General Physics Experiment Handout
Moments of Inertia
Purpose of the Experiment
Observe the rotation (circular movement) of variously shaped objects caused by torque and
calculate moments of inertia for objects of various shapes using the law of conservation of energy.
Experimental Principle
Consider a particle that rotates around a fixed axis with a radius of r, velocity of V, and mass of m,
the kinetic energy of the particle is
K mV (1)
where V rω and ω is the angular velocity. Formula (1) can be rewritten in the following manner:
K mr ω Iω
(2)
where I (I mr ) is the moment of inertia of the particle around a fixed axis.
This is the moment of inertia for a particle. If the particle that rotates around a fixed axis is replaced with
a rigid body with continuous mass distribution, the moment of inertia of such a rigid body can be
considered to be the rotation of an infinite number of small particles around a fixed axis (refer to the
Appendix). However, how can we determine the moment of inertia of an object without using theoretical
calculations, or of an object with a shape that is too complicated to be calculated? This experiment
provides an alternative method to determine an object’s moment of inertia using conservation of energy.
Figures 1 and 2 show a simple instrument or system for measuring moments of inertia.
Rotational Platform A can freely rotate around the axis OO’. When m falls, it triggers the movement of
the rotational platform. Let m be released from a stationary position (initial velocity V 0) at a
descending height h based on conservation of energy
Figure 1.
Figure 2.
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General Physics Experiment Handout
∆U
＝∆K = ∆K ＋∆K mgh = mV + Iω
(3)
where I is the moment of inertia of the rotational platform around axis OO’. The falling velocity V of m
equals the tangential velocity of the rotating platform.
V = rω
(4)
where r is the radius of the rotational platform, and
h = at = Vt.
(5)
Based on Formulas (3), (4), and (5), we obtain the moment of inertia for the empty rotational platform
I = mr మ
− 1.
(6)
When a test object is placed on the rotational platform with its center of gravity on the axis OO’, the total
moment of inertia I includes the moment of inertia of the test object I in addition to moment of inertia
I of the platform,:
I = I + I .
Therefore, the moment of inertia of the test object is
I = I − I .
I can be measured by first measuring I (object not included), and then measuring I (platform and the
object).
Laboratory Instruments
1. Rotational platform and its base (3 screws included)
2. Pulleys
3. Cotton strings
4. Hook (50 g) and weights (500 g*1, 200 g*2, 100 g*1)
5. Metal disk, cylinder (2 screws included), and ring
6. Photogate timer, photogate, stand, and base
7. Meter stick
8. Spirit level
Experimental Procedure
A. Measure the moment of inertia I of the rotational platform.
1. As shown in Figure 1, connect the strings and the hook to the rotational platform.
2. Set up the photogate timer at an appropriate position (use your own judgment), and keep the hook
stationary (initial velocity V = 0).
3. Measure the total mass attached to the hook (hook and weights).
4. Measure the time t needed for the hook to travel a distance of h. Repeat the measurement three times
and calculate the average of t and h.
5. Repeat Procedures A-3 and A-4 with three different weights attached to the hook.
6 Determine the moment of inertia I of the rotational platform.
B. Measure the moment of inertia I for the test object.
1. Use a scale to measure the weights of the test objects, including a metal disk, a ring, and a
cylinder(horizontal), and obtain the geometric measurements of these objects.
2. Place different objects on the center (axis OO’) of the rotational platform.
3. Repeat Procedures 2 to 5 in Experiment A to determine the total moment of inertia I for the various
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General Physics Experiment Handout
4.
5.
6.
test objects.
Use the moment of inertia I of the rotational platform obtained in Procedure A-6 and the total
moment of inertia I obtained in Procedure B-3 to calculate the moment of inertia I for
the test objects.
Use the Appendix to calculate the theoretical moment of inertia I for the test objects.
Compare I and I .
Notes:
1. Avoid overlapping when winding the strings onto the rotational platform and perform winding in an
orderly fashion to reduce experimental errors.
2. The radius r of the rotational platform is the radius of the winding axis below the platform.
3. The rotation axis of the rigid test body must be placed exactly on the axis OO’.
Questions for Reflection
1. Discuss the effects of friction forces in this experiment: add work resulting from friction force or
friction work w to the right-hand side of Formula (3), and infer the altered moments of inertia with
and without the friction forces being considered.
2. Explain the possible sources of friction forces in this experiment.
3. Why should the overlapping of strings be prevented when winding the platform?
4. As shown in Figure (2), consider a photogate that measures a hook that is not kept stationary when
falling (initial V ≠ 0 ), that is, consider a hook that has a velocity of V when passing Photogate
A and a velocity of V when passing Photogate B. Let the distance between A and B be h. How
should the equation ∆U
＝∆K and the I in Formula (6) be modified? How
can the measurement be achieved experimentally?
光電閘A-Photogate A, 光電閘B-Photogate B
Figure 2.
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General Physics Experiment Handout
Appendix
A rigid body that rotates around a fixed axis can be considered to be the rotation of an infinite number of
small particles around a fixed axis. The radius of rotation r and rotation velocity V for each small particle
can vary, but they have the same angular velocity ω. The total kinetic energy of all the particles is
K m V m V ⋯ m r ω m r ω ⋯ ∑ m r ω Iω
which is known as the moment of inertia of a given object around a given fixed axis. Theoretical
calculations enable the determination of the moments of inertia for various objects. As shown in the table
below, I ∑ m r :
.
物體-object, 軸-axis, 轉動慣量 moment of inertia, 圓盤-disk, 圓環-ring, 圓
柱- cylinder, 矩形棒-rectangular bar, 通過圓心-through the center, 通過圓柱
重心-through the center of gravity, 通過矩形 ab 之重心-through the center of
gravity of the rectangle ab
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