Introduction to Moments of Inertia and Rotational Kinetic Energy

Learning Goal: To understand the definition and the meaning of moment of inertia; to be able to calculate the moments of inertia for a group of particles; to relate moment of inertia to kinetic energy.

By now, you may be familiar with a set of equations describing rotational kinematics. One thing that you may have noticed was the similarity between translational and rotational formulas. Such similarity also exists in dynamics and in the work-energy domain.

For a particle of mass m moving at a constant speed v, the kinetic energy is given by the formula K=\frac{_1}{^2}mv^2. If we consider instead a rigid object of mass m rotating at a constant angular speed omega, the kinetic energy of such an object cannot be found by using the formula K=\frac{_1}{^2}mv^2 directly, since different parts of the object have different linear speeds. However, they all have the same angular speed. It would be desirable to obtain a formula for kinetic energy of rotational motion that is similar to the one for translational motion; such a formula would include the term omega^2 instead of v^2.

Such a formula can, indeed, be written: For rotational motion of a system of small particles or for a rigid object with continuous mass distribution, the kinetic energy can be written as

K=\frac{1}{2}I\omega^2.

Here, I is called the moment of inertia of the object (or of the system of particles). It is the quantity representing the inertia with respect to rotational motion.

It can be shown that for a discrete system of n particles, the moment of inertia (also known as rotational inertia) is given by

I=\sum_{i=1}^n m_i r_i^2.

In this formula, m_i is the mass of the ith particle and r_i is the distance of that particle from the axis of rotation.

Part A
On which of the following does the moment of inertia of an object depend?
Check all that apply.

linear speed
linear acceleration
angular speed
angular acceleration
total mass
shape and density of the object
location of the axis of rotation

Correct

Unlike mass, the moment of inertia depends not only on the amount of matter in an object but also on the distribution of mass in space. The moment of inertia is also dependent on the axis of rotation. The same object, rotating with the same angular speed, may have different kinetic energy depending on the axis of rotation.
Consider the system of two particles, a and b, shown in the figure. Particle a has mass m, and particle b has mass 2m.

Part B
What is the moment of inertia I_a of particle a?

mr^2
9mr^2
10mr^2
undefined: an axis of rotation has not been specified.

Correct

Part C
Find the moment of inertia I_a,x of particle a with respect to the x axis (that is, if the x axis is the axis of rotation).
I_a,x =

Part D
Find the moment of inertia I_a,y of particle a with respect to the y axis.
I_a,y =

Part E
Find the total moment of inertia I_y of the system of two particles shown in the diagram with respect to the y axis.
I_y =

For the remaining parts, suppose that both particles rotate with the same angular speed omega about the y axis while maintaining their distances from the y axis.
Part F
Using the total moment of inertia I_y of the system found in Part E, find the total kinetic energy K of the system.
K =

It is useful to see how the formula for rotational kinetic energy agrees with the formula K=\frac{_1}{^2}mv^2 for the kinetic energy of an object that is not rotating. To see the connection, let us find the kinetic energy of each particle.
Part G
Using the formula for kinetic energy of a moving particle K=\frac{_1}{^2}mv^2, find the kinetic energy K_a of particle a.
K_a =

Part H
Using the formula for kinetic energy of a moving particle K=\frac{_1}{^2}mv^2, find the kinetic energy K_b of particle b.
K_b =

Part I
This part will be visible after you complete previous part(s).

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