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# Moment of Inertia and Rotational Kinetic Energy for AP Physics C

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By — McGraw-Hill Professional
Updated on Feb 11, 2011

Practice problems for these concepts can be found at:

Rotational Motion Practice Problems for AP Physics B & C

Newton's second law states that Fnet = ma; this tells us that the greater the mass of an object, the harder it is to accelerate. This tendency for massive objects to resist changes in their velocity is referred to as inertia.

Well, spinning objects also resist changes in their angular velocity. But that resistance, that rotational inertia, depends less on the mass of an object than on how that mass is distributed. For example, a baseball player often warms up by placing a weight on the outer end of the bat—this makes the bat more difficult to swing. But he does not place the weight on the bat handle, because extra weight in the handle hardly affects the swing at all.

The moment of inertia, I, is the rotational equivalent of mass. It tells how difficult it is for an object to speed up or slow its rotation. For a single particle of mass m a distance r from the axis of rotation, the moment of inertia is

To find the moment of inertia of several masses—for example, two weights connected by a thin, light rod—just add the I due to each mass.

For a complicated, continuous body, like a sphere or a disk, I can be calculated through integration:

### Rotational Kinetic Energy

The pulley in the last example problem had kinetic energy—it was moving, after all—but it didn't have linear kinetic energy, because the velocity of its center of mass was zero. When an object is rotating, its rotational kinetic energy is found by the following equation:

Notice that this equation is very similar to the equation for linear kinetic energy. But, because we're dealing with rotation here, we use moment of inertia in place of mass and angular velocity in place of linear velocity.

If an object is moving linearly at the same time that it's rotating, its total kinetic energy equals the sum of the linear KE and the rotational KE.

Let's put this equation into practice. Try this example problem.

This is a situation you've seen before, except there's a twist: this time, when the object moves down the inclined plane, it gains both linear and rotational kinetic energy. However, it's still just a conservation of energy problem at heart. Initially, the ball just has gravitational potential energy, and when it reaches the ground, it has both linear kinetic and rotational kinetic energy.

A bit of algebra, and we find that

If the ball in this problem hadn't rolled down the plane—if it had just slid—its final velocity would have been . (Don't believe us? Try the calculation yourself for practice!) So it makes sense that the final velocity of the ball when it does roll down the plane is less than ; only a fraction of the initial potential energy is converted to linear kinetic energy.

Practice problems for these concepts can be found at:

Rotational Motion Practice Problems for AP Physics B & C

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