PHYSICS M20A LAB MANUAL
MOORPARK COLLEGE
BY: PROFESSOR H. FRED MEYER
LIST OF EXPERIMENTS
Measurements: Mass, Volume, Density
Errors and Error Propagation
Height of a Building
The Simple Pendulum
Acceleration of Gravity
The Force Table
Cantilever Beam
Friction
Center of Mass
Ballistic Pendulum
Rotational Motion and the Moment of Inertia
Young’s Modulus and Torsion Modulus
Archimedes Principle
Hooke’s Law and the Effective Mass of a Spring
MEASUREMENTS LAB
DETERMINING VOLUME , MASS AND DENSITY USING MICROMETERS,
VERNIER CALIPERS AND A LABORATORY BALANCE
INTRODUCTION
Instructional Objectives:
Learn how to use calipers, micrometers and a laboratory balance. Determine the “least count”
of an instrument.
Determine the number of significant figures in a measurement.
Determine the calculated percent error in a measurement from the errors in the
measurements.
Experimental Objectives:
Measure the dimensions and mass of various items use these measurements to determine the
density of these objects. The densities will be compared with accepted values to see if the
experimental values are within the predicted margin of error.
THE VERNIER CALIPER
Note the various measurement types listed below.
The Vernier Caliper
The Vernier Caliper
Inside Dimensions
English Units (inch)
Vernier Scales
Outside Dimensions
Depth Gauge
Metric Units (cm, mm)
Looking at the lower scale, note how measurements are read.
The least count is the smallest subdivision reading that can be read without estimating. Note
that the least count and determines the precision of the instrument. The least count shown
above is 0.01 cm. and the reading would be recorded as 3.47cm±0.01cm on the data sheet.
The ±0.01cm is called the absolute error in the measurement.
MICROMETER
The spindle of an ordinary metric micrometer has 2 threads per millimeter, and thus one complete
revolution moves the spindle through a distance of 0.5 millimeter. The longitudinal line on the frame is
graduated with 1 millimeter divisions and 0.5 millimeter subdivisions. The thimble has 50 graduations,
each being 0.01 millimeter (one-hundredth of a millimeter). Convince yourself that the reading shown
on the above micrometer is 12.93mm. (Note that the thimble is 0.43mm past the 0.50mm mark on the
sleeve. The reading would be recorded as 12.94mm±0.01mm on the data sheet.
When using the micrometer, turn the ratchet (also called the friction clutch) until it slips. This
provides the proper torque on the thimble. Also, after using the micrometers, make sure to leave the
jaws open so they don’t get sprung with temperature changes.
Whole millimeter marks.
.01 millimeter
marks
½ millimeter
marks.
Notice that the least count on the micrometer is 0.01mm.
The above readings on the instruments assumed that they read zero when closed. Before
taking a reading, close the measuring device completely and take a zero reading. This will give
you the “zero correction.” If the zero correction is positive, this value must be subtracted from
all readings. If the zero correction is negative, this value must be added to each reading.
THEORY
The mass density of an object
ρ is defined as MASS/VOLUME.
The volumes of various shapes are given by:
VSPHERE =
VCYLINDER = πR2L
THE EXPERIMENT
Caliper Measurements
DATA TABLE 1: Two measurements per partner.
Least count_________
Zero correction_______
Sphere Radius
Cylinder Radius
Cylinder Length
Reading 1
Reading 2
Reading 3
Reading4
AVERAGE
Least count______________
Sphere Mass
Cylinder Mass
Reading 1
Reading 2
Reading 3
Reading4
AVERAGE
DATA TABLE 2-two measurements per partner
Micrometer Measurements -repeat the measurements with the micrometer using a different
size sphere and different size cylinder.
Least count_________________
Zero correction_______________
Sphere Radius
Cylinder Radius
Cylinder Length
Reading 1
Reading 2
Reading 3
Reading4
AVERAGE
Sphere Mass
Cylinder Mass
Reading 1
Reading 2
Reading 3
Reading4
AVERAGE
ANALYSIS
Using the equations given for volume and the data calculate the volume of each sphere and
each cylinder. ( be sure the results are stated with the proper number of significant figures
and include the units of either grams per cubic centimeter or kilograms per cubic meter)
Using the equation for density, calculate the density ρ of each object.
We are now going to use the estimated absolute error to calculate the predicted error in the
density.
Add the absolute error (estimated from least count) to each of the measurements and calculate
+
the density again. Call this value ρ .
Now subtract the absolute error from each measurement and calculate the density of each
-
object. Call this value ρ .
Calculate the percent predicted error using the equation
converted to percent.
Look up the accepted value of the density of each object.
Calculate the percent discrepancy using the equation
converted to percent.
SUMMARY TABLE
CALIPER VALUES
MICROMETER VALUES
ρCYLINDER
ρSPHERE
PREDICTED % ERROR (CYL)
PREDICTED % ERROR (SPHERE)
% DISCREPANCY (CYL)
% DISCREPANCY (SPHERE)
DISCUSSION: If the % discrepancy is less than the % predicted error, then the results are within
the “MARGIN OF ERROR.” Note whether or not your results are within the margin of error. If
the results are not within the margin of error, can you give a reason why? Note possible
sources of error in the measurements and give suggestions how the error might be reduced.
ERRORS AND ERROR PROPAGATION
INTRODUCTION: Laboratory experiments involve taking measurements and using those
measurements in an equation to calculate an experimental result. It is also necessary to know
how to estimate the uncertainty, or error, in physical measurements and to know how to use
those uncertainties to calculate the error in the experimental result.
TYPES OF EXPERIMENTAL ERRORS
Experimental errors can generally be classified into three types: personal, systematic, and
random.
Personal Errors
These errors arise from personal bias of carelessness in reading an instrument, in recording data,
or in calculations, and parallax in reading a meter. Of these, only parallax errors can be
estimated and used in error propagation. Effort should be made to eliminate experimental errors.
(When looking at non-digital meter, there is a small distance between the needle and the scale.
As a result, the reading will change as the observer’s eye position changes from side to side.
This apparent change in reading, due to the change in position of the observer’s eye, is called
parallax.)
Systematic Errors
Errors of this type result in measured values which are consistently too high or too low.
Conditions which lead to systematic errors are as follows:
1. An improperly calibrated instrument such as a thermometer which consistently
reads 99ºC in boiling water instead of 100ºC.
2. A meter, micrometer, vernier caliper, or other instrument which was not
properly zeroed or for which the zero correction factor was not considered.
3. Theoretical errors due to a simplified mathematical model for the system
which consistently gives a calculated value different from the calculated value predicted from a
more accurate mathematical model.
Random Errors
Random errors result from unknown and unpredictable variations in experimental measurements.
Possible sources of random errors are:
1. Observational, e.g. errors when reading the scale of a measuring device to the
smallest division.
2. Environmental- unpredictable fluctuations in readings beyond the
experimenters control. Such errors can be determined statistically or can be
estimated by the experimenter.
STATISTICAL DETERMINATION OF RANDOM ERRORS
When there are many measurements of the same quantity, the average or mean value is defined
_
1 N
by x   xi where x i is the ith measured value and N is the total number of measurements.
N i 1
There are two ways to statistically calculate the uncertainty in the measured value. One method
is to calculate the deviation from the mean or “mean deviation d”
N
d
x
i 1
i
x
N
It is common to express the experimental value of the measurement as:
Measured value of x = x  d
where d a statistical estimate of the uncertainty in the measured value. As can be observed, the
mean deviation is a measure of the spread on the data.
Another method used to calculate the random error is by calculating the “standard deviation,
(s.d.)”
 x
s.d . 
i 1

2
N
i
x
N
The measures value of x can then be expressed as:
Measured value of x  x  s.d .
The statistical methods above will be used in selected lab exercises to follow such as “THE
SIMPLE PENDELUM” and “MOMENT OF INERTIA” where several measurements of time
are needed and an average or mean is calculated.
ESTIMATION OF RANDOM ERRORS
An easier method to determine random error is to estimate the random error by utilizing the
accuracy of the instrument and the judgment of the experimenter. The error in a given
instrument is determined by the smallest division on that instrument or “least count.” For
example, the smallest division on a meter stick is 1mm or 0.1cm. This is the least count for the
meter stick. In most measurements the smallest division represents the rightmost digit in the
value of that measurement and the estimated error is the measurement is  the least count. For
example, a measure value may be 78.2cm  0.1cm.
Sometimes a measurement may be made with an estimated error less than the least count. For
example, an experimenter may estimate reading on a meter stick as 78.25cm by noting that the
reading was about half way between 78.2cm and 78.3cm. The experimenter may represent the
value as 78.25cm  0.05cm. Keep in mind that rightmost digit must be estimated by the
experimenter and is thus doubtful.
Sometimes the estimated error is larger than the least count. For example, when measuring the
distance between the two spots below, the experimenter would need to estimate where the center
of each spot would be located. The error in the measured distance would be larger than the least
count and the amount of the estimated error would be up to the judgment of the experimenter.
Note how much the error estimates depend on the judgment of the experimenter. There may be
errors in judgment; however, to avoid stating a result more accurately than you probably
measured it, one should try to avoid being too conservative in estimating errors.
ERROR PROPAGATION
PARTIAL DERIVITIVES
Before we can perform error propagation calculations, we must know how to take what are
called “partial derivatives” of a function with many variables. Some may already know how to
do this; you can help the others.
Suppose we have a function f where f=f(x,y,z). The partial derivative of f with respect to x is
found by taking the ordinary derivative while treating y and z as constants. The notation for this
f
derivative is
. Likewise, the partial derivative of f with respect to y is found by taking the
x
f
ordinary derivative while treating x and z as constants and is written as
and the partial
y
derivative of f with respect to z is found by taking the ordinary derivative while treating x and y
f
as constants and is written as
.
z
As an example, let f  5x 2 yz 3 . Then
Convince yourself that
f

x 2

5 x 2 yz 3 = 5 yz 3
 10 xyz 3
x x
x


f
f
 15 x 2 yz 2 .
 5 x 2 z 3 and that
z
y
ABSOLUTE AND RELATIVE ERRORS
Absolute Error: When an error is estimated in a measured value of x it will be designated as
 x (delta x). x has the same units as x and is called the absolute error in x. For example, if
x  2.0cm  0.1cm , the absolute error is x  0.1cm .
Relative Error: The ratio of the absolute error x to the measured value x,
x
, is called the
x
relative error. It is usually represented as a percent. For example, the relative error in the
0.1cm 0.1
above example is

 0.05  5%
2.0cm 2.0
(note, there are times when it is necessary to from relative error back to absolute error:
x  errorrelative x )
COMPUTATION OF ERROR
For a function f  f x, y, z  , the absolute error in f, f , is defined as:
2
2
2

 f    f    f   

f   x   y    z  
x
y
z

         

The relative error in f would thus be
2
2
2

 f  
 f   
 f  

 x    y    z  
x
 z   

 y  
  

f
1

f
f
EXAMPLE
Using the function we used as an example for partial derivatives, we would have
f 
thus
10xyz f   5x z y  15x yz z  
3
2
3
2
2
2
2
2
2
 10 xyz 3  2  5 x 2 z 3
  15 x 2 yz 2  
  2 3 x    2 3 y    2 3 z   which when simplified
f
 5 x yz
  5 x yz
  5 x yz
 
f
becomes
2
 2x  2  y  2  3z  2 
 
     
 
f
 x   y   z  
f
Note that the quantities in the parentheses are just the percent errors multiplied by the exponent
for that particular variable.
Suppose we have the experimental values for x, y, and z as:
x  3.0cm  0.1cm , y  5.2cm  0.1cm , and z  2.4cm  0.1cm .
We would thus have the percent error in f as:
 20.1  2  0.1  2  30.1  2 
 
 
 
 =
f
 3.0   5.2   2.4  
f
0.143 ≈15%
Note that the % error is rounded up to the nearest whole number. Since it is just an estimate,
we can not justify more accuracy in the error.
ANOTHER EXAMPLE
Suppose
V 
3a 2  5b 2
where
c
a  8.2  0.1cm , b  6.5  0.1cm , and c  5.1  0.1cm
 V   2  V   2  V   2 
Thus V  
a   
b  
c  
 a    b    c   
V 6a V 10b
V
3a 2  5b 2
where
,
, and
; or,



a
c b
c
c
c2
 6a   2  10b   2  3a 2  5b 2   2 
c  
V   a   
b  
c2
  
 c    c   
V
Notice that the negative sign in
does not matter since it is squared.
c

6a

V

  2 c 2
Now
V
 3a  5b

c

2
   10b
  
c
a   
   3a 2  5b 2
  
c
  
2
   3a 2  5b 2
  
c2
b  
2
   3a  5b 2
  
c
  
 
 
c 
 
 
 
2







2
2
2

 10b  
6a


 
 c  
  2
a    2
b    
2 
2 
V
c 

 3a  5b  
 3a  5b  

V
Or,
2
2

  
   0.1 2 


68.2
106.5

0.1  

 
2
2 
 38.22  56.52 0.1   5.1  =5.5%
V






3
8
.
2

5
6
.
5






 


 
≈6%
V
The final results would be given as V  81cm  6%
PERCENT DISCREPANCY
Once an experimental value and percent error are calculated, the percent discrepancy is defined
as:
percent discrepancy in X=
X accepted  X exp erimental
X accepted
There will be agreement between the accepted value and the experimental value if the percent
discrepancy is less than the predicted percent error in the experimental value as determined by
error propagation. In other words, the experimental value is within the margin of error. This
should be addressed in your conclusion.
If there is not agreement, some sources of error may be present which may not have been
accounted for and some reasonable explanation should be included in the conclusion of your
report.
ERROR PROPAGATION EXERCISES
Determine the calculated value using the given values in the given equations. Be sure to include
the units in your answer. Using the error propagation method described above, calculate the
percent error in the calculated value. For this exercise, your percent error is to be given to two
significant figures.
Hand in this answer sheet. Work the problems neatly on scratch paper and staple your work
to this sheet.
1. A=xy, x  3.0cm  0.1cm , y  4.0cm  0.1cm
______________  ________
2. f=x+y, for x and y given in problem # 1
______________  ________
3. f=x-y, for x and y given in problem # 1
______________  ________
4. z=3x+2y, for x and y given in problem # 1
______________  ________
5. g 
2h
for h  2.00m  3% , t  0.630s  4%
t2
6. T  2
M
100 N
, M  2.5Kg  6% , k 
 2%
k
m
______________  ________
______________  ________
 5.00 
7. d   2  ML3 , M  30.0 g  2% , L  20.3  0.2cm ___________  ________
 cm g 


8. z  x 2  y 2 , x  3.0cm  2% , y  4.0cm  2%
9. z 
______________  ________
5a 3  2cm b 2
, a  2.0cm  1% , b  3.0cm  1% , C  11.0cm  2%
C
______________  ________
10. h  d sin  , d  1.00m  0.05m ,
  10  1
______________  ________
HEIGHT OF A BUILDING
INTRODUCTION: Using an angle meter and other simple measuring devices, the student will
take measurements which will be used to calculate the height of the South East corner of the
Physical Science building at Moorpark College relative to the walkway directly below the
corner.
The diagram below shows the basic geometry involved:
PS BUILDING
H
METER
STICK
METER
STICK
θ
X
A meter stick will be placed in a meter stick holder directly at the point below the overhanging
corner. This meter stick will be used by the whole class as a measurement reference. Each
group of students will have an angle meter, a ring stand and rod with a meter stick clamp and
meter stick attached to the rod, and a long tape measure.
INSTRUCTIONAL OBJECTIVES ARE TO GAIN COMPETANCE IN THE
FOLLOWING:
1. Planning an experiment
2. Deciding what data needs to be taken, how much data needs to be taken, and over what range.
3. Organizing a data table.
4. Error estimation and minimization of systematic errors.
5. Calculation of averages and standard deviations.
6. Comparing results with accepted values.
EQUIPMENT: Ring stand, meter stick clamp, meter stick, angle meter, long measuring tape.
PLANNING: Go outside and look at the height building height H which is to be determined.
Notice that the sidewalk and lawn are sloping away from the building and that θ is measured
from the horizontal. The reason for the meter stick on a stand directly under the building corner
is so when θ is at 0º, you have a reference point on the meter stick, (call it h) which can be added
to a calculated distance Y, (using θ and X) from this reference point, to determine the building
height H. You need to figure out the details for doing this.
Decide what range of values you will use to get 10 different angles and baseline values for X.
Hint: you will not get accurate results for small angles or for large angles approaching 90º.
Using a straight edge, organize a data sheet with appropriate rows and columns for your data.
Label each row and column with the appropriate variable and units. Draw a neat diagram on
your data sheet with all the variables labeled.
EXPERIMENTAL PROCEDURE:
1. Place a meter stick on a stand directly beneath the building corner with the zero end touching
the walkway. (Your instructor or lab tech will do this.)
2. Attach your meter stick to your meter stick clamp and then to your ring stand and place the
ring stand at selected distances X, then measure the corresponding angles θ.
3. Each time a new distance X and angle θ are used, a new reference height measurement must
be made at θ = 0º on the meter stick below the building corner.
ANALYSIS: Calculations must be done neatly in this section and the details must be shown.
THE SIMPLE PENDULUM
INTRODUCTION: The Simple pendulum consists of small mass, m, suspended by a string.
The period of the pendulum, T, is the time for the mass (bob) to go from one extreme position to
the other and return (the time for one complete swing). The length, L of the pendulum is the
distance from the point of suspension to the center of the bob. The amplitude, θ, of the
pendulum’s swing is the angle between the vertical position and the extreme position. The
experimental objective is to determine if the period, T, depends on the variables θ, L, and m, and
then find an empirical equation for the period, T, as a function of those variables. The empirical
equation will then be compared with the theoretical equation. The acceleration of gravity will
also be determined from the slope of the period graph.
INSTRUCTIONAL OBJECTIVES: Give the student practice in
1. Organizing a data sheet
2. Taking data when many variables are involved
3. Error estimation
4. Minimizing systematic errors
5. Representing data in graphic form
6. Obtaining equations from graphs
7. Comparing experimental results with accepted values
EQUIPMENT: Large wooden protractor, monofilament string, pendulum clamp, triple beam
balance, table clamp and rod, aluminum, lead, and steel spheres, meter stick, stop watch.
EXPERIMENTAL PROCEDURE:
1. Using a straight edge, organize a data sheet consisting of columns and rows. All of the
variables should be listed at the top of the column. Note: Reaction time is a significant source of
error. To minimize error due to reaction time, let the pendulum swing ten times. You will thus
need 10T at the top of a column as well as T.
2. Estimate the absolute error in each measurement and list it at the top of each column.
3. Set up the pendulum using the listed equipment.
4. Using a string length of about one meter and using the three masses, determine if the period
depends on the mass.
5. Keeping the length at about one meter and using the lead mass, determine if the period
depends on the angle, θ. Measure 10T for several amplitudes from 5 degrees to 80 degrees.
6. Using the lead mass and an angle of about 20 degrees, vary the length in 5cm increments
starting at one meter and decreasing the length to as small as possible as allowed by the
protractor.
ANALYSIS: From each set of data, for the three variables, find, within experimental error,
whether or not the period depends upon the variable. If the period depends on the variable, plot a
graph of T verses the variable. (T goes on the y axis.)
From the graph of T vs. L, with L in meters, find the empirical equation of T as a function of L.
Determine the proportionality constant and the units of the constant.
 2 
 L . Find the % discrepancy of your
The theoretical equation for small angles is T  
 g


2
proportionality constant compared to
. From your proportionality constant, determine a
g
value for g and compare it (% discrepancy) with the accepted value of g.
GRAPHING
BY HAND:
Plot a graph of period T versus L . Take L = 0m to be at the origin. If this graph gives a
straight line, this shows that T depends on L . When drawing your graph, choose appropriate
scales for the x and y axes so the graph comes close to filling the page while, at the same time,
keeping an easy to read scale. The graph needs to have a descriptive title, the x and y axes need
to be labeled with the variables you are graphing and the units need to be on each axis also.
Each point on the graph should be just a small point with a protective circle around it. Your line
should represent a visual best fit straight line (try to draw a line which represents an average with
about as many points on one side of the line as the other). Draw a large slope triangle on your
graph using easy to read grid points, and from this, determine the slope of the line. This slope
2
represents “
”. Draw maximum and minimum slope lines on your graph and from maximum
g
and minimum slopes on your graph, determine the % error in the slope.
QUESTION: (put calculations and answer in analysis section)
From an advanced mathematical solution for the period of the simple pendulum, the theoretical

L  1 2  9
 
expression for the period is: T  2
1  sin    sin 4    ...
g 4
 2  64
2

Where g is the acceleration of gravity and the terms in parentheses are part of an infinite series.
For small angles, the θ terms in the series are small and a good approximation for the period is
found by taking only the first term in the series. The approximation is
L
. For an angle θ of 60º, how many additional terms must be used in
T  2
g
order for the theoretical period to exceed this approximation by 7%?
REPORT: For this experiment your report will contain only the following two sections of a
formal report.
ANALYSIS: This section includes a sample calculation of each type with error propagation if
appropriate. Any graphs needed should also be in this section, as well as the calculation of
percent discrepancies.
CONCLUSION: This section includes a summary of the results along with the estimated error.
Error may be the random error as calculated from the standard deviation, or it could be an error
calculated from error estimates in your measurements. When a physical quantity is measured, as
in this lab, include in the conclusion a comparison of the measured value with the accepted value
(% discrepancy.) Note whether or not the % discrepancy is greater or less than the predicted
uncertainty; if it is greater, try to give a reasonable explanation as to why. Comments or
suggestions for improving the results are appropriate here also.
ACCELERATION OF GRAVITY
INTRODUCTION: The value of g, the acceleration of gravity, is to be determined using the
Behr free-fall apparatus. Measurements of distance intervals and time intervals will be used to
calculate speeds at various times. A graph of speed versus time will be used to calculate the
magnitude of the acceleration of a “freely falling” body.
EQUIPMENT: Behr free-fall apparatus, vernier calipers, meter stick.
INSTRUCTIONAL OBJECTIVES
1. Practice using the calipers
3. Organizing a data sheet
4. Experience drawing graphs on graph paper
5. Estimating error using maximum and minimum slopes
6. Learn to graph using Microsoft Excel
7. Become familiar with writing an ABSTRACT and an INTRODUCTION to a formal report
EXPERIMENTAL PROCEDURE
Since a free-falling body acquires a fairly large velocity in a short time interval, a special
apparatus is required to measure its position at short time intervals. The apparatus we will use is
called the “Behr Free-fall Apparatus.” It consists of a projectile which is dropped between two
vertical wires and a waxed tape. A high voltage spark timer produces sparks at the rate of 60
pulses per second between the two wires and leaves small holes in the waxed tape each time it
pulses. We thus have a waxed paper strip which has marks on it at time intervals of 1/60th of a
second between each mark.
BEHR FREE-FALL
APPARATUS
The tape appears as shown in the following figure with the points numbered.
__________________________________________________________________
.. . . . .
.
.
.
.
.
.
.
.
012 3 4 5 6
7
8
9
10
11
12
13 etc.
__________________________________________________________________
The time interval between each point is, of course, 1/60th of a second.
Operate the apparatus and obtain one tape for each partner group.
Number the points on the tape 0, 1, 2, etc. Do not start your numbering at the first points since
the first few points are too close to permit accurate measurements. Start well down the tape
where the points are at least one cm apart. (Remember that you will be drawing a graph of v vs.
t and the zero point for time is arbitrary; it just means that the beginning velocity is not zero).
Take measurements of Δx values which will be used to calculate instantaneous velocities at
various times. For example, the instantaneous velocity at point 1 is v1= Δx 0,2/Δt . This would
be the velocity at time t1=1/60th sec. v3= Δx 2,4/Δt would be the velocity at time t3= 3/60th sec.
Δt= 1/30th of a second for each interval.
As you can see, the measurements for Δx that you need to make are distances between points
(0,2), (2,4), (4,6) etc. which will be used to calculate v1, v3, v5, v7, etc. until you obtain at least
ten points for a graph of v vs. t. The average velocity over an interval, such v1= Δx 0,2/Δt gives
the instantaneous velocity v1 at the mid-time point of the interval. Using a straight edge,
construct a data table with columns and rows and label the top of the columns with Δx (i,j), Δt, t,
vi. Measure the various Δx values as accurately as possible using vernier calipers.
ANALYSIS
If the instantaneous velocities at various times are plotted on the y-axis verses the time on the xaxis, a straight line should be obtained and the slope of the line will be the acceleration.
GRAPHING
BY HAND: ( see handout on graphing)
Plot a graph of your velocities against the time. Take time t = 0sec to be at the origin. When
drawing your graph, choose appropriate scales for the x and y axes so the graph comes close to
filling the page, while, at the same time, keeping an easy to read scale. The graph needs to have
a descriptive title, the x and y axes need to be labeled with the variables you are graphing and the
units need to be on each axis also. Each point on the graph should be just a small point with a
protective circle around it. Your line should represent a visual best fit straight line (try to draw a
line which represents an average with about as many points on one side of the line as the other).
Draw a large slope triangle on your graph using easy to read grid points, and from this,
determine the slope of the line. This slope represents “g”. Draw maximum and minimum slope
lines on your graph and from maximum and minimum slopes on your graph, determine the %
error in g.
Using Microsoft Excel (see handout) plot v vs. t and on your graph. Include a TITLE, UNITS
and VARIABLE NAMES, EQUATION, and the CORRELATION COEFICIENT.
REPORT
In addition to an analysis section and conclusion as included in the last lab report, you are to
write and ABSTRACT and INTRODUCTION in your report for this experiment.
ABSTRACT: On your cover sheet to the lab, you should have the class name and time, your
name and your partner’s name and the experiment title. In addition, the cover sheet should
have an “ABSTRACT”. For complete instructions on writing an abstract, refer to
http://www.ece.cmu.edu/~koopman/essays/abstract.html. For our purposes the abstract should
be one paragraph stating the experimental objective, how the experimental objective was met,
and the results of the experiment including any comparison with accepted values. The purpose of
an abstract is to enable the reader to basically have a summary of the experiment to see if the
paper is about a topic the reader is researching. The goal is to make the abstract as complete but
as concise as possible.
INTRODUCTION: This consists of one or two sentences describing the experimental objectives
of the laboratory and what you are going to do to accomplish those objectives. An example
would be “By varying the length, mass and amplitude of a simple pendulum, how the period of
the pendulum depends upon those variables will be determined.” Notice that statements such as
“To learn how to analyze data” or “To learn how to organize a data sheet” or other “Instructional
Objectives” do not belong in the introduction or anywhere else in the report. These comments
will be found only in the laboratory manual.
FORCE TABLE
INTRODUCTION: The force table is an apparatus that allows the experimental determination
of the resultant of force vectors. It consists of a large aluminum disk with the rim graduated in
degrees. Forces are applied to a central ring by means of strings passing over pulleys and
attached to weight hangers. The magnitude of the vector is varied by adding masses to the
weight hangers and the direction of the vector is changed by moving the pulleys along the rim at
different angles. Vectors will be added experimentally using the force table. The vectors will
also be added graphically and analytically. The magnitude of the experimental resultant will be
compared to the magnitude of the resultant as found from the analytical method.
EQUIPMENT: Force table apparatus, four pulleys, weights and weight hangers, string,
protractor, ruler, bubble level, graph paper.
THEORY: Review the analytical method using components and also the graphical methods of
adding vectors as found in your text. You may also go to:
http://www.rowan.edu/colleges/lasold/physicsandastronomy/LabManual/labs/TheForceTable.pdf
EXPERIMENTAL PROCEDURE:
When two or more forces are applied to the ring, their vector sum, or resultant R, can be found
by finding the additional force needed to balance the applied force. For example, if two forces
are applied, the resultant, or vector sum, is
(1)
F1 + F2 = R
The magnitude and direction of R can be found by finding a third force E such that
F1 + F2 + E = 0 or,
(2)
F1 + F2 = -E
E is called the equilibrant and we can see when comparing equations (1) and (2) that
-E = R
Remember that taking the negative of a vector is just reversing its direction by 180º. Thus, to
find the resultant, just find the equilibrant and add 180º to the angle.
1. Level the force table.
2. Place a pin in the hole at the center of the table.
3. Attach strings, and weight hangers to the ring. Make sure the strings are free to slide on the
ring.
4. Attach the pulleys to the disk at the angles given in the vector problem.
5. Add the masses given in the vector problem (be sure to include the weight hanger in the
total.)
6. Find the equilibrant needed to center the ring around the pin. (Note: You can find the proper
direction of the equilibrant by just pulling on the string with your hand while trying different
angles until you find the right angle to center the pin. You can then just add weights)
Using the above method, find R for the following vectors.
Note: If we assign a direction to a given mass on the weight hanger, we then have a magnitude
and direction for the mass and can hence treat it as a vector. It will not then be necessary to
multiply the mass by g to get force. We shall just do everything in mass units.
VECTOR PROBLEM I:
F1
MAGNITUDE
200g
ANGLE
30º
F2
200g
120º
EI
_____
______
RI
_____
______
VECTOR PROBLEM II:
F1
MAGNITUDE
500g
ANGLE
0º
F2
300g
90º
EII
______
_____
RII
_________
________
VECTOR PROBLEM III:
F1
MAGNITUDE
200g
ANGLE
30º
F2
100g
90º
F3
300g
170º
EIII
______
______
RIII
______
______
ANALYSIS:
Using the polygon method, draw each vector to scale on graph paper. Use one sheet of paper per
problem. You do not need to draw the equilibrant, just the given vectors and the resultant.
Choose a scale so the vector diagram fills up most of the sheet (this gives more accuracy.)
Make sure each vector has arrows on the tip and show all the angles. Show the scale calculation
for the resultant.
Using the diagrams from the graphical method, break each vector into x and y components and
use the component method to find R. Use a different color to show the components.
REPORT: Hand in your initialed data sheet, the graphical and analytical calculations, and a
summary table. Calculate the percent discrepancy comparing the resultant from the force table
with the resultant as calculated using the analytical method.
When there are several results to report, a summary table should always be included in the
conclusion. Since this is your first lab with several results, a sample summary table is shown
(use excel).
PROBLEM
FORCE TABLE
R
θ
GRAPHICAL
R
θ
ANALYTICAL
R
θ
%
DISCREPANCY
R
I
II
III
QUESTIONS:
1. Why is it necessary to include the mass of the weight hangers? Since they have the same mass
shouldn’t it cancel out?
2. What are the sources of error? List them. What is the largest source of error?
3. Why is it necessary to let the string slip on the ring?
4. What would be the effect of a more massive ring?
CANTILEVER BEAM
INTRODUCTION: A wooden meter stick will be used as a cantilever beam as shown in the
diagram. As mass M is added to the end of the beam, the deflection d will increase. The
deflection d for a given mass M also depends upon the length of overhang L. By measuring
d for varying L while keeping M constant, and by measuring d for varying M while keeping
L constant, two sets of data can be obtained which can then be graphed. One graph will yield an
equation of d as a power function of M, d = k1Ma and the other graph will yield an equation of
d as a power function of L, d = k2Lb. These two equations can then be combined
mathematically to yield a single equation,
d = k3MaLb. The experimental objectives are to determine the values of k3, a, and b.
INSTRUCTIONAL OBJECTIVES: This Exercise is designed to give the student practice in
designing an experiment, determining the order in which to take data, analyzing data using
graphical techniques, and combining two equations to get one “joint variation” equation.
EXPERIMENTAL PROCEDURE:
1. Determine what you are going to use as a reference for the measurements of the deflection d.
Remember that d is the distance the meter stick deflects with a mass M placed on it compared to
its position with no mass.
2. Clamp the meter stick to the table, and, using a ruler, measure the deflection d for various
lengths L using a fixed mass M (you need to figure out what to use as a reference for measuring
d). You also need to determine what mass to use but the mass M should not be large enough to
cause a deflection of more than 10 cm when the stick is out at an L of 90 cm. If the deflection is
more than 10% of the length, the meter stick may break and it no longer behaves like a cantilever
beam. (You decide on the order of data for L values. Should you start with large or small values
of L?)
3. Now vary M for a fixed L. You need to choose a value for L. Keep in mind that small
lengths do not give much deflection and hence d would be hard to measure. Also, values of L
that are too large give a lot of deflection for very little mass and thus limit the range of values
you can use for M.
ANALYSIS: Graph your data using EXCEL. Force each graph thru (0, 0) and use “power
function” trend line for d vs. L and “linear” trend line for d vs. M. Have each equation shown
on each graph. Compare the equation on the graph of d vs. M to the equation d = k1Ma, and
determine the values of k1 and a. Compare the equation on the graph of d vs. L to the equation d
= k2Lb, and determine the values of k2 and b. Now comes the tricky part. Remember the final
equation is: d = k3MaLb , and when you varied M, L was a constant. Let’s call this constant
value L0. For this value of k3, the equation becomes : d  k3 M a Lbo . Note that the value of k1
= k 3 Lb0 ,
and by knowing the value of k1 and L0, k3 can be calculated. Calculate the value of k3,
including units and write the final equation with your values of a and b included.
TABLE
L
d
M
.
REPORT: Write a complete report with all the necessary sections as listed in “The Simple
Pendulum” lab. Pay particular attention to the analysis section and show clearly how you
obtained the final equation.
FRICTION
INTRODUCTION: The coefficient of kinetic friction between a block of wood and a wooden
plane will be determined by measuring the friction force while varying the normal force and by
measuring the angle of repose. The effect contact area of the sliding surface will also be
investigated.
THEORY:
When relatively smooth solid surfaces slide over each other, the force of kinetic friction, fk , is
proportional to the normal force FN and is directed opposite to the displacement. The
coefficient of kinetic friction is define by the equation:
fk
FN
re-arranging the equation gives (1) f k   k FN . When fk is plotted on the y-axis and FN
plotted on the x-axis, the slope of the straight line is  k .
k 
is
Before the lab period, the student shall derive the equation for the angle of repose, θR, for kinetic
friction (the angle for which the block slides down the plane at constant speed).
Draw a force diagram, use
F
x
 0 and  Fy  0 to derive the equation:
(2)  k  tan R
EQUIPMENT: Plane board, wood block, weight hangar, pulley, string, weights, table clamps,
angle meter, right angle clamps, rods, triple beam balance.
EXPERIMENTAL PROCEDURE:
Part 1: Mass the block. Place the plane board flat on the table, attach a pulley to one end and
place the block on the plane. Run a light string from the block, over the pulley, and attach a
weight hangar. The pulley should be attached to the end of the block without the hole thru it.
This will insure the grain of the wood is the same direction as in part 2 of this experiment. Notice
that the block has two sliding surfaces, one wide and one narrow. Start with the wide side down.
Notice that the pulley is adjustable in height. Make sure the pulley is adjusted so the string is
parallel to the sliding surface. The wooden blocks should only be held by the sides. Touching
the sliding surface with sticky or greasy hands will change the coefficient of friction.
Add weight to the weight hangar until the block slides with constant speed when started in
motion with a light tap. Add masses to the block in 200g increments up to 1200g. For each mass
MA added to the block, record the total mass m, hanging on the string, which makes the block
move with constant speed.
Repeat part 1 with the narrow side of the block as the sliding surface.
Part 2. Slide a small rod thru the hole in the end of the wood plane and attach a right angle
clamp to the rod. Attach the angle clamp to a vertical rod and then attach the vertical rod to a
table clamp. Keeping the grain of the block and plane the same as in part 1, place the wide side
of the block on the plane, change the angle between the plane and the table until the block slides
down the plane at constant speed when given a small tap. Measure this angle with the angle
meter. This angle is the angle of repose.
Repeat part 2 for the narrow side of the block.
ANALYSIS:
Part 1. The friction force is equal to mg where m is the mass attached to the string when the
block slides at constant speed. The normal force is equal to Mg where M is the mass of the
block plus added masses. Thus, equation (1) becomes mg  Mg or m  M . Thus, we do not
need to multiply the masses by g to get weight and we can just graph m on the y-axis and M on
the x-axis. The slope of the line will be µ.
Plot graphs of m verses M for the wide side and the narrow side. Draw the best fit line thru the
points for each graph and find the slope of these lines. The slope will be the coefficient of
friction. Put error bars for m on the graph points. Draw maximum and minimum slopes on the
graph and determine the slope of the max and min slopes. The percent error in the slope is given
max slope  min slope
by  slope 
and this is the % error in µ.
2slope 
Part 2. From the equation  k  tan R , determine the coefficient of friction for the wide and the
narrow sides.
Find the % difference for the wide side using the results from parts 1 and 2.
Find the % difference for the narrow side using the results from parts1 and 2.
Note: percent difference =
x1  x 2
x avg
Using the equation  k  tan R and differential error propagation to determine the predicted
error in µ.
REPORT: Unless asked by your instructor to do a formal report, your report shall include the
following sections:



THEORY
ANALYSIS
CONCLUSION (summarize the results and state whether or not the contact area makes
a significant difference in the coefficient of friction? In other words, is the difference
greater than predicted by the % error in the value of µ? State sources of error and how
the results could be improved)

APPENDIX (answers to questions and original data sheet)
QUESTIONS:
1. If the tangential force of the string just balances the frictional force, why does the block
move?
2. How does friction in the bearings of the pulley change the coefficient of friction?
3. (problem) Using the mass of your block and using the coefficient of friction for the wide side,
determine how much force parallel to the plane would be required to move the block up the
plane at a constant speed? Assume the plane angle is 30º
CENTER OF MASS
INTRODUCTION: Using five nearly identical meter sticks, you are to find the best systematic
procedure for stacking the sticks lengthwise, one on top of the other, out over the edge of the
table such that end of the top stick is at a maximum distance D from the edge of the table. Once
the proper method of stacking is found, measurements of the displacement of each stick, relative
to the stick immediately below it will be made and a sequence, for this displacement, determined
from these measurements. This sequence will be summed to find a series which can be used to
predict the theoretical value of the distance D. This sequence will also be used to predict the
location of the center of mass of the stacked sticks.
PROCEDURE: By trial and error, determine a systematic method for stacking the sticks.
Should you start with the bottom stick, the top stick, the middle stick, or what? We shall refer to
the top stick as stick number one, the second from the top as stick number two etc. In general,
we can have n sticks, n = 5 for our case. If the proper method of stacking has been discovered,
the distance D of the end of the top stick from the edge of the table will be over 110 cm. Once
the proper method of stacking has been determined, take measurements of D, d1, d2, d3, d4, d5 as
shown in the following diagram. Systematically shuffle the sticks and repeat the measurements
five times.
d3
d5
d2
d4
D
d1
ANALYSIS: Let L represent the length of the sticks.
1. Find the closest fractional value of d1, d2, d3, d4, d5 in terms of the length L. For example, d1
= L/2. You find the rest. These values of dn represent a sequence for the distance of the front
edge of each stick relative to the stick below it. Now find a general term for the sequence. An
example of a general term would be dn = L/n2. This, of course, is not the correct general term.
2. Represent the distance D as the sum of the sequence, D = d1 + d2 + d3 + d4 + d5 using the
general term determined from part 1 of the analysis. The sum of a sequence is called a series.
Use this series to predict the value of D and compare it to the measured value of D. For example,
5
L
using the sample sequence above, the series would be D   2
1 n
3. Using the definition for the center of mass, find the location of the center of mass of the five
sticks using the sequence from part 1 of the analysis. Choose a coordinate system with the origin
located at the edge of the table. Let x1 be the coordinate of the number 1 stick (top stick), x2 the
coordinate of stick number 2 etc. Now x1=d5+d4+d3+d2. Now substitute the value for d5, d4, etc.
using your sequence. Do the same for x2, x3, etc. Substitute these values of xn into the equation
5
5
1 M i x  1 M i xi where x is the center of mass and then solve for x . Hint, assume all


masses are equal and simplify the above expression by solving for x before doing any
substitutions. Another hint is that x2=d5+d4+d3+d2 –L/2. Where does your intuition tell you the
center of mass is located?
4. Extra credit: How far from the table would the far end of the top stick be if the number of
sticks approached ∞? Or, if you let the number of sticks get larger and larger, say approaching ∞,
what does the value of D approach? Could D approach ∞? In other words, prove that the series
diverges. Use the integral test.
REPORT: In your report, be sure to include the stacking method, detailed analysis for each
section above, and a conclusion which summarizes your results and compares your measured
results of D with the calculated value.
BALLISTIC PENDULUM AND
PROJECTILE MOTION
One period for part 1 and 2
One period for part 3
INTRODUCTION: The initial speed of a projectile fired from a spring gun will be determined
by two methods: Part 1 will use the ballistic pendulum to find the initial speed. In part 2, the
measurement of the projectile range, when it is fired horizontally from a known height, will be
used to calculate the initial speed. In part 3, the known initial speed from parts 1 or 2 or both
will be used to calculate the range of the projectile when it is fired at an angle θ from the table
and lands on the floor. The projectile will be fired and its actual range will be compared to the
calculated value. Also in part 3, the projectile will be fired at various angles at ground level and
the range versus the angle will be plotted on a graph.
EQUIPMENT:
Ballistic pendulum apparatus (Record the equipment number on your data sheet.)
Triple beam balance
Table clamp
Tape Measure
Meter stick
Plumb bob
Target paper
Carbon paper
THEORY: (to be completed by the student before class)
Part 1 – Ballistic pendulum: The ball is shot horizontally form a spring gun with speed v0 and is
caught by the pendulum bob. The ball and the pendulum rise to an angle θ, the highest point.
Let m be the mass of the projectile, L is the pendulum length, M is the mass of the pendulum
bob and shaft, h is the maximum height to which both will rise, v0 is the initial speed of the ball
(this is what we want to find.) V represents the speed of the ball and pendulum immediately
after impact.
The speed after impact, V, can be found by using conservation of mechanical energy:
(Kinetic Energy of pendulum and ball immediately after impact) = (Potential Energy of ball and
pendulum at angle θ).
From this equation the speed, V, can be found in terms of the vertical rise h. Note that masses
cancel out in this equation. Write down the equation and solve it for V. Call this Equation (1).
Next, we need to get v0 in terms of V. Using conservation of momentum for the completely
inelastic collision of m with M we have:
(momentum of the ball, m, before the collision) = (total momentum of the ball and pendulum,
M+m, after the collision)
Write down this equation and solve it for v0 in terms of V, m, and M. Call this equation (2).
Combine equations 1 and 2 and solve for v0 in terms of M, m, h, and g. Call this equation (3).
Next, it is necessary to get h in terms of the pendulum length L and the angle θ. From the
diagram and using some triangle trigonometry, show that h = L(1 - cosθ). Substitute this into
equation (3) to get the final equation.
Part 2: Calculation of v0 from the measurement of the range. The pendulum is moved up at 90º
and locked in position so it is out of the way and the projectile is fired horizontally from a known
height H above the floor. The range X is measured, and from this known range and height, the
initial speed v0 can be calculated. Using the equations of motion from lecture, derive the
equation for v0 in terms of H, X, and g. Derive this before class and include the derivation in
your report. Hint, use the y motion to find t and then substitute this time into the x equation. Let
y=0 where it lands and call y0=H
Part 3: The spring gun will now be set at an angle θ with respect to the horizontal. Using the
known velocity v0, derive an equation for the range R of the projectile.
1
The equations you will use are x  x0  v0 cos  t and y  y0  v0 sin  t  gt 2
2
The equation for R will be in terms of v0, θ, H, and g. Derive this equation before coming to
class. Hint, use the y motion to find t and then substitute this time into the x equation. Let y=0
where it lands and call y0=H. The solution of the y equation (quadratic) is a bit tricky and is
subject to algebraic errors (especially sign errors.) Work independently of your partner so you
can check each other for errors. Compare your final equation with your partner’s.
For the theory regarding the range versus the angle, you can look this up in your text.
PROCEDURE
Part 1: Initial velocity from the Ballistic Pendulum. Place the pendulum apparatus on the table.
Put a table clamp behind the launcher as a backstop (do not clamp the base to the table as it will
crack the base). Remove the ball from the launcher and mass it. Do the same for the pendulum
and rod as a unit. The length L of the pendulum is 28.5cm.
Put everything back together and put the ball back in the launcher. Use the ram rod to cock the
launcher to the desired spring tension (there are three levels). Set the angle meter to zero and fire
the projectile into the pendulum. Read the angle and repeat this at least 10 times. Calculate the
value of v0 using the equation derived in the theory section.
Part 2: On the launcher, there is a white circle with a cross on it. This is the reference point for
all height and distance measurements. Place the launcher next to the edge of the table so you can
drop a plumb bob from this point to the floor. Put a table clamp behind the launcher as a
backstop. Drop your plumb bob from the cross in the circle and put a mark on the floor, where it
touches the floor as a reference for your range measurements. Cock the launcher with the same
spring tension as in part 1 and do a test firing. Tape an 8 1/2 X 11 inch piece of paper on the
floor, centered where your test firing landed. Place a piece of carbon paper over this paper and
fire the launcher at least 10 times. Remove the carbon paper and measure to the center of the
grouping. This will be your value of X. Measure the distance from the floor to the cross on the
launcher. This will be your value of H. Calculate v0 using the equation derived in the theory
section.
Part 3: The value of θ will be provided by your instructor. Remove the launcher from its mount,
rotate it 180º and reconnect it to the launcher at the angle given. Measure the new height H from
the floor to the cross. Using equation derived in the theory section and the value you obtained for
v0, (you decide whether to use the average value of v0 obtained from parts 1 and 2 or the value of
v0 from part 1 or part 2) calculate the range R . Again, work independently of your partner and
compare results.
After calculating what R should be, tape a sheet of target paper to the floor, lengthwise along the
line of fire, centered at the calculated distance R. Place a sheet of carbon paper on top of the
target paper. Do not shoot the ball until your instructor has seen your calculated distance and is
present to watch the shot. Fire the ball several times to get a grouping of points. Remove the
carbon paper and measure the distance to the center of the grouping. Compare this distance to
the calculated distance R. Points will be assigned depending on what region of the target the ball
lands.
Now fire the projectile at ground level for 5º intervals starting at 30º and ending at 60º. Graph the
range versus the angle. Is the shape of this graph as expected? Discuss in light of the theory
from your text.
REPORT: One report for both parts. In the theory section, be sure to include all of the
derivations for each part. In the calculation section, write down the equation you derived in
the theory section. In the next step, substitute the numbers showing just one sample of each
type of calculation along with appropriate error propagation. In the conclusion, be sure to
indicate whether or not the percent discrepancy in part 3 is less than the predicted percent error.
QUESTIONS: (put answers in the appendix)
1. An experimenter measures X and H in part 2. Suppose that, unknown to the experimenter,
the floor slopes down at 1º in the direction of firing. The experimenter calculates v0 using his
measurements, assuming that the floor is level. Will the calculated value of v0 be larger or
smaller than the v0 calculated from measurements made on a level floor? Justify your answer.
2. Suppose the gun were mounted on frictionless rollers instead of being firmly anchored to the
table. Would the velocity of the ball be smaller or larger than it would be if the gun were
anchored? Justify your answer.
ROTATIONAL MOTION AND THE MOMENT OF INERTIA
Introduction: When a sphere, solid cylinder, and a hollow cylinder are
released at the top of a triple track, one finds that the sphere arrives at
the bottom first, the solid cylinder arrives second, and the hollow
cylinder arrives last. The objective of this experiment is to measure the
times it takes for each object to travel down the track and compare these
times with those predicted from theory.
Equipment: Triple beam balance, Stop watch, Angle Meter,
Meter stick, Vernier calipers, Triple track,
Table Clamp, Rods (long and short) , Rod Connector
Rolling objects (sphere, solid and hollow cylinder)
Drawing tools ( T-Square, Triangles, ruler, etc.)
Theory: From the definition of the moment of inertia  r 2dm it can be
shown that
Ihollow cyl= mr2
(thinned walled)
Ihollow cyl = 1 mr12  r22  (thick walled)
2
Isolid cyl= ½ mr2 (derive this)
Isphere =
2 2
mr
5
Notice that each of the above moments of inertia, except the thick
walled cylinder, can be expressed as
I = kmr2
where k=1 for the thinned walled cylinder, k=1/2 for the solid cylinder,
and k= 2/5 for the sphere.
To find an expression for the time it takes each object to travel down the
ramp, use conservation of energy to first find the final speed at the
bottom of the ramp.
mgH 
1 2 1 2
mv  I
2
2
where H is the height of release above the bottom of the ramp.
Substitute the expression for
v.
I  kmr2
in the above equation and solve for
Once v has been found, it can be substituted into the equation for
average speed
L
to find the time where L is the length
vavg 
t
of the ramp.
Show that:
Note that
t  2L
k 1
.
2 gH
H=Lsinθ
Note that the equation for I of a thick walled hollow cylinder cannot be
expressed as I=kmr2 and therefore the equation shown for t is not as
accurate as one derived using
Ihollow cyl = 1 mr12  r22  (thick walled)
2
Where r1 is the inside radius and r2 is the outside radius. Using this
equation for I, derive a more accurate equation for t. (does v=r1ω or
does v= r2ω ?)
L
H
θ
Procedure:
1) Adjust the incline to θ = 5º
2) Make ten measurements of t for each object
3) Measure L, H, m, θ, routside for all objects, and rinside for the hollow
cylinder.
4) Include as part of your data, an estimate of the errors for the above
measurements.
5) Repeat all measurements for θ = 10º
Analysis: Using the equation
t  2L
k 1
2 gH
, calculate the time it takes for
each object to travel down the ramp for each angle. (Use measured
values of H). Using error propagation and the error estimates in your
data, calculate the % predicted error in t when θ = 10º for the solid
cylinder. Use estimated errors in H and L to find % error in t.
Compare your measured values with the calculated values.
Use the more accurate thick walled cylinder equation derived for the
time t, to calculate new values of t for the hollow cylinder.
Report: This will be a formal report. Be sure to include all of the
sections necessary for a formal report. All of the report must be done
with a word processor including using equation editor for all of the
mathematics. Nothing except your data should be in handwriting.
The theory section must include the derivation of the moment of inertia
for a solid cylinder, using integration, as well as the derivation of the
equation for t.
In your conclusion, you should include a summary table showing the
average of the measured values of t for each angle and the calculated
value of t for each angle along with the % difference.
YOUNG’S MODULUS AND TORSION MODULUS
INTRODUCTION: The experimental objectives of this lab are to find values of Young’s
modulus and torsion (shear) modulus for steel.
EQUIPMENT:
Young’s modulus apparatus with dial indicator
Torsion modulus apparatus and torsion rod
Bubble level
Micrometers
Meter stick
Assorted large and small weights and hangars
Bow calipers
THEORY:
Young’s Modulus
For the description of the elastic properties of linear objects like wires, rods, columns which are
either stretched or compressed, a convenient parameter is the Young's modulus of the material.
Young's modulus can be used to predict the elongation or compression of an object as long as the
stress is less than the yield strength of the material.
F
L
, divided by the strain,
where F is the force
A
L
stretching the material, A is the cross-sectional, ΔL is the elongation, and L is the original length
of the material. In this lab, a wire will be stretched by adding weights to a weight hangar. Thus,
F=mg and Young’s modulus becomes
F
mg L
F L
or Y 
Y A =
L A L
A L
L
YAL
If we re-arrange this equation, we can write m 
.
gL
Young’ modulus is defined as the stress,
A graph of m vs. ΔL will then have a slope of
YA
from which Y can be determined.
gL
Torsion Modulus
Usually the sheer or torsion modulus is measured by applying a torque to one end of the rod
which is fixed at the other end and measuring the angular rotation φ. It can be shown, by using
the definition of shear modulus and integrating, that the torsion modulus is given by:
2l
(you do not need to derive this equation)
r 4
M is the shear modulus
l is the length of the rod
r is the radius of the rod
φ is the angle of twist or rotation in radians
M 
where
A weight mg will be attached to a wheel of radius R to give a torque τ = mgR and the above
2lmgR
equation then becomes M  4
.
r 
Re-arrange this equation as was done with Young’s modulus so a graph of m vs. φ gives a
straight line.
PROCEDURE:
Young’s modulus:
1. Place one kilogram on the weight hangar initially to eliminate kinks. Leave this on
throughout the experiment and do not count it as part of the load. The elongation which the
initial weight produces will not enter into your measurements since you have not zeroed your
reading of the wire length.
2. Level the stand using the bubble level
3. Measure the diameter of the steel wire using micrometers
4. Measure the length of the wire which is subject to stretching.
5. Take a “zero” elongation reading of the dial indicator
6. Add weights one kilogram at a time and record the dial indicator reading each time.
7. Once you have added as many weights as possible and taken your dial indicator readings,
remove the weights on at a time and again take readings each time a weight is removed.
Note: Be sure to record the absolute error in all of your measurements since this will be needed
for error propagation in your analysis.
Torsion Modulus
The apparatus will be set up by the lab tech at the back of the lab. When you finished, just leave
the apparatus as you found it.
1. Adjust the vernier to zero when an initial mass of 200 grams is suspended from the strap. Do
not count this initial weight in your calculations.
2. Add masses 0.50kg at a time and record the angle reading after each addition. Do not twist
the rod excessively. Do not exceed a total of 5.0kg
3. Measure the rods length and diameter.
4. Measure the diameter of the wheel on which the masses are hung.
Note that the vernier for measuring the angle is in tenths of a degree.
ANALYSIS:
Young’ modulus: Plot a graph of m verses ΔL and from the slope, calculate a value of Y and
using error propagation, calculate the predicted % error in Y. Compare your value to the
accepted value of 200 GN/m2.
Remember, “compare” means to find the % discrepancy and see if it is within the margin of
error as determined by error propagation.
Torsion modulus: Using the analysis of Young’s modulus as a guide, decide what quantities to
graph and perform an analysis similar to that, described above, for Young’s modulus.
Report: For your report, include all of the sections (abstract, introduction, etc.).
ARCHIMEDES’ PRINCIPLE
INTRODUCTION: Archimedes’ principle will be used to determine the density of a liquid, a
solid that has a density greater than that of water, and a solid with a density less than that of
water.
EQUIPMENT:
(for a group of 2)
2 Beakers
1 Triple beam balance
Table clamps and rods
1 Vernier caliper
1 Metal cube with hook
1 wood block
(for class use)
Monofilament line
Graduated cylinder filled with alcohol
Hydrometer (placed in graduated cylinder)
Analytical balance
Jug of alcohol mixture
THEORY: The mass density ρ, of a substance, is the mass per unit volume or,
m
Eqn. (1)   .
V
The mass density of water is 1.00 g/cm3 .

The specific gravity (s.g.) is defined as s.g. 
.
 water
Since the denominator has a numerical value of one when using the density of water as 1.00
g/cm3 , the specific gravity will have the same numerical value as the density of the substance but
will have no units. Physics students know that w=mg and is in units of dynes or newtons
(chemistry students may not know this.) However, since our balances “weight” things in grams,
we will use grams as though it is force and not multiply any masses by g. We will call it gram
force.
Archimedes’ principle states that the buoyant force on a body immersed in a fluid has a buoyant
force (upward force in grams) equal to the “weight” of the fluid that the body displaces. This
can be expressed as:
Eqn. (2)
buoyant force(gram force) = Vfluid displaced ρfluid
(COMPLETE THE THEORY FOR PARTS 1, 2, AND 3 BEFORE THE CLASS
MEETING)
Theory, part 1: Measurement of the density of a solid cube.
If we suspend a solid body (metal cube) from a balance and completely submerge it in a liquid,
the sum of the “gram forces” can be written as:
Eqn. (3)
Fscale + buoyant force – m = 0
Combine equations (1), (2), and (3) and derive an equation for the volume of the cube and the
density of the cube in terms of m, buoyant force, and ρwater. .
Theory, part 2: Measurement of the density of an unknown liquid.
If we now suspend the submerged cube of known volume and mass m from the balance and
measure the new buoyant force, the density of the unknown liquid can be determined.
Derive the expression  unk 
m  Funk
 water where F stands for scale reading.
m  Fwater
Theory, part 3: Measurement of the density of a wood block using Archimedes’ principle
by using a “sinker” of known volume.
Use the metal cube of mass m from part 1 to submerge the wood block and suspend the assembly
in a beaker of water. Summing the forces in the y direction gives:
Eqn. (4)
Fscale+ buoyant force –(mcube+mwood)=0
Combine equations (2) and (4) and derive an equation for the volume of the wood block in terms
of Fscale , mcube, mwood, ρwater, and Vcube .
Combine this derived equation with Eqn. (1) to get an equation for the density of the wood.
PROCEDURE:
Part 1: Density of metal cube
Measure the “weight” of the metal cube.
Using the vernier calipers, measure the dimensions of the cube.
Obtain a beaker of de-ionized water. Fasten a 1 cm diameter rod to a table clamp and mount the
triple beam balance on this vertical rod (see Figure 1 below.)
Using monofilament line, hang the cube from a notch provided in the lever arm underneath the
balance.
Fig. 1
Record the scale reading while the cube is completely submerged in the water. Do not let the
cube touch the bottom or sides of the beaker. Take at least two readings—it is difficult to get
accurate readings because of the damping effect on the submerged weight. Use Eqn. (3) to
calculate the buoyant force.
The buoyant force can also be measured by “weighting” the beaker and water without the cube
suspended in the liquid and then “weighting” the beaker and the water with the cube suspended
as shown.
Fig. 2
Convince yourself using Newton’s third law that the buoyant force is the difference between
these readings. Take two readings using this method, and average the buoyant forces with those
from the previous method.
Remember that error estimates need to be included with all of your data.
Part 2: Density of an unknown liquid.
Record the measured density of the unknown liquid using the hydrometer and the graduated
cylinder set up at the front of the classroom
Dry the cube.
Obtain a beaker of unknown liquid and repeat the measurements taken in Part 1 using the
unknown liquid instead of water.
Part 3. Density of wood.
“Weight” a wood block using the analytical balance set up in classroom.
Attach the wood block to the cube used in Part 1, and “weight” them before you submerge both
of them in water, and after you submerge both of them as shown in Fig. 1.
Calculate the buoyant force using Eqn. (4).
ANALYSIS:
Using the measured mass m and the dimensions of the metal cube, calculate the density of the
cube.
Using the measured mass m and the dimensions of the wood cube, calculate the density of the
wood cube.
For each part, re-write each equation that you derived in the theory section using Archimedes’
principle. Use each equation and calculate the density of the metal cube, the alcohol, and the
wood. Show a sample calculation of each type in your report.
Look up the density of the metal. Calculate the percent discrepancy of the density for the metal
cube.
Using the hydrometer reading as the accepted value for the unknown liquid, calculate a percent
discrepancy for the density of the unknown liquid obtained using Archimedes’ principle.
Using the density of the wood cube obtained from the dimension measurements as the accepted
value, calculate a percent discrepancy for the density obtained using Archimedes’ principle.
Using error propagation, calculate the predicted error in the density of the wood cube from its
dimensions.
REPORT:
Unless instructed by your teacher, write a standard report.
In the conclusion be sure to summarize and compare your results.
HOOKE’S LAW &
THE EFFECTIVE MASS OF A SPRING
Introduction: The equation for the period T of mass m on an ideal spring executing simple
m
harmonic motion is given by T  2
where m is the mass attached to the spring and k is the
k
spring constant. This equation only applies to a spring with zero mass. Since we don’t have any
of these in the lab, we shall use a spring with mass ms and by measuring k and the values of T
for various masses, we shall determine what fraction of the spring’s mass should be added to m
in the above equation to give the correct value for the period T.
Equipment:
One tapered spiral spring,
Stop watch
Meter stick, Meter stick clamp, and Ring stand
Triple Beam Balance
Table Clamp, Metal rod, Rod Clamp,
Pendulum Clamp
Weight hangers and assorted weights
Theory: As stated in the introduction, the equation for the period T of mass m on an ideal spring
m
executing simple harmonic motion is given by T  2
where m is the mass attached to the
k
spring and k is the spring constant. When the fractional mass of the spring is included in the
m  cms
equation for T, the equation becomes T  2
where c is the fractional value of the
k
spring’s mass, 0<c<1. By squaring both sides of the equation T  2
m  cms
, we obtain
k
4 2
4 2cms
. which gives a straight line when a graph of T2 verses m is plotted.
m
k
k
Note that equation is of the form
y = (slope)x + intercept. From the slope of the line, k can be determined . From the value of the
intercept of the graph, and knowing the values of ms and k, the value experimental value of c
can be determined.
T2 
Hooke’s law states that F=kx where F is the force required to stretch a spring a distance x form
equilibrium Another method to determine k is to vary the force F and plot a graph of F vs. x.
The slope of the graph will be k.
Procedure:
Part 1. Determine the value of k by adding masses to the spring and measuring the stretch, y.
Obtain a tapered spring and suspend it from a pendulum clamp with the small end of the spring
attached to the pendulum clamp. Attach a 50 gram weight hanger to the large end of the spring.
Place a meter stick in a meter stick clamp and attach it to the ring stand. Set the stand on the
floor, and adjust the meter stick and hanging spring/weight hanger so the zero end of the meter
stick is even with the bottom of the weight hanger. Add an amount of mass to the weight hanger
in the amount necessary to extend the spring to approximately twice its un-stretched length.
Remove this mass and divide it into roughly 10 increments. Add masses in these increments
until you have reached the mass necessary to stretch the spring to twice its un-stretched length.
Remember, you will be plotting a graph of F vs. y and you need at least 10 data points. Caution:
Excessive stretching ruins the spring and produces poor results. Never stretch the spring
more than three times the original length.
Part 2. Add an amount of mass to the weight hangar in the amount necessary to extend the
spring to approximately twice its un-stretched length. Pull the mass down slightly and set the
system in oscillation. You will need to time at least 10 oscillations in order to reduce the
absolute error to determine the period T of each oscillation. Reduce the mass and obtain a new
period T. Continue reducing the mass until you have enough data for a graph of T2 vs. m. The
mass m should change by close to a factor of three over your values. Be sure to estimate your
absolute error in all measurements, since these will be used to determine the percent error in your
calculations.
Analysis:
Part 1. Multiply the mass by g to obtain F and plot a graph of F vs. y. From the slope of the
graph, determine k.
4 2
4 2cms
which gives a
m
k
k
straight line when a graph of T2 verses m is plotted and the equation is of the form y=(slope)x +
4 2
intercept. From the slope of the line,
, determine k . From the value of the intercept of the
k
4 2 cm s
graph,
, and using the known value of ms , and the two values of k from parts 1 and
k
2, find the two experimental values of c.
Part 2. As given in the theory section, the graph of T 2 
Error analysis: The error in k can be found from the max-min slope of the F vs. y graph and the
error in ms can be found from the least count of the balance used. Using these and error
propagation, predict the experimental error in c. Calculate the percent discrepancy in c and
determine if the experimental value of c is within the predicted margin of error as determined by
error propagation.
Prior to the class meeting find the following:
The theoretical value for c.
The expression for k in terms of the graph intercept, and other constants.
The expression for c in terms of ms, k, and other constants.
CAUTION
Overstretched springs
become useless.
&
They cost $25 each to
replace.
Do not stretch them more
than 3 times the original
length
(about 60 cm total).
MOORPARK COLLEGE PHYSICS/ENGINEERING DEPARTMENT
WRITING A FORMAL REPORT
When writing a formal report, assume the reader is knowledgeable in physics, knows the basic
equipment used in experimenting, and knows how to do physics experiments. Keeping this in
mind will allow for less detail and allow you to be more concise.
Be concise. In scientific writing, it is very important to say as much as is needed while
using as few words as possible. Lab reports should be thorough, but repetition should be
avoided. The entire report should be clear and straightforward.
Write in the third person. Avoid using the words “I” or “we” when referring to the experimental
procedure. For example, instead of “I boiled 50 mL of water for 10 minutes, ”the report should
read, “50 mL of water was boiled for 10 minutes.” This can be a bit difficult to get used to, so it is
important to pay close attention to the wording in the report.
THE REPORT
TITLE PAGE: The title page should include the title of the lab experiment, your name, your
partners name(s), class name and period, and the date. The abstract should be written on the
bottom half of the title page.
ABSTRACT: The abstract should be written concisely in normal rather than highly abbreviated English.
The author should assume that the reader has some knowledge of the subject but has not read the
paper. Thus, the abstract should be intelligible and complete in itself; particularly it should not cite
figures, tables, or sections of the paper. The opening sentence or two should, in general, indicate the
subjects dealt with in the report and should state the objectives of the investigation.
The body of the abstract should summarize the results and conclusions of the experiment. In the case
of experimental results, the abstract should indicate the methods used in obtaining them. The degree
of accuracy should be given and results compared with accepted or predicted values (are the results
within the margin of error?). The abstract should be typed as one paragraph. Its optimum length will
vary somewhat with the nature and extent of the paper, but it usually does not exceed 200 words.
(since the abstract summarizes the report, it should be written last)
INTRODUCTION: This consists of one or two sentences describing the experimental objectives of the
laboratory and what you are going to do to accomplish those objectives. An example would be “By
varying the length, mass and amplitude of a simple pendulum, the empirical equation for how the
period of the pendulum depends upon those variables will be determined.” Notice that statements
such as “To learn how to analyze data” or “To learn how to organize a data sheet” or other
“Instructional Objectives” do not belong in the introduction or anywhere else in the report. These
comments will be found only in the laboratory manual.
THEORY: Some of the laboratory experiments require the derivation of equations from the fundamental
physics concepts. The derivations are to be done prior to the class meeting and are to be included in the
formal report. Written explanations of what you are doing should be included; not just the
mathematics. The equations should be typed using Microsoft Word and Equation Editor or an
equivalent program.
PROCEDURE: The procedure discusses how the experiment occurred. Documenting the procedures of
your laboratory experiment is important not only so that others can repeat your results but also so that
you can replicate the work later, if the need arises. Because your audience expects you to write the
procedures as a narrative, you should do so and not write as an outline. Achieving a proper depth in
laboratory procedures is challenging. In general, you should give the audience enough information that
they could replicate your results. For that reason, you should include those details that affect the
outcome
ANALYSIS AND DISCUSSION: In analyzing the results, you should not only analyze the results, but also
discuss the implications of those results. Moreover, pay attention to the errors that existed in the
experiment, both where they originated and what their significance is for interpreting the reliability of
conclusions. One important way to present numerical results is to show them in graphs. This section
includes the typed data table and a sample calculation of each type with error propagation if
appropriate. Any graphs needed should also be in this section, as well as the calculation of percent
discrepancies. This section shall be typed using Microsoft Word and Equation Editor or equivalent.
Graphs shall be done using Microsoft EXCEL or an equivalent program.
CONCLUSION:
Whereas the ‘ANALYSIS AND DISCUSSION” section has discussed the results individually, the
CONCLUSION section discusses the results in the context of the entire experiment. The objectives
mentioned in the "Introduction" are examined to determine whether the experiment succeeded. This
section also includes a summary of the results along with the estimated error. Error may be the random
error as calculated from the standard deviation, or it could be an error calculated from error estimates
in your measurements. When a physical quantity is measured, as in this lab, include in the conclusion a
comparison of the measured value with the accepted value (% discrepancy ). Note whether or not the %
discrepancy is greater or less than the predicted uncertainty (in other words, are the results within the
margin of error? ). If the results are not within the margin of error, try to give a reasonable explanation
as to why. If the objectives were not met, you should analyze why the results were not as predicted.
Lastly, state sources of error and give suggestions for how errors could be reduced. Suggestions for
reducing the error and improving the experiment are included here also.
APPENDIX: Attach the signed data sheet and answers to questions.