## Illustration 10.3: Moment of Inertia, Rotational Energy, and Angular Momentum

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Many objects rotate (spin) about a fixed axis. Shown is a wheel (a disk) of radius 5 cm and a mass of 200 grams rotating about a fixed axis at a constant rate **(position is given in centimeters and time is given in seconds)**. Restart.

In Illustration 10.2 we discussed how linear speed (velocity) was related to angular speed (velocity), and in the process how angular acceleration is related to the angular velocity (**α** = Δ**ω**/Δt). In this Illustration we will discuss kinetic energy of rotation, KE_{rot}, and angular momentum, **L**.

The easiest way to remember the forms for the kinetic energy of rotation and the angular momentum is by analogy with the kinetic energy of translation and the linear momentum. We recall that KE = 1/2 m v^{2} and **p** = m **v**. Can you guess what the rotational kinetic energy and angular momentum will look like?

First, what will play the role of v and **v** in the rotational expressions? If you said ω and **ω** you are right. Next we must consider what plays the role of m, and we will be all set. The property of mass describes an object's resistance to linear motion. Therefore, what we are looking for is a property of objects that describes their resistance to rotational motion. This is called the moment of inertia. The moment of inertia depends on the mass of the object, its extent, and its mass distribution. It turns out that for most simple objects the moment of inertia looks like I = *C* m R^{2}, where m is the object's mass, R is its extent (usually a radius or length), and *C* is a dimensionless constant that represents the mass distribution.

Therefore, we have that KE_{rot} = 1/2 I ω^{2} and **L** = I** ω**. What are this disk's KE_{rot} and **L**? Well, from Illustration 10.2 we know that ω = 1.256 radians/s. Since the wheel is a disk, *C* = 2. Therefore, we can calculate the moment of inertia as: 2.5 x 10^{-4} kg·m^{2}. Finally, we have that KE_{rot} = 1.97 x 10^{-4} J and **L** = 3.14 x 10^{-4} J·s (into the page or computer screen). Note that these are small values because I for this disk is small. A 1-m radius and 2-kg mass disk would have a moment of inertia of 1.0 kg·m^{2}.

Illustration authored by Mario Belloni.

Script authored by Steve Mellema, Chuck Niederriter, and Mario Belloni.