We show how the moment of inertia of an object could be experimentally determined in
Example 1. A block, attached to a massless rope, is causing a pulley to accelerate. The
angular acceleration and the net torque are stated in the problem. (The net torque could
be determined by multiplying the tension by the radius of the pulley, keeping in mind
that the tension is less than the weight of the block since the block accelerates
downward.) With these facts known, the moment of inertia of the pulley can be
determined.ȈIJ = IĮ
ȈIJ = net torque
I = moment of inertia
Į = angular acceleration
Units for I: kg·m^2
What is the moment of inertia of
the pulley?
ȈIJ = IĮ
I = ȈIJ/Į
I = (55 N·m)/(22 rad/s^2 )
I = 2.5 kg·m^2
10.3 - Calculating the moment of inertia
If you were asked whether the same amount of torque would cause a greater angular
acceleration with a Ferris wheel or a bicycle wheel, you would likely answer: the bicycle
wheel. The greater mass of the Ferris wheel means it has a greater moment of inertia. It
accelerates less with a given torque.
But more than the amount of mass is required to determine the moment of inertia; the
distribution of the mass also matters. Consider the case of a boy sitting on a seesaw.
When he sits close to the axis of rotation, it takes a certain amount of torque to cause
him to have a given rate of angular acceleration. When he sits farther away, it takes
more torque to create the same rate of acceleration. Even though the boy’s (and the
seesaw's) mass stays constant, he can increase the system’s moment of inertia by
sitting farther away from the axis.
When a rigid object or system of particles rotates about a fixed axis, each particle in the
object contributes to its moment of inertia. The formula in Equation 1 to the right shows
how to calculate the moment of inertia. The moment equals the sum of each particle’s
mass times the square of its distance from the axis of rotation.
A single object often has a different moment of inertia when its axis of rotation changes.
For instance, if you rotate a baton around its center, it has a smaller moment of inertia than if you rotate it around one of its ends. The baton is
harder to accelerate when rotated around an end. Why is this the case? When the baton rotates around an end, more of its mass on average is
farther away from the axis of rotation than when it rotates around its center.
If the mass of a system is concentrated at a few points, we can calculate its moment of inertia using multiplication and addition. You see this inMoment of inertia
Sum of each particle’s
·Mass times its
·Distance squared from the axis(^190) Copyright 2007 Kinetic Books Co. Chapter 10