Experiment 11
Moment of Inertia
A rigid body composed of concentric disks is constrained to rotate about its axis of
symmetry. The moment of inertia is found by two methods and the results are compared. In the
first method, the moment of inertia is determined theoretically by applying the formula for the
moment of inertia of a single disk to each of four disks, and adding the results. In the second
method, the moment of inertia is determined experimentally by measuring the acceleration
produced by a constant torque on the body.
The experimental determination of the moment of inertia is only valid if friction is
negligible. In Part II of the experiment, an estimate of the angular acceleration due to friction is
obtained. The validity of the above approximation is the examined.
Theory, Part I
The moment of inertia of a homogeneous disk about the axis of symmetry is
2
I disk = 12 M R ,
where M 1 is the mass of the disk and R 2 is the radius. The moment of inertia of a rigid system of
concentric disks is then
I = 12 M 1 R12 + 12 M 2 R22 +=  12 M i Ri2 ,
(1)
where the sum extends over all disks, each of which has mass M i 3 and radius R i 4. If the mass
density,  5, is uniform (i.e., constant throughout the body), the mass of each disk is given by
2
M i =  V i =  wi  Ri ,
where V i 6 is the volume of the disk and wi 7 is the width. Substituting this into (1), the moment of
inertia of each disk is then
Ii =

4
wi Ri ,
2
(2)
and the total theoretical moment of inertia is
I the =

2
 wi Ri4 .
1
(3)
If the density together with the width and the radius of each disk are known, then the moment of
inertia of the body can be computed. The moment of inertia determined in this manner will be
referred to as the theoretical value ( I the 8).
The experimental value of the
moment of inertia can be found by exerting
a constant torque on the body. A mass,
m 9, is attached to a string which is
wrapped around the body at some radius,
R 10. (Refer to Figure 1.) R 11 will be
one of the disk radii, R i 12. If m 13 is
released from rest and falls a distance,
d 14, during a time, t 15, the acceleration
of m 16 is given by
a=
2d
t
2
.
(4)
Once the value of the acceleration is
known, the moment of inertia is determined
by
g 
I exp = m R  - 1 ,
a 
Figure 1. Experimental determination of
moment of inertia.
2
(5)
assuming that friction in the supports is negligible.
Theory, Part II
A rough measure of friction in the supports can easily be found. Suppose the body (without
the mass m 17) is initially spun and N 18 revolutions occur during the time T 19 required for the
body to come to rest. Assuming the angular acceleration due to the resistance is constant, its
magnitude is then given by
f =
4N
T
2
.
(6)
This should be approximately true for the apparatus. The expression (6) represents, in some sense,
and average value of the angular acceleration due to friction.)
=If
The magnitude of the resistive torque due to friction in the supports is  f
20.
Including this torque in the derivation of the experimental moment of inertia, the result is
2
1

2 g
,
I exp = m R  - 1
 a  1+  f 


a /R 

(7)
=If
assuming the resistive torque is constant. (  f
21 is approximately true for  f 22 given by
(6) when the mass is not attached to the body. In the experimental determination of the moment of
inertia, when m 23 is attached to the body, the resistive torque will have an additional contribution
that is proportional to the tension of the string, which connects m 24 to the body. This contribution
is negligible, however, if m < < M total 25.) According to (7) the validity of (5) rests upon the degree
to which the following is true:
f
<<1 .
a /R
(8)
The percentage deviation between I exp 26 in (5) and the more accurate value, I exp 27, in (7) is then
=
f
a /R
x 100% .
(9)
Apparatus
o
o
o
o
mounted rotational body
vernier caliper
string
masking tape
o
o
o
o
50-gram short weight hanger
200-gram slotted weight
two meter stick
stopwatch
The body is made of a metal alloy and consists of four concentric disks. (Refer to Figure 2.)
The two smaller disks, which lie at either end of the axis, have identical radii.
The density of the material from which the body is made is given by the total mass divided by the
total volume. The result is
 = 7.76 x 103 kg  m-3 .
(10)
The string attached to the mass m 28 supplies a force which will be applied at each of the
three different radii of the body. At several of the radii the string can be attached to the cylinder
with some masking tape.
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Figure 2. The body consists of four concentric disks.
Procedure, Part I
1) Using the vernier caliper, measure and record the width, w 29, and diameter, d 30, of
each disk. Obtain 3 or 4 significant figures. Number the disks according to Figure 2.
2) Cut a piece of string slightly longer than two meters in length. Attach the weight hanger
to one end and place the 200-gram weight on the hanger. The mass is thus
m = 250.0 grams .
(11)
3) Attach the other end of the string to the rim of one of the disks. Ensure that the string
will not slip. Rotate the body, allowing the string to wrap around the disk without
overlapping. Using the two meter stick, position the bottom of the weight hanger so
that its vertical distance from the floor is
d = 2.000 meters .
(12)
4) Steady the weight hanger. Release the body and measure the time, t , required for the
weight hanger to strike the floor. The timing must be performed very carefully; start the
timer exactly when the body is released and stop it just when the body strikes the floor.
Do not permit the string to become tangled in the supports. Perform two more trials.
5) Repeat the procedure at each of the two other radii. The data will then consist of three
time-of-fall measurements for each of the three different radii. Note that the values of
m and d are to remain constant.
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Procedure, Part II
1) Remove the string from the body, and place a small piece of tape on the rim of the
largest disk. The tape will serve as a reference mark to be used in counting revolutions.
2) Spin the body as fast as possible while still being able to count the revolutions. The
angular velocity will then be comparable to a typical final angular velocity in Part I.
3) Start the time and count the number, N 31, of revolutions during the time, T 32,
required for the body to come to rest. Record the values of N 33 and T 34. The data
need only be approximate and only one trial is necessary here.
Analysis, Part I
Construct a data table that contains the width, diameter, radius, and moment of inertia of
each disk, using (2) and (10). Use SI units. Include the value of  35 in the title of the table.
Number the disks according to Figure 2. Determine and clearly display the value of the total
theoretical moment of inertia, I the 36, to three significant figures.
For each radius, R 37, at which a force was applied, compute the average value of the timeof-fall, t 38. Using (4) and (12), compute the acceleration for each value of R 39. Then, using (5)
and (11), compute I the 40 for each set of values of R 41 and a 42. Use SI units. Construct a data
table that contains the values of R 43, t 44, a 45, and I exp 46. (This table should be separate from
the table for the theoretical moment of inertia.) Include both the measured and averaged times-offall. Since the values of m 47 and d 48 are constant, they should be included in the title of the
table. Determine and display the average value of I exp 49.
Find the percentage difference using the equation:
I exp - I the
x 100% .
I the
Report in a table of results the experimental and theoretical values of the moment of inertia
and the percentage difference.
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Analysis, Part II
Using (6), compute and display the approximate value of the angular acceleration,  f 50,
due to friction. Express the result in units of rad/sec2. For each value of R 51, compute the angular
acceleration, a /R 52, and the relative measure of the friction,  53. Include these values in the
I exp 54 data table in Part I.
Questions
1) Draw free body diagrams for m 55 and the disk, and consequently derive (5).
2) Derive (6).
3) Derive (7).
4) How does the presence of friction in the supports of the body affect the ideal
=
<
>
relationship I exp I the 56? That is, does friction cause I exp I the 57, I exp I the 58, or
I exp = I the 59? Explain.
5) Place a raw egg and a hard-boiled egg on a smooth horizontal surface. Spin them,
giving each approximately the same initial angular velocity. Describe and explain the
difference in their subsequent motions.
6) If a raw egg and a hard-boiled egg are simultaneously released from rest at the top of an
incline, and both subsequently roll without slipping, which one will reach the bottom
first? Explain.
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Figure 3. Physical pendulums shown with their various axes of rotation.
The position of the center of masses must be calculated.
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