wait until it has dropped down to close to

room temperature. Note the final temperature. Each time you compress the air in this

sequence, wait until the temperature returns back down close to this value.

4. Compress the plunger to 35 cc and hold it at this position. Watch the temperature, and

hold the plunger at 35 cc until the temperature has dropped to the value you noted in

step 3. Do not release the plunger.

5. Compress the plunger to 30 cc, and wait until the temperature drops as before.

6. Repeat for 25 cc and 20 cc.

7. Stop recording data.

4.3.2 Analysis

1. Look at your pressure and temperature graphs. Correlate the changes in pressure and

temperature to the movement of the plunger.

2. Highlight the area on the graph when the plunger was at 40 cc. Use the data table to

determine the equilibrium pressure and temperature.

3. Repeat for all other volumes. For each position, pick the pressure that has the

temperature closest to the “equilibrium” value. It does not matter what this temperature

is, as long as all pressures are measured at the same temperature. Record all your values

in a table.

4. For each of the pressures, calculate the inverse pressure (1/P). Graph Volume vs 1/P.

1 1 2

2 2 1

V T P

V T P



Why does this give a straight line? Use the Ideal gas law to show that a graph of

Volume vs 1/P results in a straight line with a slope given by

Slope = nRT

5. Determine the slope of this line from your graph of Volume vs. 1/P. Use your values to

determine the number of moles (n) of air in the syringe. Pay attention to the units!

6. Look carefully at the graph. Why is there an offset in the axis for the volume? How do

you account for this extra volume?

4.4 Further Investigations

1. Disconnect the white plastic pressure coupler from the sensor. Set the plunger at 60 cc,


2. Repeat the procedure, taking pressure and temperature data at each of the volumes

(40cc, 35cc, etc.) as you did before.

3. Put this new data on the same graph. Why is this slope different? Is the volume offset

about the same as before?

4.5 Adiabatic Compression

4.5.1 Procedure

1 Set the sample rate at 50 Hz

2 Disconnect the white plastic coupler from the pressure sensor. Set the plunger at 60 cc,


3 Start recording data. As quickly as possible, compress the plunger from 60 cc down to

20 cc. Do this in one quick motion, bottoming out the piston on the mechanical stop.

4 Stop recording data.

4.5.2 Analysis

1. Measure the initial temperature (T1) and pressure (P1) of the gas from your data just

before you compressed the syringe. This data corresponds to an initial volume (V1) of

60 cc.

2. For an adiabatic compression, the initial pressure and volume (P1, V1) are related to the

final pressure and volume (P2, V2) by

where (gamma) is the ratio of specific heats, and for air has a value of 1.40. Use your

values to calculate the theoretical peak pressure if the compression was adiabatic.

3. Measure the peak pressure (P2) after compression. Was this truly adiabatic?

1 1 2 2 = PV PV



4. Using the Ideal Gas Law, calculate the theoretical peak temperature.

5. Measure the peak temperature after compression. Why did it not occur at the same time

as the peak pressure? Why is this temperature so much lower than the theoretical?

LAB 12: GAY-LUSSAC’S LAW

1. EQUIPMENT

REQUIRED:

Pressure Sensor CI-6532A

Temperature Sensor CI-6527A

Absolute Zero Apparatus TD-8595

Plastic Containers 740-183


Hot Water

Cold Water or Ice

Calipers

2. INTRODUCTION

The Absolute Zero Apparatus is used to experimentally determine the temperature of

absolute zero (in degrees Celsius). Absolute zero, by definition, is the point at which a gas

exerts zero pressure. With a computer, the Absolute Zero Apparatus can help students to

observe the relationship between temperature and pressure and use DataStudio to

mathematically extrapolate to find absolute zero.

3. THEORY

For an ideal gas, the absolute pressure is directly proportional to the absolute

temperature of the gas.

VT P

nR (12.1)

Thus a plot of temperature vs. pressure will result in a straight line.

y ( slope )x b (12.2)

0V

T PnR

(12.3)

The slope of the line depends on the amount of gas in the thermometer, but regardless

of the amount of gas, the intercept of the line with the temperature axis should be at



absolute zero. If we instead plot the temperature in degrees Celsius, the intercept will not be

zero, but rather the temperature of absolute zero in degrees Celsius.




4. EXPERIMENT

4.1 Constant volume (Gay-Lussac’s law)

4.1.1 Description of apparatus

The Absolute Zero

Apparatus consists of a hollow

sphere that acts as a container of

constant volume as the

apparatus is placed in different

temperature water baths. Plug

the mini stereo jack into the temperature sensor to measure the temperature using the

thermistor imbedded in the wall of the sphere. Connect the white plastic coupling to the

port of the pressure sensor to measure the pressure inside the sphere. You will need a

source of hot water and a source of cold water (or ice).

4.1.2 Procedure

1. Set the sample rate at 10 Hz. Set up a graph in DataStudio of pressure vs. temperature.

Select Manual Sampling from the sampling options in the Setup menu. It is also helpful

to set up digit displays of temperature and pressure.

2. Fill one of the plastic containers with enough hot water to cover the sphere. Use the

other container for a supply of cold water (or ice). Make sure the white plastic pressure

coupler is connected to the sensor before putting the sphere in the hot water.

3. Start data collection. Completely submerge the sphere and wait for the pressure and

temperature to equalize. Click on keep in the top menu bar to save that data pair.

4. Add cold water (or ice) to decrease the temperature of the water 5 to 10oC. Stir the

water to get an even temperature and click on keep when the pressure and temperature

equalize.

5. Keep decreasing the temperature and taking data until you have the system as cold as

you can get it. You may need to dump out some of your water.

4.1.3 Analysis

1. Use calipers to measure the diameter of the sphere, and calculate its volume. Is this

measurement more or less than the actual volume of the sphere? Why?

2. Use the Ideal Gas Law to show that a graph of Pressure vs. Temperature results in a

straight line with a slope given by:



V

nRSlope

3. Determine the slope of this line from the Pressure vs. Temperature graph. Use your

values to determine the number of moles (n) of air in the sphere. Pay attention to the

units!

4.2 Further Investigations

1. Re-fill the container with hot water.

2. Disconnect the white plastic pressure coupler from the sensor and place the sphere in

the hot water. Does air flow in or out of the sphere? Re-connect the coupler to the

sensor.

3. Repeat the procedure, taking pressure data at different temperatures as you did before.

4. Put this new data on the same graph. Why is this slope different? Calculate the new

number of moles of air.

4.3 Conclusion

1. Describe what happens to the pressure of a gas at constant temperature when its

volume is changed.

2. Describe what happens to the volume of a gas at constant pressure when its

temperature is changed.

3. Describe what happens to the temperature of a gas at constant volume when its

pressure is changed.

4.4 Determining Absolute Zero while Keeping the Number of Gas Moles (n)

Constant

1. Start with the water as hot as possible.

2. Connect the hose fitting from the Absolute Zero Apparatus to the Pressure Sensor.

Connect the stereo plug from the apparatus to the Temperature Sensor.

3. Set up your experiment in DataStudio. In DataStudio, open a Digits display and a

temperature vs. pressure graph. Click the Start button.

4. Place the sphere of the apparatus in the water bath, and keep the sphere completely

submerged.

5. Watch the Digits display of temperature. When the display stops changing (in the

hundredths place), click on the Keep button. Do not stop recording.

6. Cool the water bath by adding cold water or some ice cubes. When the container

becomes too full, dump out some of the water, but always have enough water to keep



the apparatus completely submerged. Cool the bath by about 10oC, and repeat step (4).

7. Repeat steps 4 through 6, for temperatures down as low as you can go, and then click on

the Stop button to end recording.

8. In the Graph display, click on the Fit button and select a linear curve fit. The y-intercept

is your value for absolute zero.



LAB 13: HEAT ENGINE CYCLES

1. EQUIPMENT

INCLUDED:

1 Heat Engine/Gas Law Apparatus TD-8572


1 Ohaus Slotted Mass Set (Need 200 g mass only) SE-8726

1 Drilled Mass (10g) 648-06508

1 Drilled Mass (20g) 648-06509

1 Mass Hanger 648-04857

2 Plastic containers (3 liters) for hot and cold water 740-183

1 thread 699-011


1 Rotary Motion Sensor CI-6538

2 Temperature Sensor CI-6505B

1 Low Pressure Sensor CI-6534A


1 ScienceWorkshop 500 or 750 Interface CI-6400


2. INTRODUCTION

A heat engine is a device that does work by extracting thermal energy from a hot

reservoir and exhausting thermal energy to a cold reservoir. In this experiment, the heat

engine consists of air inside a cylinder which expands when the attached can is immersed in

hot water. The expanding air pushes on a piston and does work by lifting a weight. The

heat engine cycle is completed by immersing the can in cold water, which returns the air

pressure and volume to the starting values.



3. THEORY

The Laws of Thermodynamics

First law: Energy is conserved; it can be neither created nor

destroyed.

Second law: In an isolated system, natural processes are

spontaneous when they lead to an increase in disorder, or entropy.

Third law: The entropy of a perfect crystal is zero when the

temperature of the crystal is equal to absolute zero (0 K).

The theoretical maximum efficiency of a heat engine depends only on the temperature

of the hot reservoir, TH, and the temperature of the cold reservoir, TC. The maximum

efficiency is given by

1 100C

H

Te %

T

(13.1)

The actual efficiency is defined as

100H

We %

Q (13.2)

where W is the work done by the heat engine on its environment and QH is the heat

extracted from the hot reservoir.

1. At the beginning of the cycle, the air is held at a constant temperature while a weight is

placed on top of the piston. Work is done on the gas and heat is exhausted to the cold

reservoir. The internal energy of the gas V( )U nC T does not change since the

temperature does not change. According to the First Law of Thermodynamics,

U Q W where Q is the heat added to the gas and W is the work done by the gas.

2. In the second part of the cycle, heat is added to the gas, causing the gas to expand,

pushing the piston up, and doing work by lifting the weight. This process takes place at

constant pressure (atmospheric pressure) because the piston is free to move. For an

isobaric process, the heat added to the gas isP PQ nC T , where n is the number of

moles of gas in the container, CP is the molar heat capacity for constant pressure, and

T is the change in temperature. The work done by the gas is found using the First Law

of Thermodynamics, W Q U , where Q is the heat added to the gas and U is the

internal energy of the gas, given by VU nC T , where CV is the molar heat capacity



for constant volume. Since air consists mostly of diatomic molecules, V

5

2C R and

P

7

2C R .

3. In the third part of the cycle, the weight is lifted off the piston while the gas is held at

the hotter temperature. Heat is added to the gas and the gas expands, doing work.

During this isothermal process, the work done is given by

f

i

lnV

W nRTV

(13.3)

where Vi is the initial volume at the beginning of the isothermal process and Vf is the

final volume at the end of the isothermal process. Since the change in internal energy is

zero for an isothermal process, the First Law of Thermodynamics shows that the heat

added to the gas is equal to the work done by the gas:

0U Q W (13.4)

4. In the final part of the cycle, heat is exhausted from the gas to the cold reservoir,

returning the piston to its original position. This process is isobaric and the same

equations apply as in the second part of the cycle.




4. EXPERIMENT

4.1 Set up

1. Put the rod in the rod stand. Attach the

Heat Engine to the rod by sliding the Heat

Engine’s rod clamp onto the rod. The Heat

Engine should be oriented with the piston

end up and the Heat Engine should be

positioned close to the bottom of the rod

stand (see Figure 13.1).

2. Attach the Rotary Motion Sensor to the top

of the rod stand and align the medium

groove of the pulley of the Rotary Motion

Sensor so a string coming from the center

of the Heat Engine’s piston platform will

pass over the pulley.

3. Thread one end of a piece of string through

the hole in the top of the piston platform and tie that end of the string to the shaft of the

piston under the piston platform. See Figure 13.2. Pass the

other end of the string over the medium step of Rotary

Motion Sensor pulley and attach the mass hanger and masses

totaling 35 grams. This mass acts as a counterweight for the

piston.

4. Position the piston about 2 or 3 cm from the bottom of the

cylinder and attach the tube from the can to one port on the

Heat Engine and attach the tube from the pressure sensor to

the other port on the Heat Engine.

5. Connect the Pressure Sensor to Channel A, the two

Temperature Sensors to Channels B and C, and the Rotary Motion Sensor to Channels 1

and 2 on the computer interface.

6. Put hot water (about 80oC) into one of the plastic containers (about half full). Put ice

water in the other plastic container. The large (about 3 liter) containers keep the hot and

cold temperatures constant during the heat engine cycle.

Figure 13.1: Set up the experiment

Figure 13.2: Attaching

string to piston



7. Place one temperature sensor in the hot water and place the other temperature sensor in

the cold water. Note that the temperature sensors are labeled hot and cold in the

software program so you will have to pay attention to which sensor you put in the hot

water and which is in the cold water.

4.2 Software Set Up

1. Run DataStudio.

2. Load the file called "HeatEngineCycle". This file is set up to record a graph of air

pressure inside the cylinder versus volume (i.e., a P-V diagram).

4.3 Procedure

1. Perform the following cycle without hesitating between steps. You may want to practice

a few times before recording a data run. Start with the can in the cold water. This starting

point will be called point A. Record the height of the bottom of the piston. Start

recording data on the computer.

A: No weight on piston, cylinder in cold bath.

AB: Place the 200g mass on the platform gradually, cylinder in cold bath.

B: All weight on piston, cylinder in cold bath.

BC: All weight on piston, move the can from the cold bath to the hot bath.

C: All weight on piston, cylinder in hot bath.

CD: Remove the 200g mass from the platform gradually, cylinder in hot bath.

D: No weight on cylinder, cylinder in hot bath.

DA: No weight on piston, move the can from the hot bath to the cold bath.

Figure 13.3: Figure adapted from PASCO Scientific instruction manual for Heat

Engine/ Gas Law Apparatus.

2. Print the graph of the cycle. Label the four corners of your graph as A, B, C, and D.

Identify the temperatures at points A, B, C, and D. Put arrows on the cycle to show the



direction of the process.

3. Identify the types of processes (i.e., isothermal, etc.) and the actual physical

performance (put mass on, put in hot bath, etc.) for A to B, B to C, C to D, and D to A.

4. Identify and label the two processes in which heat is added to the gas.

4.4 Analysis

1. Calculate the ideal (maximum) efficiency for a heat engine operating between the two

temperatures using Equation (13.1).

2. Calculate QH, the heat added to the gas by the hot reservoir during the isobaric

expansion from B to C and the isothermal expansion from C to D. You will need to

calculate the following:

a) We do not know the initial volume, VA, but we can calculate it by measuring the

volume of the can and adding the initial volume of air in the cylinder. We will

ignore the volume in the tubes.

2

can 0 cylinderV ( r h ) ( Ah )

where A is the cross-sectional area of the piston.

Calculate VD using an isobar and the Ideal Gas Law: A D

A D

V V.

T T

b) Calculate VC using an Isotherm and the Ideal Gas Law:

PCVC =PDVD

c) Calculate QC D. For an isotherm, Q = nRT ln(Vf /Vi), and since PV = nRT,

QC D = PDVDln(VD/VC)

Remember that Absolute P = (Gauge P) + (Atmospheric P)

d) Calculate QB C. For an isobar, Q = nCpT, and since air is a diatomic gas Cp =



5/2 R, and nR = PV/T,

)(2

7DC

D

DDCB TT

T

VPQ

e) Calculate QH = QB C + QC D.

3. Calculate the work done by the gas by measuring the Area inside the curve.

4. Calculate the efficiency e = work done by gas/ heat extracted from hot reservoir

(Equation (13.2).

100H

We %

Q

How does this compare with the ideal efficiency from step 5?

5. Calculate the actual work done on the 200 g mass using W = mgh. Be careful to use

only the change in height of the mass. How does this compare to the work done by the

gas from step 7? Does the gas do any work other than lifting the 200 g mass?

6. Mix some of the ice water with the hot water and vice versa so the two reservoirs are

closer to the same temperature. Perform the cycle again. How high is the weight lifted

now? What is the theoretical efficiency using the new temperatures?



LAB 14: BLACKBODY RADIATION

1. EQUIPMENT

INCLUDED:

1 Prism Spectrophotometer Kit OS-8544

1 Optics Bench (60 cm) OS-8541

1 Spectrophotometer Accessory Kit OS-8537

1 Aperture Bracket OS-8534

1 Broad Spectrum Light Sensor CI-6630

1 Rotary Motion Sensor CI-6538

1 Voltage Sensor CI-6503

1 Power Amplifier II CI-6552A

1 Replacement Bulb (10 pk) SE-8509

1 Banana Plug Cord-Black (5 pack) SE-9751


1 ScienceWorkshop 750 Interface CI-7650

1 DataStudio CI-6870

2. INTRODUCTION

A blackbody is defined as an object that perfectly absorbs all (and thus reflects none) of

the radiation incident on its surface. When a blackbody is in thermal equilibrium with its

surroundings, it must also be a perfect emitter so that the temperature of the blackbody

stays the same. But this emitted light is not at the same frequency as the light that was

initially absorbed; rather it is distributed between different frequencies in a characteristic

pattern called the blackbody spectrum.

The measurement of the blackbody spectrum was the center of a crisis in physics during

the early 20th

century known as the ultraviolet catastrophe. Different classical models could

explain the blackbody spectrum over some frequency ranges, but broke down (in one case

predicting infinite radiation at some frequencies). Max Planck eventually resolved the crisis

by introducing the quantization of energy, giving birth to the quantum revolution in the

process.

In this lab you will use an incandescent light bulb and a prism spectrometer to measure

the blackbody spectrum. Although a light bulb is not a blackbody (it emits much more



radiation than it absorb!) it is a good approximation of a grey body: an object that emits a

fraction of the blackbody spectrum with the same frequency distribution. Due to this

approximation and the simplicity of the apparatus, your intensity data will not

quantitatively match that of a blackbody, but the shape of the intensity curve should be

qualitatively the same.

Classical physics suggested that all modes had an equal chance of being produced, and

that the number of modes went up proportional to the square of the frequency. The

predicted continual increase in radiated energy with frequency was named “the ultraviolet

catastrophe” (Fig.14.1).

Figure 14.1: The amount of radiation as a function of frequency: classical and quantum

theories.

3. THEORY

The intensity (I) of radiation emitted by a body is given by Planck's Radiation Law:

= ℎ𝜆 ℎ𝑐𝜆𝑘𝑇− (14.1)

where c is the speed of light in a vacuum, h is Planck's constant, k is Boltzmann's constant,

T is the absolute temperature of the body, and is the wavelength of the radiation.

The wavelength with the greatest intensity is given by



= 𝑇 = . ⋅𝑇 (14.2)

where T is the absolute temperature of the body. The temperature of the blackbody light

filament can be calculated using the resistance of the filament while it is lit. The resistivity

of the tungsten filament is a nonlinear function of the temperature. A function that

approximates the calibration curve (CRC Handbook, 45th

edition, page E-110) for the

resistivity of tungsten is used in the DataStudio setup file to calculate the temperature.

The resistance of the filament is found using

𝑅 = 𝑉𝐼 (14.3)

where V is the voltage applied to the lamp and I is the current through the lamp. The

temperature dependence of the resistance is used to determine the temperature of the hot

filament. See the Appendix for an explanation of how this is done in the DataStudio file.




4. EXPERIMENT

The spectrum of an incandescent light bulb is scanned by hand using a prism

spectrophotometer that measures relative light intensity as a function of angle. A Broad

Spectrum Light Sensor is used with a prism so the entire spectrum from approximately 400

nm to 2500 nm can be scanned without the overlapping orders caused by a grating. The

wavelengths corresponding to the angles are calculated using the equations for a prism

spectrophotometer. The relative light intensity can then be plotted as a function of

wavelength as the spectrum is scanned, resulting in the characteristic blackbody curve. The

intensity of the light bulb is reduced, reducing the temperature, and the scan is repeated to

show how the curves nest with a shift in the peak wavelength.

The temperature of the filament of the bulb can be estimated indirectly by determining

the resistance of the bulb from the measured voltage and current. From the temperature, the

theoretical peak wavelength can be calculated and compared to the measured peak

wavelength.


1. Place the Spectrophotometer (Rotary motion sensor + bench + disk) on the optics track.

2. Attach the Broad Spectrum Light Sensor and the aperture plate to the arm of the

spectrophotometer using the black rod (figure 14.2). Plug the Broad Spectrum Light

Sensor into Analog Channel A on the ScienceWorkshop interface.

Figure 14.2: Broad Spectrum Light Sensor attached to the arm of the spectrophotometer

3. Place the focusing lens on the spectrophotometer arm in between the light sensor and

the prism, inside of the white angled markings.



Figure 14.3: Focusing lens position

4. Plug in the power cable for the power amplifier and connect its cable to Analog

Channel C on the ScienceWorkshop interface.

5. Place the incandescent lamp source on the track and connect to the power amplifier

outputs with the banana plugs.

6. Attach the Voltage Sensor (banana plugs on one end and analog channel input on the

other) to the terminals of the lamp and Analog Channel B. You can plug the banana

plugs into the back of the ones coming from the power amplifier. This will allow the

computer to measure the voltage across the lamp terminals.

Figure 14.4: Incandescent lamp source and Voltage Sensor set up

7. Place the collimating slit holder and then the collimating lens in front of the

incandescence lamp. Make sure that the collimating lens is about 12 cm from the

collimating slits. The lamp should slide into the back of the collimating slit holder.

Have someone with 20/20 vision (corrected with glasses is ok) look through the

collimating lens at the slits. Adjust the collimating lens until the slits are in sharp focus.

The collimating lens should be about 10 cm from the collimating slits.



Figure 14.5: Collimating slit holder and collimating lens together with Incandescent lamp

source

8. Move the spectrophotometer close to the collimating lens, the focusing lens should now

be about 10 cm from the collimating lens.

9. Position the Aperture Bracket so that you can see the thin beam of white light. Move the

focusing lens so that you get the most in focus beam of light on the Bracket (This

should be towards the rear of the angled box).

Figure 14.6: Complete Set up

1. The complete set up will be as in Figure 14.6.

The light sensor used in this experiment is the

Broad Spectrum Light Sensor.

2. Check that the prism is oriented as shown in

Figure 14.7 with the apex facing the light

source. Figure 14.7: Prism Orientation



3. The collimating lens must be 10 cm (the lens’ focal length) from the collimating slits.

4. Plug the Blackbody Light into the Power Amplifier. Plug the Power Amplifier into

Channel C of the Science Workshop 750 interface.

5. Plug the Broad Spectrum Light Sensor into Channel A of the interface. Plug the Voltage

Sensor into Channel B. Connect the leads of the Voltage Sensor to the Blackbody Light.

Plug the Rotary Motion Sensor into Channels 1 and 2.

6. Open the DataStudio setup file called "Blackbody.ds". See the Appendix for an

explanation of the equations in the setup file.

4.2 Procedure

1. Set the collimating slits on Slit #4. Set the Light Sensor mask on Slit #4.

2. In DataStudio, click the Signal Generator window and turn the generator ON at 10 V

DC.

Caution: If 10 volts is applied to the blackbody light for an extended amount of time, the

life of the bulb will be reduced. Only turn on the bulb when taking measurements.

3. Look at the light coming from the Blackbody Light

Source. Observe the color.

4. Look at the spectrum on the Light Sensor screen. Are

all the colors (from red to violet) present?

5. Rotate the scanning arm until it touches the stop. This

will be the starting position for all the scans.

6. Set the Broad Spectrum Light Sensor gain switch to

“x100” and press the tare button. Click START in DataStudio.

7. As the scan begins, check that the angle is positive. If not, reverse the Rotary Motion

Sensor plugs in the interface and re-start the data run. Slowly rotate the scanning arm

through the spectrum and continue all the way past zero degrees (the position where the

light sensor is directly opposite the light source). The graph of intensity vs. wavelength

is set to automatically stop at about 2500 nm because the glass in the spectrophotometer

optics does not transmit wavelengths greater than 2500 nm.

8. There will be a peak on the intensity vs. angle graph where the light sensor is aligned

with the light source because some light passes by the prism instead of going through

the prism. This peak enables the initial angle to be exactly determined. Use the Smart

Cursor to determine the angle from the starting position to the central peak where the

light sensor is directly opposite the light source. Click the calculator in DataStudio and

Figure 14.3: Spectrum

on Light Sensor Mask



enter this angle called "init".

9. Repeat Steps 2 through 7 for voltages of 7 V and 4 V. On these scans, it is not necessary

to scan all the way past the center as before because the angle has already been

calibrated.

4.3 Analysis

1. Does the peak shift toward shorter or longer wavelengths as the temperature is lowered?

2. How does the intensity change as the temperature is changed?

3. Use the Smart Cursor on the Temperature graph to find the temperatures of the

filament. Calculate the peak wavelength for each temperature using Equation (2). Do

these theoretical values correspond to peak wavelengths on the intensity graphs?

4. Planck's formula (Equation (14.1)) is already in the DataStudio setup file. Click and

drag the calculation from the Data List on the left to the graph of intensity vs.

wavelength. Change the amplitude in the calculator so it matches the tallest curve. Does

the shape of the curve match the theoretical curve? Can the bulb really be considered a

blackbody?

5. How did the color of the bulb change with temperature? How did the color composition

of the spectrum change with temperature? Considering the peak wavelengths, why is a

bulb’s filament red at low temperatures and white at high temperatures?

6. At about what wavelength is the peak wavelength of our Sun? What color is our Sun?

Why?

7. For the highest temperature, is more of the intensity (area of the intensity vs.

wavelength graph) in the visible part of the spectrum or in the infrared part of the

spectrum? How could a light bulb be made more efficient so it puts out more light in the

visible?



5. APPENDIX

Explanations of the Calculations in the DataStudio Setup File

Wavelength Calculation: The index of refraction of the prism glass varies with the

wavelength of the light. To determine the wavelength as a function of the angle, the

relationship between the index of refraction and the angle is determined using Snell’s Law

at each face of the prism.

= √(√ 𝑖 𝜃 + ) +

The Cauchy equation gives the relationship between the index of refraction and the

wavelength: = 𝜆 + , where A and B are specific to the type of glass and are determined

experimentally.

Solving this for wavelength gives = √ − . However, this equation is an approximation

which does not fit well in the region of interest. The following table summarizes the

dependence of the index of refraction on wavelength for the prism (provided by the supplier

of the prism):

Index of Refraction Wavelength ( nm )

1.68 2325.40

1.69 1970.10

1.69 1529.60

1.70 1060.00

1.70 1014.00

1.71 852.10

1.72 706.50

1.72 656.30

1.72 643.00

1.72 632.80

60o

θ

θ2

θ3

60o



1.73 589.30

1.73 546.10

1.75 486.10

1.76 435.80

1.78 404.70

The wavelength is calculated using an equation derived from a curve fit for the prism,

where the coefficients are determined experimentally for the type of flint glass the prism

was made from. It is not necessary to change this calculation unless you use a prism

different from the one supplied in the experiment. 𝑎 ℎ = ×√ + + + + + + + + = − . × = . × = − . × = − . × = . = . = − . = . = . = = 𝑖 𝑎 𝑖

From Snell’s Law:

= √(√ 𝑖 𝜃 + ) +

This equation is expressed in the setup file as:

= 𝑖 . , , √ . sin 𝑖 − 𝑖 , ,𝑅𝑎 𝑖 + . + .

where the purpose of the filters is to keep the function from getting out of range. The initial

angle is determined by measuring the angle of the central maximum from the origin: 𝑖 = 𝑖 𝑖𝑎 = . . This initial angle will be about the same for all apparatus



because the procedure calls for starting with the scanning arm against the stop. =𝑎 𝑃 𝑖 𝑖 , ℎ & , the measurement from the Rotary Motion Sensor.

The angle measured by the Rotary Motion Sensor is greater than the angle through

which the scanning arm moves because the pin rotating against the disk acts as a gear. The

ratio of the disk diameter to the pin diameter is approximately 60. However, you can rotate

the disk through 180 degrees as read from the marks on the disk and compare to the reading

from the Rotary Motion Sensor to get a more precise measurement of this ratio (𝑅𝑎 𝑖 =. ).

Light Intensity Calculation: The light intensity is smoothed by eight points to eliminate

noise: 𝑖 = ℎ , − 𝑉 = 𝑉 𝑎 , ℎ 𝑉 = . . This is an optional offset in case you neglect to tare the light sensor before

recording data.

Theoretical Intensity Calculation: = ℎ ℎ𝑐𝜆𝑘𝑇 −

This equation is expressed in the setup file as: 𝑇ℎ = . ×( ( . × 7𝑇 ) − )

where x is the wavelength in nanometers. To model this function, the wavelength (x) is

varied from 300 nm to 2800 nm in 50 increments. = , = . (This is an arbitrary constant to adjust the amplitude of the function because the

Light Sensor is not calibrated.) 𝑇 = (This is the temperature of the light bulb filament in Kelvin.)

Finding the Temperature:

The temperature of the filament is calculated using the temperature dependence of its

resistivity.

𝑖 𝑖 𝑖 = = ( + 𝛼 𝑇 − 𝑇 ) → 𝑇 = 𝑇 + 𝑜⁄ −𝛼



However, the value of the temperature coefficient (α) is not constant over the range

from room temperature to the temperature of the hot bulb. The table shows the temperature

coefficient for a broader range of temperatures.

A curve is fit to this data and a filter is used to keep the temperature calculation in the range

of interest: 𝑎 = 𝑖 ( , , + . − . + . × − )

The resistance is determined using the voltage across the filament and the current through

the filament (𝑅 = 𝑉𝐼 ).

𝑜 = 𝑅𝑅𝑜 = 𝑅−𝑅 𝑖 𝑒𝑅𝑜 → = 𝑅−𝑅 𝑖 𝑒𝑅𝑜 where Rwires is the approximate resistance

of the wires connecting the voltage source to the bulb.

resistivity

(×10-8Ω∙m)

Temperature

(K)

resistivity

(×10-8Ω∙m)

Temperature

(K)

5.65 300 60.06 2100

8.06 400 63.48 2200

10.56 500 66.91 2300

13.23 600 70.39 2400

16.09 700 73.91 2500

19.00 800 77.49 2600

21.94 900 81.04 2700

24.93 1000 84.70 2800

27.94 1100 88.33 2900

30.98 1200 92.04 3000

34.08 1300 95.76 3100

37.19 1400 99.54 3200

40.36 1500 103.3 3300

43.55 1600 107.2 3400

46.78 1700 111.1 3500

50.05 1800 115.0 3600

53.35 1900 115.0 3600

56.67 2000 115.0 3600



In the calculation, = = 𝑖 𝑖 𝑖 ℎ 𝑖 𝑎 𝑖 Ω ⋅ . 𝑖 𝑖 𝑖 = . 𝑉𝐼 − .𝑅

= . Ω ⋅ (This is the resistivity of tungsten at room temperature.) 𝑉 = 𝑉 𝑎 , ℎ = , ℎ 𝑅 = . Ω (This is measured with a multimeter when the bulb filament is at

room temperature.)

GENERAL PHYSICS 1 LABORATORY - hcmiu.edu.vn guide.pdf · ... 6 4. EXPERIMENT ... International...

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