This document contains instructions for a physics lab experiment on moment of inertia. The experiment has two parts: Part I measures the moment of inertia of a disk by applying a torque from a hanging mass and measuring the angular acceleration. Part II measures the moment of inertia of a rod with two movable masses by varying the mass positions and amounts and again measuring angular acceleration from a hanging torque source. Equations are provided to calculate moment of inertia from experimental measurements.

1. Part II:
When r= 0.7r
Part II:
When r= 0
Reflection on this week’s Objectives
Discuss this week’s objectives:

2. Objectives / Competencies:
· Analyze internal organizational dynamics and the influence on
business continuity.
· Describe cultural, structural, leadership considerations that
must be incorporated into strategy implementation.
· Determine the resources needed for strategy implementation
Prepare a 350- to 1,050- word paper detailing the findings of
your discussion.
Part ! :
Part II :
When r = ro

3. Part II:
When r= .8ro
Physics161
Moment of Inertia
Introduction
In this experiment we will study the effect of a constant torque
on a symmetrical body. In Part I you will determine the angular
acceleration of a disk when a constant torque is applied to the
disk. From this we will measure its moment of inertia, which we

4. will compare with a theoretical value. In Part II, you will
observe the relationship between torque, moment of inertia and
angular acceleration for a rotating rod with two masses on
either end. You will vary the mass connected (and therefore the
torque applied) to the rod by two pulleys. You will also change
the moment of inertia of the rod system by changing the
distance of the masses from the center of mass of the rod.
Reference
Young and Freedman, University Physics, 13th Edition:
Chapter 3, section 4; Chapter 9, sections
1-4Theory
Moment of inertia is a measure of the distribution of mass in a
body and how difficult that body is to accelerate angularly. For
both parts of the experiment, a falling mass will accelerate a
rotating object in the horizontal plane. In Part I, the object will
be a disk. In Part II, you will find the moment of inertia of a rod
with two masses attached to it.
The basic equation for rotational motion is:
(1)
where is angular acceleration in units of rad/s2, is applied
torque in N m, and finally I is the moment of inertia or
rotational inertia in units of kg m2. For a uniform disk pivoted
about the center of mass, the theoretical moment of inertia is
(2)
where M is the mass of the disk and R is the radius of the disk.

5. In Part I we measure the angular acceleration, α, and use this to
calculate moment of inertia, I, which we will compare with the
theoretical value of I.
In Part II the moment of inertia is the sum of the moments of
inertia of the two masses and the rod. For the masses that slip
onto the rod, we will assume point masses. Thus, the moment
of inertia for one of the two masses is:
(3)
where r is the distance of the center of mass from the axis of
rotation located at the center of the rod. Because the masses can
be moved along the rod, r will be adjusted to change their
moment of inertia. The moment of the inertia of the rod with
mass M and a length L is:
(4)
The moment of inertia for a rod with length L and two masses
on each end at a distance r is simply the sum of the components
as defined by Equations 3 and 4:
(5)

6. The first term in equation (3) is multiplied by two because there
are two point masses. Given the moment of inertia of the entire
system, I, and the torque,, applied by the mass M hanging from
the pulley, angular acceleration, can be found. For translational
motion, Fnet = ma , while for rotational motion we have the
analogous equation
(6)
Torque () is a rotational “force” that causes rotation with an
angular acceleration,, in an object with a moment of inertia, I.
The rotating object will be attached to a pulley placed at the
center of the rotary motion sensor which has a string wrapped
around it. The string has a mass tied to one end (the mass will
vary) and is laid over an additional pulley which allows the
falling motion of the mass to be converted to a torque on the
rotary motion sensor. For the purposes of this lab, the applied
torque will be due to a mass accelerated by gravity acting on the
pulley of the rotary motion sensor. You might expect the torque
to be given by the weight of the hanging mass (mhangg) times
the moment arm (Rpulley): . However, the torque actually arises
from the tension in the string, not the weight of the hanging
mass. The analysis using Newton’s laws (see Appendix) shows
that the torque is actually slightly less than this:

7. (8)
where mhang is the mass hanging on the pulley. The
experimental moment of inertia can be found using the
following equation:
(9)
In Part II, by adjusting the masses on the rod, we can observe
how an increased moment of inertia (where either the mass is
distributed farther from the center of mass or the total mass is
increased) will result in a decreased angular acceleration for the
same torque. It is the same type of relationship as the one you
observed for Newton's Second Law.Procedure
Part I: Moment of Inertia for a Disk
The experiment uses a mass hanging on a string over a pulley to
generate a constant torque on the system resulting in a constant
angular acceleration of the system.
1. Set up the apparatus as shown in Figure 9.1, making sure to
connect the rotary motion sensor to the 850 Universal Interface.
Connect a disk to the rotary motion sensor.
Figure 9.1

8. 2. Make sure that the pulley is set up to give positive values for
angular position. This means that the rotary motion sensor will
turn in a counterclockwise direction as the mass on the pulley
drops. (The motion sensor may have an indication of which
direction is positive taped to it.)
3. Adjust the measurement rate to 10 Hz on the Rotary Motion
Sensor.
4. Choose angular velocity,, in radians/s under the y-axis of the
graph.
5. Measure and record the radius of the rotary motion sensor
pulley around which the string is wound, and the radius of the
disk. Hang a mass of 50 grams from the pulley, press Record,
and release it.
6. Press Stop just before the hanging mass reaches the ground.
7. Using a linear trend line fit, determine the angular
acceleration and the uncertainty in.
8. Make the following table and find the experimental moment
of inertia for the disk (using your torque and angular
acceleration results) and compare it to its theoretical value
(using equation (2)).
Rdisk(m)
Rpulley(m)

9. mhang(kg)
mdisk(kg)
σα
0.048
0.027
0.0496
0.1194
66.8
0.1071
0.11
0.00016031
0.00013755
Part II: Moment of Inertia of a Rod with Two Masses Attached
The experiment uses a mass hanging on a string over a pulley to
generate a constant torque on the system resulting in a constant
angular acceleration of the system.
1. Weigh the rod and the point masses and record these masses.
2. Set up the apparatus as shown in Figure 9.2, making sure to
connect the rotary motion sensor to the interface box.

10. Figure 9.2
3. Make sure that the pulley is set up to give positive values for
angular position. This means that the rotary motion sensor will
turn in a counterclockwise direction as the mass on the pulley
drops. (The motion sensor may have an indication of which
direction is positive taped to it.)
4. Adjust the measurement rate to 10 Hz on the Rotary Motion
Sensor.
5. Measure angular position,, in radians, angular velocity,, in
radians/s and angular acceleration,, in radians/s². Create graphs
of these quantities vs. time.
6. Adjust the masses so they are as far from the axis as possible
without falling off. Measure and record this distance (r0). Hang
a mass of 50 grams from the pulley, press Record, and release
it.
7. Press Stop just before the hanging mass reaches the table.
You will repeat this for four trials with different positions for
the masses and create a table with the following format:

11. Run
r(m)
Rpulley(m)
Mhang(kg)
0.188
0.027
0.05
0.00529
0.00563
2.15
0.01315
0.006116
0.1504
0.027
0.05
0.00339
0.0037313

12. 2.66
0.01313
0.004973
0.1316
0.027
0.05
0.00259
0.0029313
3.33
0.01311
0.003937
0
0.027
0.05
0
0.0003413
27.3
0.01224
0.000448
For the case , remove the masses completely from the rod.
IMPORTANT: An important distinction: r is the distance
between the center of the point mass and the axis of rotation
(the center of the rod), while Rpulley is the radius of the pulley
on the rotary motion sensor around which the string is wrapped.

13. For Your Lab Report:
Include a sample calculation of the torques and moments of
inertia Itheoretical and Iexperimental for Part I. Estimate the
uncertainty of the measurements for the radius of the disk
(Rdisk), and the mass. Use the uncertainty in angular
acceleration as recorded by Capstone. Assume 2% uncertainty
for g. Use these values to propagate your uncertainty to find
uncertainties for Itheoretical and Iexperimental for Part I. (You
can ignore the second term in eq. (8) when propagating
uncertainty for the torque.) Do Itheoretical and Iexperimental
agree within their uncertainties?
For Part II, compare Itheoretical and Iexperimental without
uncertainty calculations. Find the % difference for each case
and discuss possible sources of systematic error. Are your plots
of angular acceleration consistent with the assumption that
angular acceleration is constant?Appendix
The calculation of the torque applied to the rotary motion
sensor is as follows:
The rotary motion sensor has a radius, Rpulley, which is the
distance from the axis of rotation at which the force, T, acts. T
is the tension in the string attached to the pulley and is due in
part to the weight on the string, Mg, where M is the mass of the
hanging mass and g is the acceleration due to gravity. Figure
9.3 below shows a rough sketch of the set up.
T
Figure 9.3

14. The application of Newton's Second Law to the hanging mass
results:
The downward direction is taken to be positive. We can
substitute for the linear acceleration, a:
Where and Rpulley are the angular acceleration and the radius
of the rotary motion sensor, respectively. Continuing to solve
the equation gives us:
T is the tension in the string. It is a force, not a torque. To
find the torque acting on the pulley, we must multiply by the
distance from the axis of rotation at which the force acts:

15. For the set up of this experiment, take the following example
where = 0.06 kg, m and rad/sec². Substituting these values
results in:
From this example, we see that so we can approximate.
7
2
2
1
MR
I
disk
=
2
mr
I
mass
=
2
12
1
L
M

16. I
rod
rod
=
2
12
1
2
2
L
M
mr
I
rod
+
=
t
a
a
t
I
net
=
t
a
pulley
hang
gR
m
=
t
a
t
2
pulley
hang

17. pulley
hang
R
m
gR
m
-
=
a
t
/
experiment
=
I
w
a
a
a
t
experiment
I
theory
I
q
w
a
2
2
mr
I
masses
=
rod
theory
I
mr

18. I
+
=
2
2
a
t
a
t
/
experiment
=
I
0
r
r
=
0
8
.
r
r
=
0
7
.
0
r
r
=
0
=
r
0
=
r

19. a
t
2
pulley
hang
pulley
hang
R
m
gR
m
-
=
a
m
T
g
m
F
a
m
F
hang
hang
y
hang
y
=
-
=
S
=
S
å
=
a

20. t
I
a
pulley
R
a
=
a
a
a
pulley
hang
hang
pulley
hang
hang
R
m
g
m
T
R
m
T
g
m
-
=
=
-
a
t
t
2
pulley
hang

21. pulley
hang
pulley
R
m
gR
m
T
R
-
=
=
hang
m
025
.
0
»
pulley
R
58
.
2
»
a
Nm
R
m
Nm
gR
m
pulley
hang
pulley
hang
3

22. 2
3
10
0969
.
0
10
7
.
14
-
-
´
=
´
=
a
a
2
pulley
hang
pulley
hang
R
m
gR
m
>>
pulley
hang
gR
m
=
t