Every object we have studied so far moved from one place to another. A spinning wheel does not do that. When you hold a bicycle wheel off the ground and spin it, its center stays exactly where it was while something obviously happens to the wheel. This topic is about describing that new thing.

The good news is that there is not a lot of new stuff. Every quantity on this page has a linear (translational) partner you already know, and every equation has the same format as its counterpart. What changes is what the symbols stand for.

1. The angle is the position

A rotating wheel has no position to report. A random point you pick on the wheel will still change its position. If you draw a line from that point to the center, then choose a direction as your reference point, the angle between the line and your reference point shows you everything about the motion of the wheel. When you turn the wheel, that angle changes. That change is called the angular displacement.

\(\Delta\theta = \theta – \theta_0\)radians

The angle is measured in radians, and that is not to make your life more difficult than it already is. One radian is the angle that wraps an arc equal to the radius around the circle, so one full turn is \(2\pi\) radians, a number a little over six. Every equation in this unit uses radians. A number in degrees used in any rotational motion equation is wrong by a factor of about fifty seven.

A wheel is not a point-like mass. Until now, you have been allowed to shrink a cart to a single point carrying its mass. A rotating wheel will not survive that treatment: the top of it moves one way while the bottom moves the other, so no single point can be used to calculate the entire motion. The rotational motion approach calls this a rigid system. It maintains its shape, and different parts of it move in different directions at the same moment.

One direction of turning is positive and the other is negative, and which one you pick is your call as long as you maintain it for the whole problem. Counterclockwise positive is the usual choice since it is based on the unit circle. The sign is part of the answer, exactly as it was on a straight track.

In AP Physics C you will meet these angular quantities written as vectors, and the exam asks you to manipulate their magnitudes. The directions of those vectors are not assessed. On this page, clockwise and counterclockwise about a stated axis is all that direction means.

2. Angular velocity is how fast the angle changes

Angular velocity is the rate at which the angle changes, and it is defined exactly how linear velocity was defined.

\(\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}\)radians per second

\(\omega = \frac{d\theta}{dt}\)the instantaneous value, at one moment

The average over an interval is the first expression. The angular velocity at a single instant is the second, and the two agree whenever the angle changes at a steady rate. When it does not, the average tells you nothing about any particular moment inside the interval, which is the same warning you met in Unit 1.

The unit is read as radians per second. A record player turning at thirty three and a third revolutions per minute is turning at about 3.5 radians per second, and converting it is the first step in solving rotational motion questions.

Every point on a rigid system shares one angular velocity. This is crucial, because the next topic depends on it. The rider at the rail of a merry-go-round and the rider near the middle do not have anything in common except for this quantity. Both go around once in the same time, so both sweep the same angle every second.

3. Angular acceleration is how fast the angular velocity changes

If the angular velocity is changing, that means there is an angular acceleration, and it is defined the way you can probably guess.

\(\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}\)radians per second squared

\(\alpha = \frac{d\omega}{dt} = \frac{d^{2}\theta}{dt^{2}}\)the instantaneous value

The sign rule is what tricks a lot of students, and you might remember it from Unit 1. Angular acceleration does not mean speeding up. It means the angular velocity is changing. When \(\alpha\) carries the same sign as \(\omega\), the spin gets faster. When the signs are opposite, the spin slows down, and if nothing stops it the wheel will halt and then turn back the other way.

A wheel can also have a large angular velocity and no angular acceleration at all. A fan at its pre-set speed turns steadily, so \(\omega\) is large and constant and \(\alpha\) is zero. Nothing about a fast spin requires an angular acceleration to go with it unless it starts spinning from rest.

4. Three equations you already know, kinda

When the angular acceleration is constant, the three relationships between angle, angular velocity and time are the three you memorized in Unit 1 with the letters exchanged.

\(\omega = \omega_0 + \alpha t\)the counterpart of \(v = v_0 + at\)

\(\theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^{2}\)the counterpart of \(x = x_0 + v_0 t + \tfrac{1}{2}at^{2}\)

\(\omega^{2} = \omega_0^{2} + 2\alpha(\theta – \theta_0)\)the counterpart of \(v^{2} = v_0^{2} + 2a(x – x_0)\)

There is nothing to derive and nothing new to memorize. You only need to replace \(x\) with \(\theta\), \(v\) with \(\omega\), and \(a\) with \(\alpha\). The condition holds true: all three require the angular acceleration to be constant, and none of the equations work when it is not constant.

5. The graphs carry the same three facts

When you plot the angle against time, the slope of that graph gives you the angular velocity. When you plot the angular velocity against time, the slope of that graph gives you the angular acceleration. It is the same approach you used in Unit 1, and it runs the same way back down: the area under an angular velocity graph is the angular displacement.

\(\theta = \theta_0 + \int \omega \, dt \qquad \omega = \omega_0 + \int \alpha \, dt\)the way back down

When you write the trip back down as an integral, you are no longer limited to a constant angular acceleration. These integrals give you the three equations from the last section whenever \(\alpha\) does not change. When it does change, the integral still works and those three equations do not.

This is how a graph question is answered with no equation at all. A straight rising line on an angle against time graph means a constant angular velocity. A curve that steepens means the angular velocity is growing, so there is an angular acceleration, and the sign of the curvature is the sign of that angular acceleration.

Watch what the three graphs do together when you move the cursor. Nothing on the second graph can be identified from the first graph without calculating the slope, and nothing on the first graph can be identified from the second graph without taking an area. Do not try to read a value straight across from one graph to another since it does not give you anything.