Champlain – St. Lawrence
Winter2017
Physics 203-NYA-05
Mechanics Lab
Moment of Inertia
Background Theory
Torque is defined as the physical quantity required to make an object rotate about an axis
of rotation. Its magnitude is given by the formula
τ = r F sinθ
where τ is the torque, F is the magnitude of the applied force, r is the distance between
the axis of rotation and the point where the force is applied, and θ is the angle between
the extension of r and the force F as shown below. The units of torque (in the MKS
system) are N . m.
When a torque acts on a rigid object that can rotate around a fixed axis, it creates an
angular acceleration according to Newton’s second law for rotations:
τ=Iα
where τ is the torque (or total torque if there is more than one), I is the moment of inertia
of the object, and α is the angular acceleration (in rad/s2). The moment of inertia is the
resistance that an object has to being rotated. Its value depends on the mass and the shape
of the object as well as the axis around which it rotates. From the above equation, one
finds that the MKS units of moment of inertia are kg . m2.
Moment of Inertia
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Champlain – St. Lawrence
Winter2017
Physics 203-NYA-05
Mechanics Lab
If a system is made up of many parts, its moment of inertia is the sum of the moments of
inertia of its parts:
Itot = I 1 + I 2 + … + I n
The values of moments of inertia can be derived theoretically from the distribution of
mass in the system with respect to the axis of rotation. For example, a point particle of
mass m revolving at a distance L from an axis of rotation has a (theoretical) moment of
inertia given by
Itheor = mL2
The moment of inertia of any object (in particular, an irregularly-shaped object and/or an
object which has a highly complex, or unknown, mass distribution) can be obtained
experimentally if the applied torque and the resulting angular acceleration are measured.
From Newton’s second law for rotations, one finds:
Iexp = τ / α
In this laboratory exercise, a rotational apparatus will be subjected to a torque. By
measuring the magnitude of this torque and the resulting angular acceleration, its
experimental moment of inertia will be calculated. By changing the mass distribution on
the apparatus, the formula for the theoretical moment of inertia of a point mass (Itheor =
mL2) will be verified.
Procedure
The instructor will give you a detailed explanation of how to operate the rotationalmotion apparatus shown in the diagram below.
Moment of Inertia
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Champlain – St. Lawrence
Winter2017
Physics 203-NYA-05
Mechanics Lab
Two experiments will be conducted:
Experiment A: with masses m 1a and m 1b both equal to 500g.
Experiment B: with m 1a and m 1b removed.
Manipulations, Measurements and Calculations (MKS units)
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Level the apparatus.
Measure the diameter d of the vertical shaft with a vernier caliper. Compute the
radius r.
(For experiment A): Measure the masses m 1a and m 1b . Neglect the uncertainty on
the mass. Place m 1a and m 1b at the same distance L from the center of the shaft.
Measure L (from the center of the shaft to the middle of the masses).
Wrap a string around the shaft, run it over the pulley and attach it to a falling mass
m 2 . The motion of the horizontal rod and the falling mass will begin from rest. The
value of m 2 must be chosen so as to give a falling time of about 5-10 seconds.
Measure m 2 (neglect its uncertainty).
Measure the height h that m 2 falls.
Measure the time it takes m 2 to fall with a stopwatch. Do this three times and
compute the average time.
From the average time and the height h, compute the linear acceleration “a” of m 2 (in
m/s2).
Calculate the angular acceleration of the system (in rad/s2) from the formula α = a/r.
Calculate the torque exerted on the shaft by the string.
Calculate the (experimental) moment of inertia of the rotating system.
In experiment A, the moment of inertia that you will find (I A ) will be the one of the
“point masses” m 1a and m 1b plus the moment of inertia of the frame that rotates with
them. In experiment B, you will find the moment of inertia (I B ) of the frame only.
Afterward, you will compute the difference between the moment of inertia of experiment
A and that of experiment B to obtain the total moment of inertia of the point masses only.
You will then compare this experimental moment of inertia (Iexp) with the value obtained
from the theoretical formula: Itheor = (m 1a + m 1b )L2 .
Moment of Inertia
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