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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.

HTML updated for JavaScript by Ricky Davidson.

Physlets were developed at Davidson College and converted from Java to JavaScript
using the
SwingJS
system developed at St. Olaf College.