Practice problems for these concepts can be found at:

Rotational Motion Practice Problems for AP Physics B & C

For an object moving in a straight line, we defined five variables.

Now consider a fixed object spinning, like a compact disc. The relevant variables become the following.

These variables are related via the following three equations. Obviously, these equations differ from the "star equations" used for kinematics … but they're nonetheless very similar:

So try this example:

In any linear kinematics problem the units should be in meters and seconds; in rotational kinematics, the units MUST be in RADIANS and seconds. So convert revolutions per minute to radians per second. To do so, recall that there are 2π radians in every revolution:

- 200 rev/min × 2π rad/rev × 1 min/60 s = 21 rad/s.

Now identify variables in a chart:

We want to solve for Δθ because if we know the angle through which the wheel turns, we can figure out how far the edge of the wheel has gone. We know we can solve for Δθ, because we have three of the five variables. Plug and chug into the rotational kinematics equations:

What does this answer mean? Well, if there are 2π (that is, 6.2) radians in one revolution, then 105 radians is about 17 revolutions through which the wheel has turned.

Now, because the wheel has a radius of 0.50 m, the wheel's circumference is 2π*r* = 3.1 m; every revolution of the wheel moves the bike 3.1 meters forward. And the wheel made 17 revolutions, giving a total distance of about 53 meters.

Is this reasonable? Sure—the biker traveled across about half a football field in 10 seconds.

There are a few other equations you should know. If you want to figure out the linear position, speed, or acceleration of a spot on a spinning object, or an object that's rolling without slipping, use these three equations:

where *r* represents the distance from the spot you're interested in to the center of the object.

So in the case of the bike wheel above, the top speed of the bike was *v* = (0.5 m) (21 rad/s) = 11 m/s, or about 24 miles per hour—reasonable for an average biker. *Note: To use these equations, angular variable units must involve radians, not degrees or revolutions!!!*

The rotational kinematics equations, just like the linear kinematics equations, are only valid when acceleration is constant. If acceleration is changing, then the same calculus equations that were used for linear kinematics apply here:

Practice problems for these concepts can be found at:

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