Physics Laboratory
Manual
Third Edition
David H. Loyd
Angelo State University
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Physics Laboratory Manual, Third Edition
David H. Loyd
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Contents
For each laboratory listed below the symbol
preceding the laboratory means that lab requires a calculation
preceding the laboratory
of the mean and standard deviation of some repeated measurement. The symbol
means that the laboratory requires a linear least squares fit to two variables that are presumed to be linear. The
symbol WWW preceding the laboratory indicates a computer-assisted laboratory available to purchasers of this
manual at www.thomsonedu.com/physics/loyd
Preface
xi
Acknowledgements
xiii
General Laboratory Information
1
Purpose of laboratory, measurement process, significant figures, accuracy and precision,
systematic and random errors, mean and standard error, propagation of errors, linear least
squares fits, percentage error and percentage difference, graphing
L A B O R AT O R Y
1
Measurement of Length
13
Measurement of the dimensions of a laboratory table to illustrate experimental uncertainty,
mean and standard error, propagation of errors
L A B O R AT O R Y
2
Measurement of Density
23
Measurement of the density of several metal cylinders, use of vernier calipers, propagation
of errors
L A B O R ATO R Y
3
Force Table and Vector Addition of Forces
33
Experimental determination of forces using a force table, graphical and analytical
theoretical solutions to the addition of forces
L A B O R AT O R Y
4
Uniformly Accelerated Motion
43
Analysis of displacement versus (time)2 to determine acceleration, experimental value for
acceleration due to gravity g
WWW
L A B O R AT O R Y
4A
Uniformly Accelerated Motion Using a Photogate
Measurement of velocity versus time using a photogate to determine acceleration for a cart
on an inclined plane
iii
iv
Contents
L A B O R AT O R Y
5
Uniformly Accelerated Motion on the Air Table
53
Analysis to determine the average velocity, instantaneous velocity, acceleration of a puck on
an air table, determination of acceleration due to gravity g
L A B O R AT O R Y
6
Kinematics in Two Dimensions on the Air Table
63
Analysis of x and y motion to determine acceleration in y direction, with motion in the
x direction essentially at constant velocity
L A B O R AT O R Y
7
Coefficient of Friction
73
Determination of static and kinetic coefficients of friction, independence of the normal
force, verification that s > k
WWW
L A B O R AT O R Y
7A
Coefficient of Friction Using a Force Sensor and a Motion Sensor
Measurement of coefficients of static and kinetic friction using a force sensor and a motion
sensor
L A B O R AT O R Y
8
Newton’s Second Law on the Air Table
85
Demonstration that F ¼ ma for a puck on an air table and determination of the frictional
force on the puck from linear analysis
L A B O R AT O R Y
9
Newton’s Second Law on the Atwood Machine
95
Demonstration that F ¼ ma for the masses on the Atwood machine and determination of the
frictional force on the pulley from linear analysis
L A B O R AT O R Y
10
Torques and Rotational Equilibrium of a Rigid Body
105
Determination of center of gravity, investigation of conditions for complete equilibrium,
determination of an unknown mass by torques
L A B O R AT O R Y
11
Conservation of Energy on the Air Table
117
Spring constant, spring potential energy, kinetic energy, conservation of total mechanical
energy (kinetic þ spring potential)
L A B O R AT O R Y
12
Conservation of Spring and Gravitational Potential Energy
127
Determination of spring potential energy, determination of gravitational potential energy,
conservation of spring and gravitational potential energy
WWW
L A B O R AT O R Y
12 A
Energy Variations of a Mass on a Spring Using a Motion Sensor
Determination of the kinetic, spring potential, and gravitational potential energies of a mass
oscillating on a spring using a motion sensor
Contents
L A B O R AT O R Y
13
The Ballistic Pendulum and Projectile Motion 137
Conservation of momentum in a collision, conservation of energy after the collision,
projectile initial velocity by free fall measurements
L A B O R ATO RY
14
Conservation of Momentum on the Air Track
149
One-dimensional conservation of momentum in collisions on a linear air track
WWW
L A B O R AT O R Y
14 A
Conservation of Momentum Using Motion Sensors
Investigation of change in momentum of two carts colliding on a linear track
L A B O R ATO RY
15
Conservation of Momentum on the Air Table
159
Vector conservation of momentum in two-dimensional collisions on an air table
L A B O R ATO RY
16
Centripetal Acceleration of an Object in Circular Motion
169
Relationship between the period T, mass M, speed v, and radius R of an object in circular
motion at constant speed
L A B O R AT O R Y
17
Moment of Inertia and Rotational Motion 179
Determination of the moment of inertia of a wheel from linear relationship between the
applied torque and the resulting angular acceleration
L A B O R ATO RY
18
Archimedes’ Principle
189
Determination of the specific gravity for objects that sink and float in water, determination
of the specific gravity of a liquid
L A B O R AT O R Y
19
The Pendulum—Approximate Simple Harmonic Motion
197
Dependence of the period T upon the mass M, length L, and angle y of the pendulum,
determination of the acceleration due to gravity g
L A B O R AT O R Y
20
Simple Harmonic Motion—Mass on a Spring
207
Determination of the spring constant k directly, indirect determination of k by the analysis
of the dependence of the period T on the mass M, demonstration that the period is
independent of the amplitude A
WWW
L A B O R AT O R Y
20A
Simple Harmonic Motion—Mass on a Spring Using a Motion Sensor
Observe position, velocity, and acceleration of mass on a spring and determine the
dependence of the period of motion on mass and amplitude
v
vi
Contents
L A B O R AT O R Y
21
Standing Waves on a String
217
Demonstration of the relationship between the string tension T, the wavelength l,
frequency f, and mass per unit length of the string r
L A B O R AT O R Y
22
Speed of Sound—Resonance Tube
225
Speed of sound using a tuning fork for resonances in a tube closed at one end
L A B O R AT O R Y
23
Specific Heat of Metals
235
Determination of the specific heat of several metals by calorimetry
L A B O R AT O R Y
24
Linear Thermal Expansion
243
Determination of the linear coefficient of thermal expansion for several metals by direct
measurement of their expansion when heated
L A B O R AT O R Y
25
The Ideal Gas Law
251
Demonstration of Boyle’s law and Charles’ law using a homemade apparatus constructed
from a plastic syringe
L A B O R AT O R Y
26
Equipotentials and Electric Fields
259
Mapping of equipotentials around charged conducting electrodes painted on resistive
paper, construction of electric field lines from the equipotentials, dependence of the electric
field on distance from a line of charge
L A B O R AT O R Y
27
Capacitance Measurement with a Ballistic Galvanometer
269
Ballistic galvanometer calibrated by known capacitors charged to known voltage, unknown
capacitors measured, series and parallel combinations of capacitance
L A B O R AT O R Y
28
Measurement of Electrical Resistance and Ohm’s Law
279
Relationship between voltage V, current I, and resistance R, dependence of resistance on
length and area, series and parallel combinations of resistance
L A B O R AT O R Y
29
Wheatstone Bridge
289
Demonstration of bridge principles, determination of unknown resistors, introduction to
the resistor color code
L A B O R AT O R Y
30
Bridge Measurement of Capacitance
299
Alternating current bridge used to determine unknown capacitance in terms of a known
capacitor, series and parallel combinations of capacitors
Contents
L A B O R AT O R Y
31
Voltmeters and Ammeters
307
Galvanometer characteristics, voltmeter and ammeter from galvanometer, and comparison
with standard voltmeter and ammeter
L A B O R AT O R Y
32
Potentiometer and Voltmeter Measurements of the emf of a Dry Cell
319
Principles of the potentiometer, comparison with voltmeter measurements, internal
resistance of a dry cell
L A B O R AT O R Y
33
The RC Time Constant
329
RC time constant using a voltmeter as the circuit resistance R, determination of an unknown
capacitance, determination of unknown resistance
WWW
L A B O R AT O R Y
33A
RC Time Constant with Positive Square Wave and Voltage Sensors
Determine the time constant, and time dependence of the voltages across the capacitor and
resistor in an RC circuit using voltage sensors
L A B O R ATO RY
34
Kirchhoff’s Rules
339
Illustration of Kirchhoff’s rules applied to a circuit with three unknown currents and to a
circuit with four unknown currents
L A B O R AT O R Y
35
Magnetic Induction of a Current Carrying Long Straight Wire
349
Induced emf in a coil as a measure of the B field from an alternating current in a long
straight wire, investigation of B field dependence on distance r from wire
WWW
L A B O R AT O R Y
35A
Magnetic Induction of a Solenoid
Determination of the magnitude of the axial B field as a function of position along the axis
using a magnetic field sensor
L A B O R AT O R Y
36
Alternating Current LR Circuits
359
Determination of the phase angle f, inductance L, and resistance r of an inductor
WWW
L A B O R AT O R Y
36A
Direct Current LR Circuits
Determination of the phase relationship between the circuit elements and the time constant
for an LR circuit
L A B O R AT O R Y
37
Alternating Current RC and LCR Circuits
369
Phase angle in an RC circuit, determination of unknown capacitor, phase angle
relationships in an LCR circuit
vii
viii
Contents
L A B O R AT O R Y
38
Oscilloscope Measurements
379
Introduction to the operation and theory of an oscilloscope
L A B O R AT O R Y
39
Joule Heating of a Resistor
391
Heat (calories) produced from electrical energy dissipated in a resistor (joules), comparison
with the expected ration of 4.186 joules/calorie
L A B O R AT O R Y
40
Reflection and Refraction with the Ray Box
401
Law of reflection, Snell’s law of refraction, focal properties of each
L A B O R AT O R Y
41
Focal Length of Lenses 413
Direct measurement of focal length of converging lenses, focal length of a converging lens
with converging lens in close contact
L A B O R AT O R Y
42
Diffraction Grating Measurement of the Wavelength of Light
421
Grating spacing from known wavelength, wavelengths from unknown heated gas,
wavelength of colors from continuous spectrum
WWW
L A B O R AT O R Y
42A
Single-Slit Diffraction and Double-Slit Interference of Light
Light sensor and motion sensor measurement of the intensity distribution of laser light for
both a single slit and a double slit
L A B O R AT O R Y
43
Bohr Theory of Hydrogen—The Rydberg Constant
431
Comparison of the measured wavelengths of the hydrogen spectrum with Bohr theory to
determine the Rydberg constant
WWW
L A B O R AT O R Y
43A
Light Intensity versus Distance with a Light Sensor
Investigate the dependence of light intensity versus distance from a light source using
a light sensor
L A B O R AT O R Y
44
Simulated Radioactive Decay Using Dice ‘‘Nuclei’’
441
Measurement of decay constant and half-life for simulated radioactive decay using 20-sided
dice as ‘‘nuclei’’
L A B O R AT O R Y
45
Geiger Counter Measurement of the Half-Life of
137
Ba
451
Geiger counter plateau, half-life from activity versus time measurements
Contents
L A B O R AT O R Y
46
Nuclear Counting Statistics
463
Distribution of series of counts around the mean, demonstration that
uncertainty in the count N
L A B O R AT O R Y
pﬃﬃﬃﬃ
N is a measure of the
47
Absorption of Beta and Gamma Rays
473
Comparison of absorption of beta and gamma radiation by different materials,
determination of the absorption coefficient for gamma rays
Appendix I 483
Appendix II
Appendix III
485
487
ix
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Preface
This laboratory manual is intended for use with a two-semester introductory physics course, either calculusbased or noncalculus-based. For the most part, the manual includes the standard laboratories that have been
used by many physics departments for years. However, in this edition there are available some laboratories
that use the newer computer-assisted data-taking equipment that has recently become popular. The major
change in the current addition is an attempt to be more concise in the Theory section of each laboratory to
include only what is required to prepare a student to take the needed measurements. As before, the
Instructor’s Manual gives examples of the best possible experimental results that are possible for the data for
each laboratory. Complete solutions to all portions of each laboratory are included. All of the laboratories are
written in the same format that is described below in the order in which the sections occur.
OBJECTIVES
Each laboratory has a brief description of what subject is to be investigated. The current list of objectives
has been condensed compared to the previous edition.
EQUIPMENT
Each laboratory contains a brief list of the equipment needed to perform the laboratory.
THEORY
COPYRIGHT ª 2008 Thomson Brooks/Cole
This section is intended to be a description of the theory underlying the laboratory to be performed,
particularly describing the variables to be measured and the quantities to be determined from the
measurements. In many cases, the theory has been shortened significantly compared to previous editions.
EXPERIMENTAL PROCEDURE
The procedure given is usually very detailed. It attempts to give very explicit instructions on how to
perform the measurements. The data tables provided include the units in which the measurements are to
be recorded. With few exceptions, SI units are used.
xi
xii
Preface
CALCULATIONS
Very detailed descriptions of the calculations to be performed are given. When practical, actual data are
recorded in a data table, and calculated quantities are recorded in a calculations table. This is the preferred
option because it emphasizes the distinction between measured quantities and quantities calculated from
the measured quantities. In some cases it is more practical to combine the two into a data and calculations
table. That has been done for some of the laboratories.
Whenever it is feasible, repeated measurements are performed, and the student is asked to determine the
mean and standard error of the measured quantities. For data that are expected to show a linear relationship
between two variables, a linear least squares fit to the data is required. Students are encouraged to do these
statistical calculations with a spreadsheet program such as Excel. It is also acceptable to do them on a
handheld calculator capable of performing them automatically. Use of the statistical calculations is included
in 35 of the 47 laboratories.
GRAPHS
Any graphs required are specifically described. All linear data are graphed and the least squares fit to the
data is shown on the graph along with the data.
PRE-LABORATORY
Each laboratory includes a pre-laboratory assignment that is based upon the laboratory description. We
intend to prepare students to perform the laboratory by having them answer a series of questions about
the theory and working numerical problems related to the calculations in the laboratory. The questions in
the pre-laboratory have been changed somewhat to include more conceptual questions about the theory
behind the laboratory. However, there remains an emphasis on preparing students for the quantitative
processes needed to perform the laboratory.
LABORATORY REPORT
The laboratory includes the data and calculations tables, a sample calculations section, and a list of
questions. Usually the questions are related to the actual data taken by the student. They attempt to
require the student to think critically about the significance of the data with respect to how well the data
can be said to verify the theoretical concepts that underlie the laboratory.
COMPUTER-ASSISTED LABORATORIES
The Table of Contents lists 10 laboratories, prefaced by a symbol WWW that use computer-assisted data
collection and analysis. DataStudio software and compatible sensors are to be used for these laboratories.
The laboratories are available to purchasers of this manual at www.thomsonedu.com/physics/loyd.
Options for including these computer-assisted laboratories in a customized version of the lab manual are
available through Thomson’s digital library, Textchoice. Visit www.textchoice.com or contact your local
Thomson representative.
CONTACT INFORMATION FOR AUTHOR
Please contact me at david.loyd@angelo.edu if you find any errors or have any suggestions for improvements in the laboratory manual. I will keep an updated list of errors and suggestions at the Thomson
website.
Acknowledgements
I wish to acknowledge the mutual exchange of ideas about laboratory instruction that occurred among
H. Ray Dawson, C. Varren Parker and myself for over 30 years at Angelo State University. I also thank the
following users of previous editions of the manual for helpful comments: (1) Charles Allen, Angelo State
University (2) William L. Basham, University of Texas at Permian Basin (3) Gerry Clarkson, Howard
Payne University (4) Carlos Delgado, College of Southern Nevada (5) Poovan Murgeson, San Diego City.
I am grateful to all the highly professional and talented people of Thomson Brooks/Cole for their
excellent work to improve this third edition of the laboratory manual. I especially want to acknowledge
the help and encouragement of Rebecca Heider and Chris Hall in this rather lengthy process. Their
comments and suggestions about the changes and additions that were needed were very beneficial.
I wish to thank the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S.,
and to Longman Group Ltd., London, for permission to reprint the table in Appendix I from their book
Statistical Tables for Biological, Agricultural and Medical Research. (6th edition, 1974)
I thank Melissa Vigil, Marquette University and Marllin Simon, Auburn University for conversations
we have had about laboratory instruction. I am particularly indebted to Marllin Simon for his permission
to use the procedures and other aspects from several of his laboratories that use computer assisted data
acquisition techniques.
My final and most important acknowledgement is to my wife of 47 years, Judy. Her encouragement
and help with proof-reading have been especially important during this project. Her good humor and
practical advice are always appreciated.
COPYRIGHT ª 2008 Thomson Brooks/Cole
David H. Loyd
xiii
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Physics Laboratory Manual
n
Loyd
L ABOR AT ORY
General Laboratory
Information
PURPOSE OF LABORATORY
The laboratory provides a unique opportunity to validate physical theories in a quantitative manner.
Laboratory experience demonstrates the limitations in the application of physical theories to real physical
situations. It teaches the role that experimental uncertainty plays in physical measurements and introduces
ways to minimize experimental uncertainty. In general, the purpose of these laboratory exercises is both to
demonstrate some physical principle and to teach techniques of careful measurement.
DATA-TAKING PROCEDURES
Original data should always be recorded directly in the data tables provided. Avoid the habit of recording
the original data on scratch sheets and transferring them to the data tables later.
When working in a group, all partners should contribute to the actual process of taking the measurements. If time and other considerations permit, each partner should perform a separate set of measurements as a check on the procedure. Each partner should record data separately even if only one set of data
is taken by the group.
SIGNIFICANT FIGURES
COPYRIGHT ª 2008 Thomson Brooks/Cole
The number of significant figures means the number of digits known in some number. The number of
significant figures does not necessarily equal the total digits in the number because zeros are used as place
keepers when digits are not known. For example, in the number 123 there are three significant figures. In
the number 1230, although there are four digits in the number, there are only three significant figures
because the zero is assumed to be merely keeping a place. Similarly, the numbers 0.123 and 0.0123 both
have only three significant figures. The rules for determining the number of significant figures in a
number are:
.
The most significant digit is the leftmost nonzero digit. In other words, zeros at the left are never
significant.
.
In numbers that contain no decimal point, the rightmost nonzero digit is the least significant digit.
.
In numbers that contain a decimal point, the rightmost digit is the least significant digit, regardless of
whether it is zero or nonzero.
.
The number of significant digits is found by counting the places from the most significant to the least
significant digit.
ª 2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work
covered by the copyright hereon may be reproduced or used in any form or by any meansçgraphic, electronic, or mechanical,including photocopying, recording, taping,web distribution, information
storage and retrievalsystems,or in any other mannerçwithout the written permission of the publisher.
1
2
Physics Laboratory Manual n Loyd
As an example, the numbers in the following list of numbers all have four significant figures. An
explanation for each is given.
.
3456: All four nonzero digits are significant.
.
135700: The two rightmost zeros are not significant because there is no decimal point.
.
0.003043: Zeros at the left are never significant.
.
0.01000: The zero at the left is not significant, but the three zeros at the right are significant because there
is a decimal point.
.
1030.: There is a decimal point, so all four numbers are significant.
.
1.057: Again, there is a decimal point, so all four are significant.
.
0.0002307: Zeros at the left are never significant.
READING MEASUREMENT SCALES
For the measurement of any physical quantity such as mass, length, time, temperature, voltage, or current,
some appropriate measuring device must be chosen. Despite the diverse nature of the devices used to
measure the various quantities, they all have in common a measurement scale, and that scale has a
smallest marked scale division. All measurements should be done in the following very specific manner.
All meters and measuring devices should be read by interpolating between the smallest marked scale
division. Generally the most sensible interpolation is to attempt to estimate 10 divisions between the
smallest marked scale division. Consider the section of a meter stick pictured in Figure 1 that shows the
region between 2 cm and 5 cm. The smallest marked scale divisions are 1 mm apart. The location of
the arrow in the figure is to be determined. It is clearly between 3.4 cm and 3.5 cm, and the correct
procedure is to estimate the final place. In this case a reading of 3.45 cm is estimated. For this measurement
the first two digits are certain, but the last digit is estimated. This measurement is said to contain three
significant figures. Much of the data taken in this laboratory will have three significant figures, but
occasionally data may contain four or even five significant figures.
MISTAKES OR PERSONAL ERRORS
All measurements are subject to errors. There are three types of errors, which are classified as personal,
systematic, or random. Random errors are sometimes called statistical errors. This section deals with
personal errors. Systematic and random errors will be discussed later. In fact, personal errors are not
really errors in the same sense as the other two types of errors. Instead, they are merely mistakes made by
the experimenter. Mistakes are fundamentally different from the other two types of errors because
mistakes can be completely eliminated if the experimenter is careful. Mistakes can be made either while
taking the data or later in calculations done with the original data. Either type of mistake is bad, but a
mistake made in the data-taking process is probably worse because often it is not discovered until it is too
late to correct it.
The correct attitude toward all data-taking processes is one of skepticism about all the procedures that
are carried out in the laboratory. Essentially, this amounts to assuming that things will go wrong unless
2
Figure 1
3
4
5
Laboratory n General Laboratory Information
3
constant attention is given to making sure that no mistakes are made. For every measurement taken, all
aspects of the process must be checked and rechecked. Everyone in the group must be convinced that they
know exactly what is supposed to be measured, what the correct procedure is to measure it, and that the
group is making no mistakes in carrying out that procedure.
ACCURACY AND PRECISION
The central point to experimental physical science is the measurement of physical quantities. It is assumed
that there exists a true value for any physical quantity, and the measurement process is an attempt to
discover that true value. It is expected that there will be some difference between the true value and the
measured value. The terms accuracy and precision are used to describe different aspects of the difference
between the measured value and the true value of some quantity.
The accuracy of a measurement is determined by how close the result of the measurement is to the true
value. For example, in several of the experiments, we will determine a value for the acceleration due to
gravity. For this case, the accuracy of the result is decided by how close it is to the true value of 9.80 m/s2.
For many laboratory experiments, the true value of the measured quantity is not known, and we cannot
determine the accuracy of the experiment from the available data.
The precision of a measurement refers essentially to how many digits in the result are significant. It
indicates also how reproducible the results are when measurements of some quantity are repeated. The
smaller the variations of the individual repeated measurements of a quantity, the more precise the quoted
value of the measurement is considered to be. We will elaborate upon and quantify this idea about the
relationship between the size of the variations in the measurements and the precision of the measurement
in a later section on statistical methods.
COPYRIGHT ª 2008 Thomson Brooks/Cole
SYSTEMATIC ERRORS
Systematic errors are errors that tend to be in the same direction for repeated measurements, giving
results that are either consistently above the true value or consistently below the true value. In many cases
such errors are caused by some flaw in the experimental apparatus. For example, a voltmeter could be
incorrectly calibrated in such a way that it consistently gives a reading that is 80% of the true voltage
across its input terminals. It is also possible to have a voltmeter with a zero offset on its scale, which is
assumed for this discussion to be 0.50 volts. In the first case, the error is a constant fraction of the true
value (in this case, 20%), and in the second case, the error is a constant absolute voltage. Either of these is a
systematic error, and the answer to the question of which one is worse depends upon the magnitude of the
voltage to be measured. If the voltage to be measured is l.00 volts, then the meter with absolute error of
0.50 volts causes an error of 50%, whereas the meter with relative error causes an error of 20%. On the
other hand, if the voltage to be measured is l00 volts, the meter with absolute error of 0.50 volts causes
only a 0.5% error, and the other meter still causes a 20% error, or in this case, 20 volts. If this measured
voltage is used to calculate some other quantity, it too will show a systematic error in the results.
A second common type of systematic error is failure to consider all of the variables that are important
in the experiment. In some cases one may be aware that some other factors need to be considered, but
might not have the ability to do so quantitatively. For example, when using an air table to validate
Newton’s Laws, it is common to ignore friction. This is done because friction is assumed to be small, but
also because often there is no easy way to determine its contribution. It is expected, therefore, that
neglecting friction might introduce a systematic error.
For purposes of this laboratory, the concern with systematic errors will usually be twofold—to
attempt to eliminate any obvious systematic errors to the extent possible, and to attempt to identify any
data that show systematic error, and suggest possible reasonable causes for such error.
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Physics Laboratory Manual n Loyd
RANDOM ERRORS
The final class of errors is those that are produced by unpredictable and unknown variations in the total
experimental process even when one does the experiment as carefully as is humanly possible. The
variations caused by an observer’s inability to estimate the last digit the same way every time will
definitely be one contribution. Other variations can be caused by fluctuations in line voltage, temperature
changes, mechanical vibrations, or any of the many physical variations that may be inherent in the
equipment or any other aspect of the measurement process. It is important to realize the following
difference between random errors and personal and systematic errors. In principle all personal and
systematic errors can be eliminated, but there will always remain some random errors in any
measurement. Even in principle the random errors can never be completely eliminated.
Random errors, on the other hand, can be determined in a prescribed way. It has been found
empirically that random errors often are distributed according to a particular statistical distribution
function called the Gauss distribution function, which is also referred to as the normal error function.
Random measurement errors are said to be normally distributed when a histogram of the frequency
distribution of the results of a large number of repeated measurements produces a bell-shaped curve with
a peak at the mean of the measurements. The histogram of the frequency distribution is simply a graph of
the number of times the measurements fall within a certain range versus the measured values.
MEAN AND STANDARD DEVIATION
Assume a series of repeated measurements is made in which there are no systematic or personal errors,
and thus only random errors are present. Assume that there are n measurements made of some quantity x,
and the ith value obtained is xi where i varies from 1 to n. If it is true that the errors are normally
distributed, statistical theory says that the mean is the best approximation to the true value. In formal
mathematical terms, the mean (which has a symbol of x ) is given by the equation
X
1 n
x¼
xi
n 1
ðEq: 1Þ
For example, assume that four measurements are made of some quantity x, and that the four results
are 18.6, 19.3, 17.7, and 20.4. Equation 1 is simply shorthand notation for the averaging process given by
x ¼ ð1=4Þ ð18:6 þ 19:3 þ 17:7 þ 20:4Þ ¼ 19:0
ðEq: 2Þ
It is not surprising that the mean is the best approximation to the true value. It seems intuitively
reasonable. We can prove mathematically that the mean is indeed the proper choice by something called
the principle of least squares, which we state in the following way. The most probable value for some
quantity determined from a series of measurements is that value that minimizes the sum of the squares of
the deviations between the chosen value and the measured values. We can demonstrate that the proper
choice to produce this minimum sum of deviations is simply the mean of the measurements. This idea can
be usefully generalized later for the case of two variables.
Statistical theory, furthermore, states that the precision of the measurement can be determined by
calculating a quantity called the standard deviation from the mean of the measurements. The symbol for
standard deviation from the mean is sn1, and it is defined by the equation
sn1
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
1 X
¼
½xi x2
n1 1
ðEq: 3Þ
Laboratory n General Laboratory Information
5
For the data given, the standard deviation is calculated from Equation 3 to be the following:
sn1
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
¼
ðð18:6 19:0Þ2 þ ð19:3 19:0Þ2 þ ð17:7 19:0Þ2 þ ð20:4 19:0Þ2 Þ ¼ 1:1
41
The quantity sn1, which is actually called the sample standard deviation, is a measure of the
precision of the measurement in the following statistical sense. It gives the probability that the
measurements fall within a certain range of the measured mean. From the sample standard deviation and
tables of the standard error function, it is possible to determine the probability that the measurements fall
within any desired range about the mean. The common range to be quoted is the range of one standard
deviation as calculated by Equation 3.
Probability theory states that approximately 68.3% of all repeated measurements should fall within a
range of plus or minus sn1 from the mean. Furthermore, 95.5% of all repeated measurements should
fall within a range of 2sn1 from the mean. For the example given above, 68.3% should fall in the range
19.0 1.1 (from 17.9 to 20.1), and 95.5% should fall in the range 19.0 2.2 (from 16.8 to 21.2).
As a final note on the expected distribution for measurements that follow a normal error curve, 99.73%
of all measurements should fall within 3sn1 of the mean. This implies that if one of the measurements is
3sn1 or farther from the mean, it is very unlikely that it is a random error. It is much more likely to be the
result of a personal error.
A second issue that can be addressed by these repeated measurements is the precision of the mean.
After all, this is what is really of concern, because the mean is the best estimate of the true value. The
precision of the mean is indicated by a quantity called the standard error. The standard error, which has a
symbol of a, is defined by
COPYRIGHT ª 2008 Thomson Brooks/Cole
sn1
a ¼ pﬃﬃﬃ
n
ðEq: 4Þ
For the example given above with sn1 ¼ 1.1 and n ¼ 4, the value is a ¼ 0.55. The significance of a is
that if several groups of n measurements are made, each producing a value for the mean, 68.3% of the
means should fall in the range 19.0 0.6. In other words, there is a 68.3% probability that the true value
lies in this range. Of course, all these statements are valid only if there are no other errors present other
than random errors.
In this laboratory, students will often be asked to make repeated measurements of some quantity and
to determine the mean. Assuming that a represents the uncertainty in the value of the mean, a crucial
question is the appropriate number of significant figures to retain in a. In this laboratory, the convention to
be followed is to retain one significant figure in a and to make the least significant figure in the mean be in
the same decimal place as a. In this context the appropriate procedure is to originally calculate the mean and
sn1 to more significant figures than it is assumed are needed, and then allow the value of a to determine
the significant figures to be retained in the mean. In the example given above, the result should be stated
as 19.0 0.6. Notice that as described above, only one significant figure has been retained in a, and the
mean has its least significant digit in the same decimal place as a.
To illustrate how the concepts of the mean and standard error apply to accuracy and precision,
consider the following sets of three measurements of the acceleration due to gravity made by four students
named Alf, Beth, Carl, and Dee. The results for each measurement, the means, the sample standard
deviations sn1, and the standard errors a are given for each student.
The accuracy of each student’s data is determined by comparing the mean with the true value of 9.80.
Dee’s value of 9.76 is the most accurate, Alf’s value of 9.43 is second, Beth’s value of 9.26 is third, and Carl’s
value of 8.74 is the least accurate. Using the values of the standard errors of the mean as a criterion for
precision, Carl’s value is the most precise, Dee’s is second, Beth’s is third, and Alf’s value is the least precise.
In fact, the situation is not quite so simple as has been presented. There is an interplay between the
concepts of accuracy and precision that we must consider. If a measurement appears to be very accurate,
6
Physics Laboratory Manual n Loyd
Table 1
Measurement 1
Measurement 2
Measurement 3
Mean
Standard Deviation
Standard Error
Alf
Beth
Carl
Dee
7.83
11.61
8.85
9.43
1.96
1
9.53
9.38
8.87
9.26
0.35
0.2
8.70
8.75
8.77
8.74
0.036
0.02
9.72
9.86
9.70
9.76
0.087
0.05
but the precision is poor, we do not know if the results are meaningful. Consider Alf’s mean of 9.43, which
differs from the true value of 9.80 by only 0.37, and thus appears to be quite accurate. However, all of his
measurements have large deviations from the true value, and his standard error is very large. It seems much
more likely, then, that Alf’s mean of 9.43 is due to luck rather than to a careful measurement. In contrast, it
seems likely that Dee’s mean of 9.76 is meaningful because the value of her standard error is small.
Carl’s results are an example of a situation that is common in the interplay between accuracy and
precision. Carl’s precision is extremely high, yet his accuracy is not very good. When a measurement has
high precision but poor accuracy, it is often the sign of a systematic error, and in this case it seems very likely
that Carl has some systematic error in his measurements.
PROPAGATION OF ERRORS
Consider the following set of data that was taken by measuring the coordinate position d of some object as a
function of time t.
Table 2
d (m)
t (s)
7.57
11.97
16.58
21.00
25.49
1.00
2.00
3.00
4.00
5.00
From these data the average speed over each time interval can be calculated. The average speed n over
some time interval Dt during which a distance interval Dd was traveled is given by
n¼
Dd
Dt
ðEq: 5Þ
For the data of Table 2 there are four intervals for the five data points, and for the first two intervals the
results are:
n1 ¼
11:97 7:57
16:58 11:97
¼ 4:40 m=s n2 ¼
¼ 4:61 m=s
2:00 1:00
3:00 2:00
The other two intervals give average speeds of 4.42 m/s and 4.49 m/s. A basic question is on what basis
was the decision made to express n1 ; for example, as 4.40 rather than 4.4 or 4.400? We derive the answer by
Laboratory n General Laboratory Information
7
further extending the rules for significant figures to include calculations. Use the following rules to
determine the number of significant figures to retain at the end of a calculation:
.
When adding or subtracting, figures to the right of the last column in which all figures are significant
should be dropped.
.
When multiplying or dividing, retain only as many significant figures in the result as are contained in
the least precise quantity in the calculation.
.
The last significant figure is increased by 1 if the figure beyond it (which is dropped) is 5 or greater.
These rules apply only to the determination of the number of significant figures in the final result.
In the intermediate steps of a calculation, one more significant figure should be kept than is kept in the
final result.
Consider these examples of addition, multiplication, and division of numbers:
753:1
37:08
0:697
56:3
847:177
753:1
37:1
0:7
56:3
847:2
327:23
36:73
12019:158
8:90906
36:73 327:23
Following the above rules for addition strictly implies rewriting each number as shown in the second
addition where the first digit beyond the decimal is the least significant digit. This is true because that
column is the rightmost column in which all digits are significant. Note that one gets the same result if the
numbers are added on the calculator (as done at the left), and then it is noted that the first digit beyond the
decimal is the last one that can be kept. Therefore 847.177 is rounded off to 847.2. A similar process is used
for multiplication and division, as shown in the third and fourth part above. In each case, the result is
rounded to four significant figures because the least significant number in each calculation (36.73) has
only four significant figures. For the multiplication the result is 12020, and for the division it is 8.909.
LINEAR LEAST SQUARES FITS
Often measurements are taken by changing one variable (call it x) and measuring how a second variable
(call it y) changes as a function of the first variable. In many cases of interest it is assumed that there exists
a linear relationship between the two variables. In mathematical terms one can say that the variables obey
an equation of the form
COPYRIGHT ª 2008 Thomson Brooks/Cole
y ¼ mx þ b
ðEq: 6Þ
where m and b are constants. This also implies that if a graph is made with x as the horizontal axis and y as
the vertical axis, it will be a straight line with m equal to the slope (defined as Dy/Dx) and b equal to the y
intercept (the value of y at x ¼ 0).
The question is how to best verify that the data do indeed obey Equation 6. One way is to make a
graph of the data, and then try to draw the best straight line possible through the data points. This will
give a qualitative answer to the question, but it is possible to give a quantitative answer to the question by
the process described below.
The measurements are repeated measurements in the sense that they are to be considered together in
the attempt to determine to what extent the data obey Equation 6. It is possible to generalize the idea of
minimizing the sum of squares of the deviations described earlier for the mean and standard deviation to
the present case. The result of the generalization to two-variable linear data is called a linear least squares
fit to the data. It is also sometimes referred to as a linear regression.
8
Physics Laboratory Manual n Loyd
The aim of the process is to determine the values of m and b that produce the best straight-line fit to the
data. Any choice of values for m and b will produce a straight line, with values of y determined by the
choice of x. For any such straight line (determined by a given m and b) there will be a deviation between
each of the measured y’s and the y’s from the straight-line fit at the value of the measured x’s. The least
squares fit is that m and b for which the sum of the squares of these deviations is a minimum. Statistical
theory states that the appropriate values of m and b that will produce this minimum sum of squares of the
deviations are given by the following equations:
n
m¼
n
X
n
X
xi yi
1
!
xi
1
n
X
x2i
n
n
X
1
n
X
b¼
!
yi
n
X
n
X
1
ðEq: 7Þ
xi
!
x2i
n
X
!
xi yi
1
n
X
1
yi
1
!2
1
1
n
!
x2i
n
X
!2
n
X
1
!
xi
ðEq: 8Þ
xi
1
Refer again to the data of Table 2 for coordinate position versus time. The question to be answered is
whether or not the data are consistent with constant velocity. If the speed v is constant, the data can be fit
by an equation of the form
d ¼ nt þ do
ðEq: 9Þ
Equation 9 is of the form of Equation 6 with d corresponding to y, t corresponding to x, v corresponding
to m, and do corresponding to b. Thus v will be the slope of a graph of d versus t, and do will be the intercept,
which is the coordinate position at the arbitrarily chosen time t ¼ 0.
Calculating some of the individual terms gives:
P
ti ¼ 1:00 þ 2:00 þ 3:00 þ 4:00 þ 5:00 ¼ 15:00
P
ðti Þ2 ¼ ð1:00Þ2 þð2:00Þ2 þð3:00Þ2 þð4:00Þ2 þð5:00Þ2 ¼ 55:00
P
di ¼ 7:57 þ 11:97 þ 16:58 þ 21:00 þ 25:49 ¼ 82:61
P
ti di ¼ ð1:00Þð7:57Þ þ ð2:00Þð11:97Þ þ ð3:00Þð16:58Þ þ ð4:00Þð21:00Þ þ ð5:00Þð25:49Þ ¼ 292:70
P
ðdi Þ2 ¼ ð7:57Þ2 þð11:97Þ2 þð16:58Þ2 þð21:00Þ2 þð25:49Þ2 ¼ 1566:22
Using these values in Equations 7 and 8 with the appropriate correspondence of variables gives
v ¼ 4.49 and do ¼ 3.06. Thus the velocity is determined to be 4.49 m/s, and the coordinate at t ¼ 0 is found to
be 3.06 m.
At this point, the best possible straight-line fit to the data has been determined by the least squares fit
process. A second goal remains, to determine how well the data actually fit the straight line that we have
obtained. Again, we derive a qualitative answer to this question by making a graph of the data and the
straight line and qualitatively judging the agreement between the line and the data.
There is, however, a quantitative measure of how well the data follow the straight line obtained by the
least squares fit. It is given by the value of a quantity called the correlation coefficient, r. This quantity is a
measure of the fit of the data to a straight line with r ¼ 1.000 exactly signifying a perfect correlation, and
r ¼ 0 signifying no correlation at all. The equation to calculate r in terms of the general variables x and y is
given by
Laboratory n General Laboratory Information
n
n
X
xi yi
1
n
X
!
xi
1
n
X
!
yi
1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
r ¼ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
!2ﬃv
!2ﬃ
u n
u n
n
n
u X
u X
X
X
tn
x2i
xi tn
y2i
yi
1
1
1
9
ðEq: 10Þ
1
Making the substitutions for the variables of the problem of the fit to the displacement versus time by
substituting t for x and d for y in the above equation and using the appropriate numerical values calculated
earlier gives r ¼ 0.99998. Thus the data show an almost perfect linear relationship because r is so close to
1.000. In calculations of r keep either three significant figures, or else enough until the last place is not a 9.
When performing a least squares fit to data, particularly when a small number of data points are
involved, there is some tendency to obtain a surprisingly good value for r even for data that do not
appear to be very linear. For those cases, we can determine the significance of a given value of r by
comparing the obtained value of r with the probability that that value of r would be obtained for n values
of two variables that are unrelated. A table for such comparisons is given in Appendix I in a table entitled
Correlation Coefficients.
STATISTICAL CALCULATIONS
A very high percentage of the laboratories in this course will involve two variables that are linearly
related. These cases usually will require a least squares fit to the data. Although the least squares fit
calculations and mean and standard deviation calculations are not difficult in principle, they are tedious
and time-consuming. The use of a spreadsheet computer program such as Excel is highly recommended.
As an alternative, many handheld calculators have automatic routines built in that allow the calculation of
these quantities simply by inputting the data points one after another. Note that most calculators will
calculate two different standard deviations. The one needed is usually denoted sn1, and it is the sample
standard deviation. Also available on most calculators is a quantity that is usually denoted as sn. It applies
to the case when the population is known, and it will never be appropriate for data taken in this
laboratory. Always be sure to choose the quantity sn1, which is the one defined by Equation 3.
PERCENTAGE ERROR AND PERCENTAGE DIFFERENCE
COPYRIGHT ª 2008 Thomson Brooks/Cole
In several of the laboratory exercises, the true value of the quantity being measured will be considered to
be known. In those cases, the accuracy of the experiment will be determined by comparing the
experimental result with the known value. Normally this will be done by calculating the percentage error
of your measurement compared to the given known value. If E stands for the experimental value, and K
stands for the known value, then the percentage error is given by
Percentage error ¼
jE Kj
100%
K
ðEq: 11Þ
In other cases we will measure a given quantity by two different methods. There will then be two
different experimental values, E1 and E2, but the true value may not be known. For this case, we will
calculate the percentage difference between the two experimental values. Note that this tells nothing
about the accuracy of the experiment, but will be a measure of the precision. The percentage difference
between the two measurements is defined as
Percentage difference ¼
jE2 E1 j
100%
½E1 þ E2 =2
ðEq: 12Þ
10
Physics Laboratory Manual n Loyd
PREPARING GRAPHS
It is helpful to represent data in the form of a graph when interpreting the overall trend of the data. Most
of the graphs for this laboratory will use rectangular Cartesian coordinates. Note that it is customary to
denote the horizontal axis as x and the vertical axis as y when developing general equations, as was done
in the development of the equations for a linear least squares fit. However, any two variables can be
plotted against each other.
When preparing a graph, first choose a scale for each of the axes. It is not necessary to choose the same
scale for both axes. In fact, rarely will it be convenient to have the same scale for both axes. Instead, choose
the scale for each axis so that the graph will range over as much of the graph paper as possible, consistent
with a convenient scale. Choose scales that have the smallest divisions of the graph paper equal to
multiples of 2, 5, or 10 units. This makes it much easier to interpolate between the divisions to locate the
data points when graphing.
The student is expected to bring to each laboratory a supply of good quality linear graph paper.
A very good grade of centimeter by centimeter graph paper with one division per millimeter is the best
choice. Do not, for example, ever use 1/4 inch by 1/4 inch sketch paper or other such coarse scaled paper as
graph paper. In some cases special graph paper like semilog or log-log graph paper may be required.
Figure 2 is a graph of the data for displacement versus time from Table 2 for which the least squares fit
was previously made. Note that scales for each axis have been chosen, to spread the graph over a reasonable
portion of the page. Also note that because the data have been assumed linear, a straight line has been
drawn through the data points. The straight line is the one obtained from the least squares fit to the data.
For most experiments, the variables will take on only positive values. For that case the scales should
range from zero to greater than the largest value for any data point. For example, in Figure 2 the
displacement is chosen to range from 0 to 30 meters because the largest displacement is 25.49, and the time
scale has been chosen to range from 0 to 6 seconds because the largest time is 5.00 seconds. Also note that
the scales should not be suppressed as a means to stretch out the graph. For example, if a set of data
contains ordinates that range from 60 to 90, do not choose a scale that shows only that range. Instead a
scale from 0 to 100 should be chosen, and there is nothing that can be done in that case to make the graph
range over more than about 30% of the graph paper. Scales should always be chosen to increase to the
right of the origin and to increase above the origin.
Graph of Displacement Versus Time
30
Displacement (m)
25
20
15
10
5
0
0
Figure 2
1
2
3
Time (s)
4
5
6
Laboratory n General Laboratory Information
11
COPYRIGHT ª 2008 Thomson Brooks/Cole
Graphs should always have the scales labeled with the name and units of each variable along each
axis. Major scale divisions should be labeled with the appropriate numbers defining the scale. Always
include a title for each graph, keeping in mind that it is customary to state the vertical axis versus the
horizontal axis.
All graphs should be plotted as points with no attempt to connect the data with a smooth curve. Do
not write the coordinates on the graph next to the data point, as is common practice in mathematics
classes. The only time it is appropriate to draw any continuous line to represent the trend of the data is
when it is assumed that the mathematical form of the data is known. In practice, the only time this will be
true will be when linearity is assumed, and in that case, it is appropriate to draw the straight line that has
been obtained by the least squares fitting procedure.
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Physics Laboratory Manual
n
Loyd
L ABOR AT ORY
1
Measurement of Length
OBJECTIVES
o Demonstrate the speciﬁc knowledge gained by repeated measurements of the length and width
of a table.
o Apply the statistical concepts of mean, standard deviation from the mean, and standard error to
these measurements.
o Demonstrate propagation of errors by determining the uncertainty in the area calculated from
the measured length and width.
EQUIPMENT LIST
.
2-meter stick
.
Laboratory table
THEORY
In this laboratory it is assumed that the uncertainty in the measurement of the length and width of the
table is due to random errors. If this assumption is valid, then the mean of a series of repeated measurements represents the most probable value for the length or width.
Consider the general case in which n measurements of the length and width of the table are made.
We will make 10 measurements, so n = 10 for this case, but we will develop equations for the case in which
n can be any chosen value. If Li and Wi stand for the individual measurements of the length and width,
and L and W stand for the mean of those measurements, the equations relating them are
COPYRIGHT ª 2008 Thomson Brooks/Cole
L¼
n
1 X
Li
n i
W¼
n
1 X
Wi
n i
ðEq: 1Þ
We get information about the precision of the measurement from the variations of the individual
measurements using the statistical concept of the standard deviation. The values of the standard
deviation from the mean for the length and width of the table, sLn1 and sW
n1 ; are given by the equations:
sLn1
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
1 X
¼
ðLi LÞ2
n1 1
sW
n1
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
1 X
¼
ðWi WÞ2
n1 1
ðEq: 2Þ
ª 2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work
covered by the copyright hereon may be reproduced or used in any form or by any meansçgraphic, electronic, or mechanical,including photocopying, recording, taping,web distribution, information
storage and retrievalsystems,or in any other mannerçwithout the written permission of the publisher.
13
14
Physics Laboratory Manual n Loyd
If errors are only random, it should be true that approximately 68.3% of the measurements of length
should fall in the range L sLn1 ; and that approximately 68.3% of the measurements of width should fall
within the range W sW
n1 : Furthermore, 95.5% of the measurements of both length and width should
fall within 2 sn–1 of the mean, and 99.73% should fall within 3 sn–1 of the mean.
The precision of the mean for L and W is given by quantities called the standard error, aL and aW.
These quantities are defined by the following equations:
sLn1
ﬃﬃﬃ
aL ¼ p
n
sW
n1
ﬃﬃﬃ
aW ¼ p
n
ðEq: 3Þ
The meaning of aL and aW is that, if the errors are only random, there is a 68.3% chance that the true value
of the length lies within the range L aL and the true value of the width lies within the range W aW .
An important problem in experimental physics is to determine the uncertainty in some quantity
that is derived by calculations from other directly measured quantities. For this experiment, consider the
area A of the table as calculated from the measured values of the length and width L and W by the
following:
A¼L W
ðEq: 4Þ
For the case of an area that is the product of two measured quantities, the uncertainty in the area is related
to the uncertainty of the length and width by:
aA ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðLÞ2 ðaW Þ2 þ ðWÞ2 ðaL Þ2
ðEq: 5Þ
EXPERIMENTAL PROCEDURE
1. Place the 2-meter stick along the length of the table near the middle of the width and parallel to one
edge of the length. Do not attempt to line up either edge of the table with one end of the meter stick or
with any certain mark on the meter stick.
2. Let X stand for the coordinate position in the length direction. Read the scale on the 2-meter stick that
is aligned with one end of the table and record that measurement in Data and Calculations Table 1 as
X1. Read the scale that is aligned at the other end of the table and record that measurement in Data and
Calculations Table 1 as X2. A 35 note card held next to the edge of the table may help to determine
where the 2-meter stick is aligned with the table for each measurement. Note that the stick has
1 millimeter as the smallest marked scale division. Therefore, each coordinate should be estimated to the
nearest 0.1 millimeter (nearest 0.0001 m).
3. Repeat Steps 1 and 2 nine more times for a total of 10 measurements of the length of the table. For each
measurement place the 2-meter stick on the table with no attempt to align either end of the stick or any
particular mark on the stick with either end of the table. Make the measurements at 10 different
places along the width of the table so that any variation in the length of the table is included in the
measurements.
4. Perform Steps 1 through 3 for 10 measurements of the width of the table. Let the coordinate for the
width be given by Y and record the 10 values of Y1 and Y2 in Data and Calculations Table 2. Again
place the stick along the different lines each time, but make no attempt to align any particular mark on
the stick with either edge of the table.
Laboratory 1 n Measurement of Length
15
CALCULATIONS
1. After all measurements are completed, perform the subtractions of the coordinate positions to
determine the 10 values of the length Li, and the 10 values of width Wi. Record the 10 values of Li and
Wi in the appropriate table.
2. Use Equations 1 to calculate the mean length L and the mean width W and record their values in the
appropriate table. Keep five decimal places in these results. For example, typical values might be
L ¼ 1:37157 m and W ¼ 0:76384 m.
3. For each measurement of length and width, calculate the values of Li L and Wi W and record them
in the appropriate table. Then for each value of the length and width, calculate and record the values
of ðLi LÞ2 and ðWi WÞ2 in the appropriate table.
4. Perform the summations of the values of ðLi LÞ2 and the summations of the values of ðWi WÞ2 and
record them in the appropriate box in the tables.
5. Use the values of the summations of ðLi LÞ2 and of ðWi WÞ2 in Equations 2 to calculate the values of
sLn1 and sW
n1 and record them in the appropriate table.
W
6. Calculate L sLn1 ; L þ sLn1 ; W sW
n1 ; and W þ sn1 and record the values in the appropriate table.
7. Use the values of sLn1 and sW
n1 in Equations 3 to calculate the values of aL and aW and record them in
the appropriate table.
COPYRIGHT ª 2008 Thomson Brooks/Cole
8. Use the values of L and W in Equation 4 to calculate the value of A, the area of the table, and record it
in the appropriate table. Use Equation 5 to calculate the value of aA and record it in the appropriate
table.
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Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Measurement of Length
L A B O R A T O R Y 1
PRE-LABORATORY ASSIGNMENT
1. State the number of significant figures in each of the following numbers and explain your answer.
(a) 37.60__________
(b) 0.0130__________
(c) 13000__________
(d) 1.3400__________
2. Perform the indicated operations to the correct number of significant figures using the rules for
significant figures.
(a)
37:60
1:23
3:765
(c) þ 1:2
þ 37:21
(b) 6:7 j 8:975
COPYRIGHT ª 2008 Thomson Brooks/Cole
3– 6. Three students named Abe, Barb, and Cal make measurements (in m) of the length of a table
using a meter stick. Each student’s measurements are tabulated in the table below along with the
mean, the standard deviation from the mean, and the standard error of the measurements.
Student
L1
L2
L3
L4
L
sn–1
a
Abe
1.4717
1.4711
1.4722
1.4715
1.4716
0.00046
0.0002
Barb
1.4753
1.4759
1.4756
1.4749
1.4754
0.00043
0.0002
Cal
1.4719
1.4723
1.4727
1.4705
1.4719
0.00096
0.0005
Note that in each case only one significant figure is kept in the standard error a, and this determines
the number of significant figures in the mean. The actual length of the table is determined by very
sophisticated laser measurement techniques to be 1.4715 m.
17
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Physics Laboratory Manual n Loyd
3. State how one determines the accuracy of a measurement. Apply your idea to the measurements
of the three students above and state which of the students has the most accurate measurement.
Why is that your conclusion?
4. Apply Equations 1, 2, and 3 to calculate the mean, standard deviation, and standard error for Abe’s
measurements of length. Confirm that your calculated values are the same as those in the table.
Show your calculations explicitly.
5. State the characteristics of data that indicate a systematic error. Do any of the three students have
data that suggest the possibility of a systematic error? If so, state which student it is, and state how
the data indicate your conclusion.
6. Which student has the best measurement considering both accuracy and precision? State clearly
what the characteristics are of the student’s data on which your answer is based.
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Lab Partners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
L A B O R A T O R Y 1
Measurement of Length
LABORATORY REPORT
Data and Calculations Table 1 (nearest 0.0001 m, which is 0.1 mm)
Trial
X1 (m)
X2 (m)
Li L (m)
Li = X2 – X1 (m)
n
X
ðLi LÞ2 (m2 )
ðLi LÞ2 ¼
COPYRIGHT ª 2008 Thomson Brooks/Cole
1
L¼
sLn1 ¼
L sLn1 ¼
L þ sLn1 ¼
aL ¼
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Physics Laboratory Manual n Loyd
Data and Calculations Table 2 (nearest 0.0001 m, which is 0.1 mm)
Trial
Y1 (m)
Y2 (m)
Wi W (m)
Wi = Y2 – Y1 (m)
n
X
ðWi WÞ2 (m2 )
ðWi WÞ2 ¼
1
W¼
sW
n1 ¼
W sW
n1 ¼
A¼LW ¼
SAMPLE CALCULATIONS
1. L1 ¼ X21 X11 ¼
2. W1 ¼ Y12 Y11 ¼
10
1 X
Li ¼
3. L ¼
10 1
10
1 X
4. W ¼
Wi ¼
10 1
5. L1 L ¼
6. ðL1 LÞ2 ¼
7. W1 W ¼
8. ðW1 WÞ2 ¼
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
1 X
ðLi LÞ2
9. sLn1 ¼
n1 1
10. L sLn1 ¼
11. L þ sLn1 ¼
W þ sW
n1 ¼
sA ¼
aW ¼
Laboratory 1 n Measurement of Length
12.
sW
n1
21
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
1 X
¼
ðWi WÞ2
n1 1
13. W sW
n1 ¼
14. W þ sW
n1 ¼
15. A ¼ L W ¼
16. sA =
QUESTIONS
1. According to statistical theory, 68% of your measurements of the length of the table should fall in
the range from L sLn1 to L þ sLn1 . About 7 of your 10 measurements should fall in this range. What
is the range of these values for your data? From __________m to __________m. How many of your 10
measurements of the length of the table fall in this range? __________? State clearly the extent to which
your data for the length agree with the theory. What is your evidence for your statement?
W
2. Answer the same question for the width. Range of W sW
n1 to W þ sn1 is from __________m to
__________m. The number of measurements that fall in that range is __________. Do your data for the
width of the table agree with the theory reasonably well? State your evidence for your opinion.
COPYRIGHT ª 2008 Thomson Brooks/Cole
3. According to statistical theory, if any measurement of a given quantity has a deviation greater than
3sn–1 from the mean of that quantity, it is very unlikely that it is statistical variation, but rather is more
likely to be a mistake. Calculate the value of 3sLn1 . Do any of your measurements of length have a
deviation from the mean greater than that value? If so, calculate how many times larger than sLn1 it is.
Do any of your measurements of the length appear to be a mistake, and, if so, which ones?
4. For the width measurements calculate 3sW
n1 . Do any of your measurements of width have a deviation from the mean greater than that value? If so, calculate how many times larger than sW
n1 it is.
Do any of your measurements of width appear to be a mistake, and, if so, which ones?
22
Physics Laboratory Manual n Loyd
5. If possible, state the accuracy of your measurements of the length and width and give your reasoning.
If this cannot be done, state why it is not possible. If possible, state the precision of your measurement
of the length and width and give your reasoning. If this cannot be done, state why it is not possible.
Physics Laboratory Manual
n
Loyd
L ABOR AT ORY
2
Measurement of Density
OBJECTIVES
o Determine the mass, length, and diameter of three cylinders of different metals.
o Calculate the density of the cylinders and compare with the accepted values of the density of the
metals.
o Determine the uncertainty in the value of the calculated density caused by the uncertainties in
the measured mass, length, and diameter.
EQUIPMENT LIST
.
Three solid cylinders of different metals (aluminum, brass, and iron)
.
Vernier calipers
.
Laboratory balance and calibrated masses
THEORY
The most general definition of density is mass per unit volume. Density can vary throughout the body if
the mass is not distributed uniformly. If the mass in an object is distributed uniformly throughout the
object, the density r is defined as the total mass M divided by the total volume V of the object. In equation
form this is
M
V
ðEq: 1Þ
pd2 L
4
ðEq: 2Þ
r¼
For a cylinder the volume is given by
COPYRIGHT ª 2008 Thomson Brooks/Cole
V¼
where d is the cylinder diameter, and L is its length. Using Equation 2 in Equation 1 gives
r¼
4M
pd2 L
ðEq: 3Þ
We will determine the quantities M, d, and L by measuring each of them four times and calculating
the mean and standard error for each quantity. Using the mean of each measured quantity in Equation 3
leads to the best value for the measured density r.
ª 2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work
covered by the copyright hereon may be reproduced or used in any form or by any meansçgraphic, electronic, or mechanical,including photocopying, recording, taping,web distribution, information
storage and retrievalsystems,or in any other mannerçwithout the written permission of the publisher.
23
24
Physics Laboratory Manual n Loyd
An important question in experimental physics is how the uncertainty in a quantity calculated from
other measured quantities is related to the uncertainty in those measured quantities. For this laboratory,
the uncertainty in the density (standard error) is related to the standard errors in the mass, length, and
diameter by:
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
aM
aL
ad
ðEq: 4Þ
ar ¼ r ð Þ2 þ ð Þ2 þ 4ð Þ2
M
L
d
The form of this equation is stated here without proof, but it can be derived from the relationship
between the measured quantities and the density described by Equation 3.
We determine the mass of the cylinders with a laboratory balance, which balances the weight of an
unknown mass m against the weight of a known mass mk. Although the balance is between two forces
(the weight of the masses), the scales can be calibrated in terms of mass assuming that the force per unit
mass is the same for both the known and unknown mass. The unknown mass on a pan at the left is
balanced against the sum of all the known masses placed on the right pan plus the mass equivalent of
the permanent sliding mass on the beam. Figure 2-1 shows a picture of a Harvard Trip balance, which has
a calibrated beam along which a permanent sliding mass can be moved in units of 0.1 gram up to
10 grams.
The length and diameter of the metal cylinder will be measured with a vernier caliper. A caliper is
actually any device used to determine thickness, the diameter of an object, or the distance between two
surfaces. Often calipers are in the form of two legs fastened together with a rivet, so they can pivot about
the fastened point. The vernier caliper used in this laboratory consists of a fixed rule that contains one jaw,
and a second jaw with a vernier scale that slides along the fixed rule scale as shown in Figure 2-2.
Vernier is the name given to any scale that aids in interpolating between marked divisions.
Image not available due to copyright restrictions
Image not available due to copyright restrictions
Laboratory 2 n Measurement of Density
25
Alignment of
vernier with
fixed scale
2
3
Vernier
Zero Mark
Figure 2-3 Illustration of vernier caliper reading of 2.06 cm.
The caliper has marked on the main scale major divisions of 1 cm for which there are both a mark and
a number. On the main scale are also marked 10 divisions, each 1 mm apart between the 1 cm divisions.
The 1 mm marks are not labeled with a number. This vernier is marked with a scale that, when aligned
with different marks on the fixed rule scale, allows interpolation between the 1 mm marks on the fixed
scale to 0.1 mm accurately. A vernier caliper can measure distances accurately to the nearest 0.01 cm.
A measurement is made by closing the jaws on some object and noting the position of the zero mark
on the vernier and which one of the vernier marks is aligned with some mark on the fixed rule scale.
This is illustrated in Figure 2-3. The position of the zero mark of the vernier scale gives the first two
significant figures (2.0 cm in Figure 2-3). We derive the interpolation between 2.0 cm and 2.1 cm for this
case from the fact that the sixth mark beyond the vernier zero is best aligned with a mark on the fixed
rule scale. The reading in this example is 2.06 cm.
Before making any measurements, determine whether or not the vernier calipers read zero when
the jaws are closed. If the calipers do not read zero when the jaws are closed, they are said to have a zero
error. A correction is necessary for each measurement performed with the calipers. If the vernier zero is to
the right of the fixed scale zero when the jaws are closed, the zero error is positive. Note the mark on the
vernier scale that is aligned with the fixed scale, and subtract that number of units of 0.01 cm from each
measurement. For example, if the third mark to the right of the vernier zero is aligned with the fixed scale
when the jaws are closed, then each measurement should have 0.03 cm subtracted from it. If the vernier
zero is to the left of the fixed scale zero, then the zero error is negative. In that case, find which vernier
mark is aligned with the fixed scale. Then determine how far to the left of the 10 mark on the vernier scale
the alignment occurs. For example, if the alignment occurs at the 7 mark on the vernier scale, you will
add 0.03 cm to the reading.
EXPERIMENTAL PROCEDURE
1. Zero the laboratory balance according to directions given by your laboratory instructor.
COPYRIGHT ª 2008 Thomson Brooks/Cole
2. Use the laboratory balance and calibrated masses to determine the mass of each of the three cylinders.
Make four independent measurements for each of the cylinders and record the results in the Data
Table.
3. Make four separate readings of the zero correction for the vernier calipers. Record the four values in
the Data Table. Record the zero correction as positive if the vernier zero is to the right of the fixed scale
zero and record it as negative if the vernier zero is to the left of the fixed scale zero.
4. Use the vernier calipers to measure the lengths of the three cylinders. Make four separate trials of the
measurement of the length of each cylinder. Measure the length at different places on each cylinder for
the four trials to sample any variation in length of the cylinders. Record the results in the Data Table.
5. Use the vernier calipers to measure the diameters of the three cylinders. Make four separate trials of
the measurement of the diameter of each cylinder. Measure the diameter at four different positions
along the length of the cylinders to sample any variation in diameter of the cylinders. Record the
results in the Data Table.
26
Physics Laboratory Manual n Loyd
CALCULATIONS
1. Calculate the mean M, the standard deviation sn–1, and the standard error aM for the four
measurements of the mass of each cylinder and record the results in the Data Table. For this and all
subsequent calculations keep one significant figure only for all standard errors, and then keep the
number of decimal places in the mean that coincides with the decimal place of the standard error.
2. Determine the measured length and diameter for each trial by making the appropriate zero correction
to each measurement and then calculating the means L and d, the standard deviations, and the
standard errors aL and ad for each cylinder. Record the results in the Data Table.
3. Use Equation 3 to calculate the density r of each of the cylinders. Use the mean values for the mass,
diameter, and length. Use Equation 4 to calculate the standard error of the density. Record the results
in the Data Table.
4. For purposes of this laboratory, assume that the density of aluminum is 2.70 gram/cm3, the density of
brass is 8.40 gram/cm3, and the density of iron is 7.85 gram/cm3. Calculate the percentage error in
your results for the density of each of these metals.
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
L A B O R A T O R Y 2
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Measurement of Density
PRE-LABORATORY ASSIGNMENT
1. A cylinder has a length of 3.23 cm, a diameter of 1.75 cm, and a mass of 65.3 grams. What is the density
of the cylinder? Based on its density, of what kind of material might it be made? Material is likely to
be: ______________________________ (Show your work.)
2. Figure 2-4 shows a vernier caliper scale set to a particular reading. What is the reading of the scale?
Reading = ______________________________cm
2
3
COPYRIGHT ª 2008 Thomson Brooks/Cole
Figure 2-4 Example of a reading of a vernier caliper.
3. The caliper in Figure 2-5 has its jaws closed. If the caliper has a zero error, what is its value? Is it
positive or negative? Error = __________cm
0
1
Figure 2-5 Vernier caliper with its jaws closed. Does it have a zero error?
27
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Physics Laboratory Manual n Loyd
4. A series of four measurements of the mass, length, and diameter are made of a cylinder. The results of
these measurements are:
Mass—20.6, 20.5, 20.6, and 20.4 grams
Length—2.68, 2.67, 2.65, and 2.69 cm
Diameter—1.07, 1.05, 1.06, and 1.05 cm
Find the mean, standard deviation, and standard error for each of the measured quantities and
tabulate them below. Keep only one significant figure in each standard error and then keep decimal
places in the mean to coincide with the standard error.
M¼
sn1 ¼
aM ¼
L¼
sn1 ¼
aL ¼
d¼
sn1 ¼
ad ¼
Calculate the density and the standard error of the density using Equations 3 and 4. Keep only one
significant figure in the standard error and then keep decimal places in the density to coincide with the
standard error.
r¼
ar ¼
5. Because ar has only one digit, it determines the place of the least significant digit kept in the
calculation of the density. From that information, how many significant figures are there in the
density for the above calculation? State clearly the reasoning for your answer.
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Lab Partners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
L A B O R A T O R Y 2
Measurement of Density
LABORATORY REPORT
Mass Data and Calculations Table
M1 (kg)
M2 (kg)
M3 (kg)
M4 (kg)
M (kg)
sn–1 (kg)
aM (kg)
Aluminum
Brass
Iron
Zero Reading of the calipers ______________________________
Length Data and Calculations Table
L1 (m)
L2 (m)
L3 (m)
L4 (m)
L (m)
sn–1 (m)
aL (m)
d2 (m)
d3 (m)
d4 (m)
d (m)
sn–1 (m)
ad (m)
Aluminum
Brass
Iron
COPYRIGHT ª 2008 Thomson Brooks/Cole
Diameter Data and Calculations Table
d1 (m)
Aluminum
Brass
Iron
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Physics Laboratory Manual n Loyd
Density Data and Calculations Table
rexp (kg/m3)
ar (kg/m3)
rknown (kg/m3)
Err (kg/m3)
% Error
Aluminum
Brass
Iron
SAMPLE CALCULATIONS
1. M ¼
2. sn1 ¼
sn1 sn1
3. aM ¼ pﬃﬃﬃ ¼ pﬃﬃﬃ ¼
n
4
4M
4. r ¼ 2 ¼
pd L
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
aM 2 aL 2
ad 2
5. ar ¼ r
þ
þ4
¼
M
L
d
6. Error = r – rknown =
rrknown
100% ¼
7. % Error ¼
rknown
QUESTIONS
1. Consider the uncertainty in the measured value of r to be given by ar. Taking the one decimal place
of ar as the least significant digit in r, how many significant figures are indicated for each of the
measurements of r?
2. What is the accuracy of your determination of the density for each metal? State clearly what quantity
describes the accuracy of your measurements of the density.
Laboratory 2 n Measurement of Density
31
3. Consider the value of the standard error ar as an indication of the precision of your measurements.
Express the standard error as a percentage of the measured value of the density, and relate it to the
accuracy of each of your measurements.
4. Considering your answers to Question 3, is there evidence for a systematic error in any of your
measurements of the density of any of the metals? State clearly your evidence either for or against the
presence of a systematic error.
COPYRIGHT ª 2008 Thomson Brooks/Cole
5. For the same percentage error in each of the three quantities, mass, diameter, and length, which would
contribute the most to the error in the density? (Hint—Consider the form of Equation 4.)
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Physics Laboratory Manual
n
Loyd
L ABOR AT ORY
3
Force Table and Vector
Addition of Forces
OBJECTIVES
o Demonstrate the addition of several vectors to form a resultant vector using a force table.
o Demonstrate the relationship between the resultant of several vectors and the equilibrant of
those vectors.
o Illustrate and practice graphical and analytical solutions for the addition of vectors.
EQUIPMENT LIST
.
Force table with pulleys, ring, and string
.
Mass holders and slotted masses
.
Protractor and compass
COPYRIGHT ª 2008 Thomson Brooks/Cole
THEORY
Physical quantities that can be completely specified by magnitude only are called scalars. Examples of
scalars include temperature, volume, mass, and time intervals. Some physical quantities have both
magnitude and direction. These are called vectors. Examples of vector quantities include spatial displacement, velocity, and force.
Consider the case of several forces with different magnitudes and directions that act at the same point.
The single force, which is equivalent in its effect to the effect produced by the several applied forces, is
called the resultant force. This resultant force can be found theoretically by a special addition process
known as vector addition.
One process of vector addition is by graphical techniques. Figure 3-1(a) shows the case of two vectors,
F1 of magnitude 20.0 N, and F2 of magnitude 30.0 N. A scale of 1.00 cm = 10.0 N is used, and these vectors
are shown as 2.00 cm and 3.00 cm in length, respectively. The forces are assumed to act at the same point,
but 608 different in direction as shown. Figure 3-1(b) shows the graphical addition process called the
parallelogram method. Two lines are constructed, each one parallel to one of the vectors having the length
of that vector as shown. The resultant FR of the vector addition of F1 and F2 is found by constructing the
straight line from the point at the tails of the two vectors to the opposite corner of the parallelogram formed
by the original vectors and the constructed lines. A measurement of the length of FR in Figure 3-1(b) shows
it to be 4.35 cm in length, and a measurement of the angle between FR and F1 shows it to be about 378.
Because the scale is 1.00 cm = 10.0 N, the value of the resultant FR is 43.5 N, and it acts in a direction 378 with
respect to the direction of F1.
ª 2008 Thomson Brooks/Cole, a part of TheThomson Corporation.Thomson,the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTSRESERVED.No part of this work
covered by the copyright hereon may be reproduced or used in any form or by any meansçgraphic, electronic, or mechanical,including photocopying, recording, taping,web distribution, information
storage and retrievalsystems,or in any other mannerçwithout the written permission of the publisher.
33
34
Physics Laboratory Manual n Loyd
F R F1 F2
F2
F R F1 F2
F2
60°
F2
37°
F1
F1
(a)
(b)
60°
37°
F1
(c)
Figure 3-1 Illustration of the parallelogram and triangle addition of vectors.
In the graphical vector addition process known as the polygon method one of the vectors is first drawn
to scale. Each successive vector to be added is drawn with its tail starting at the head of the preceding
vector. The resultant vector is the vector drawn from the tail of the first arrow to the head of the last arrow.
Figure 3-1(c) shows this process for the case of only two vectors (for which the polygon method is the
triangle method). The second vector, F2, must be drawn at the proper angle relative to F1 by extending a
line in the direction of F1 and constructing F2 relative to that line. In Figure 3-1(c) the length of FR is 4.35 cm
corresponding to 43.5 N, and it acts at 378 with respect to F1.
The polygon method is illustrated for the case of three vectors in Figure 3-2. Vector F1 is drawn, F2 is
drawn at the proper angle a relative to F1, and F3 is drawn at the proper angle b relative to F2. The resultant
FR is the vector connecting the tail of F1 and the head of F3.
The analytical process of vector addition uses trigonometry to express each vector in terms of its
components projected on the axes of a rectangular coordinate system. Figure 3-3 shows a vector, a coordinate
system superimposed on the vector, and the components jFjcos y and jFjsin y into which the vector is
resolved. When the analytical process for multiple vectors is used, each vector is resolved into components in
that manner. The components along each axis are then added algebraically to produce the net components of
the resultant vector along each axis. Those components are at right angles, and the magnitude of the resultant
can be found from the Pythagorean theorem. The case of three vectors, F1, F2, and F3, is shown in Figure 3-4.
F3

FR F1 F2 F3
F3
F2

F2
␣
␣
F1
F1
(a)
(b)
Figure 3-2 Illustration of the polygon method for vector addition.
y
F
y
F
F sin
(a)
x
(b)
F cos
(c)
Figure 3-3 Illustration of analytical resolution of a vector.
x
Laboratory 3 n Force Table and Vector Addition of Forces
y
35
y
FRy 16.0
15.0
135°
12.0
30°
10.0
x
(a)
FR 21.6
FRx 14.5
x
(b)
Figure 3-4 Illustration of the analytical addition of vectors.
Image not available due to copyright restrictions
Taking the algebraic sum of each of the components of the three vectors and combining the components to find the resultant and its direction leads to the following:
FRx ¼ F1x þ F2x þ F3x ¼ 10:0 cos ð0 Þ þ 15:0 cos ð30 Þ þ 12:0 cos ð135 Þ ¼ 14:5
FRy ¼ F1y þ F2y þ F3y ¼ 10:0 sin ð0 Þ þ 15:0 sin ð30 Þ þ 12:0 sin ð135 Þ ¼ 16:0
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
jF R j ¼ FR ¼ ðFRx Þ2 þ ðFRy Þ2 ¼ ð14:5Þ2 þ ð16:0Þ2 ¼ 21:6
COPYRIGHT ª 2008 Thomson Brooks/Cole
y ¼ arc tan ðFRy =FRx Þ ¼ arc tan ð16:0=14:5Þ ¼ arc tan ð1:10Þ ¼ 47:7
The force table (Figure 3-5) provides a force from the gravitational attraction on masses attached to a
ring by a string passing over a pulley. Each force is applied over a separate pulley, and the pulley
positions can be adjusted to any desired position around a circular plate. Experimentally the applied
forces are balanced by the application of a single force that is equal to the magnitude of the resultant of the
applied forces and acts opposite of the resultant. This balancing force (called the equilibrant) is what is
determined by the measurements. The resultant is the same magnitude as the equilibrant and 1808
different in direction.
EXPERIMENTAL PROCEDURE
Part 1. Two Applied Forces
1. Place a pulley at the 20.08 mark on the force table and place a total of 0.100 kg (including the mass
holder) on the end of the string. Calculate the magnitude of the force (in N) produced by the mass.
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Physics Laboratory Manual n Loyd
Assume that g = 9.80 m/s2. Assume three significant figures for this and for all other calculations of
force. Record the value of this force as F1 in Data Table 1.
2. Place a second pulley at the 90.08 mark on the force table and place a total of 0.200 kg on the end of the
string. Calculate the force produced and record as F2 in Data Table 1.
3. Determine by trial and error the magnitude of mass needed and the angle at which it must be located
for the ring to be centered on the force table. Jiggle the ring slightly to be sure that this equilibrium
condition is met. Attach all strings to the ring so that they are directed along a line passing through the
center of the ring. All the forces will then act through the point at the center of the table. Record this
value of mass in Data Table 1 in the row labeled Equilibrant FE1.
4. Calculate the force produced (mg) on the experimentally determined mass. Record the magnitude and
direction of this equilibrant force FE1 in Data Table 1.
5. The resultant FR1 is equal in magnitude to FE1, and its direction is 1808 from FE1. Record the magnitude
of the force FR1, the mass equivalent of this force, and the direction of the force in Data Table 1 in the
row labeled Resultant FR1.
Part 2. Three Applied Forces
1. Place a pulley at 30.08 with 0.150 kg on it, one at 100.08 with 0.200 kg on it, and one at 145.08 with
0.100 kg on it.
2. Calculate the force produced by those masses and record them as F3, F4, and F5 in Data Table 2.
3. Determine the equilibrant force and the resultant force by following a procedure like that in Part 1,
Steps 3 through 5 above. Record the magnitudes of the forces, the associated values of mass, and the
directions in Data Table 2 in the rows labeled FE2 and FR2.
CALCULATIONS
Part 1. Two Applied Forces
1. Find the resultant of these two applied forces by scaled graphical construction using the parallelogram
method. Use a ruler and protractor to construct vectors with scaled length and direction that represent
F1 and F2. A convenient scale might be 1.00 cm = 0.100 N. All directions are given relative to the force
table. Account for this in the graphical construction to ensure the proper angle of one vector to
another. Determine the magnitude and direction of the resultant from your graphical solution and
record them in the appropriate section of Calculations Table 1.
2. Use trigonometry to calculate the components of F1 and F2 and record them in the analytical solution
portion of Calculations Table 1. Add the components algebraically and determine the magnitude of
the resultant by the Pythagorean theorem. Determine the angle of the resultant from the arc tan of the
components. Record those results in Calculations Table 1.
3. Calculate the percentage error of the magnitude of the experimental value of FR compared to the
analytical solution for FR. Also calculate the percentage error of the magnitude of the graphical
solution for FR compared to the analytical solution for FR. For each of those comparisons, also calculate
the magnitude of the difference in the angle. Record all values in Calculations Table 1.
Part 2. Three Applied Forces
1. Use the polygon scaled graphical construction method to find the resultant of the three applied forces.
Determine the magnitude and direction of the resultant from your graphical solution and record them
in Calculations Table 2.
2. Use trigonometry to calculate the components of all three forces, the components of the resultant, and
the magnitude and direction of the resultant, and record them all in Calculations Table 2.
3. Make the same error calculations for this problem as described in Step 3 of Part 1 above. Record the
values in Calculations Table 2.
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
L A B O R A T O R Y 3
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Force Table and Vector Addition of Forces
PRE-LABORATORY ASSIGNMENT
1. Scalars are physical quantities that can be completely specified by their ______________________________.
2. A vector quantity is one that has both ______________________________ and ______________________________.
3. Classify each of the following physical quantities as vectors or scalars:
(a) Volume __________
(b) Force __________
(c) Density __________
(d) Velocity __________
(e) Acceleration __________
Answer Questions 4–7 with reference to Figure 3-6 below.
4. If F1 stands for a force vector of magnitude 30.0 N and F2 stands for a force vector of magnitude 40.0 N
acting in the directions shown in Figure 3-6, what are the magnitude and direction of the resultant
obtained by the vector addition of these two vectors using the analytical method? Show your work.
Magnitude = __________N Direction(relative to x axis) = __________degrees
5. What is the equilibrant force that would be needed to compensate for the resultant force of the vectors
F1 and F2 that you calculated in Question 4?
COPYRIGHT ª 2008 Thomson Brooks/Cole
Magnitude = __________N
Direction(relative to x axis) = __________degrees
y
F2
60°
F1
x
Figure 3-6 Addition of two force vectors.
37
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Physics Laboratory Manual n Loyd
6. Figure 3-6 has been constructed to scale with 1.00 cm = 10.0 N. Use the parallelogram graphical
method to construct (on Figure 3-6) the resultant vector FR for the addition of F1 and F2. Measure the
length of the resultant vector and record it below. State the force represented by this length. Measure
with a protractor the angle that the resultant makes with the x axis.
Resultant vector length = __________cm
Force represented by this length = __________N
Direction of resultant relative to x axis = __________degrees
7. Use the polygon method of vector addition to construct on the axes below a graphical solution to the
problem in Figure 3-6. Use the scale 1.00 cm = 10.0 N.
Resultant vector length = __________cm
Force represented by this length = __________N
Direction of resultant relative to x axis = __________degrees
y
x
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Section . . . . . . . . . . . . . . . .
Date . . . . . . . . . . . . . . . .
Lab Partners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
L A B O R A T O R Y 3
Force Table and Vector Addition of Forces
LABORATORY REPORT
Data Table 1
Force
Mass (kg)
Force (N)
Direction
F1
0.100
20.08
F2
0.200
90.08
Equilibrant FE1
Resultant FR1
Data Table 2
Force
Mass (kg)
Force (N)
Direction
F3
0.150
30.08
F4
0.200
100.08
F5
0.100
145.08
Equilibrant FE2
COPYRIGHT ª 2008 Thomson Brooks/Cole
Resultant FR2
Calculations Table 1
Graphical Solution
Force
Mass (kg)
Force (N)
Direction
F1
0.100
20.08
F2
0.200
90.08
Resultant FR1
39
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Physics Laboratory Manual n Loyd
Analytical Solution
Force
Mass (kg)
Force (N)
Direction
F1
0.100
20.08
F2
0.200
90.08
x-component
y-component
Resultant FR1
PART 1. ERROR CALCULATIONS
Percent Error magnitude Experimental compared to Analytical = __________%
Percent Error magnitude Graphical compared to Analytical = __________%
Absolute Error in angle Experimental compared to Analytical = __________degrees
Absolute Error in angle Graphical compared to Analytical = __________degrees
Calculations Table 2
Graphical Solution
Force
Mass (kg)
Force (N)
Direction
F3
0.150
30.08
F4
0.200
100.08
F5
0.100
145.08
Resultant FR2
Analytical Solution
Force
Mass (kg)
Force (N)
Direction
F3
0.150
30.08
F4
0.200
100.08
F5
0.100
145.08
Resultant FR2
x-component
y-component
Laboratory 3 n Force Table and Vector Addition of Forces
41
PART 2. ERROR CALCULATIONS
Percent Error magnitude Experimental compared to Analytical = __________%
Percent Error magnitude Graphical compared to Analytical = __________%
Absolute Error in angle Experimental compared to Analytical = __________degrees
Absolute Error in angle Graphical compared to Analytical = __________degrees
SAMPLE CALCULATIONS
1. F = mg =
F
¼
g
3. Direction FE opposite FR so direction FR = direction FE –1808 =
2. m ¼
4. F1x = F1 cos (208) =
5. F1y = F1 sin (208) =
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
6. FR1 ¼ ðFRx Þ2 þ ðFRy Þ2 ¼
Fy
1
7. y ¼ tan
¼
Fx
jExperimental Analyticalj
8. % Error Exp ¼
100% ¼
Analytical
9. Absolute Err = y (exp) – y (analytical) =
QUESTIONS
COPYRIGHT ª 2008 Thomson Brooks/Cole
1. To determine the force acting on each mass it was assumed that g = 9.80 m/sec2. The value of g at the
place where the experiment is performed may be slightly different from that value. State what effect
(if any) it would have on the percentage error calculated for the comparisons. To test your answer to the
question, leave g as a symbol in the calculation of the percentage error.
2. Two forces are applied to the ring of a force table, one at an angle of 20.08, and the other at an angle of
80.08. Regardless of the magnitudes of the forces, choose the correct response below.
The equilibrant will be in the (a) first quadrant (b) second quadrant (c) third quadrant (d) fourth
quadrant (e) cannot tell which quadrant from the available information.
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Physics Laboratory Manual n Loyd
The resultant will be in the (a) first quadrant (b) second quadrant (c) third quadrant (d) fourth
quadrant (e) cannot tell which quadrant from the available information.
3. Two forces, one of magnitude 2 N and the other of magnitude 3 N, are applied to the ring of a force
table. The directions of both forces are unknown. Which best describes the limitations on R, the
resultant? Explain carefully the basis for your answer.
(a) R # 5 N (b) 2 N # R # 3 N (c) R $ 3 N (d) 1 N # R # 5 N (e) R # 2 N.
4. Suppose the same masses are used for a force table experiment as were used in Part 1, but each pulley
is moved 1808 so that the 0.100 kg mass acts at 2008, and the 0.200 kg mass...

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