To describe the motion of rotating or spinning objects, we need a more natural set of variables than the x's, v's, and a's we have been using so far. For instance, every point on a spinning object has a different velocity, but we can define an angular velocity that is the same for all points. v is proportional to r, so all we really need to do it divide out the r.

Angular variables

Angular displacement (angle): θ = s/r

Angular velocity: ω = v/r

Angular acceleration: α = a_t/r

Note that the angular acceleration is connected to the tangential acceleration, not the centripetal acceleration. There is an angular acceleration only when the rotation rate changes.

If the angular velocity is constant, then the speed of a point on the rotating object is:

v = 2πr / T

where T is the period, the time to go around once.

The angular speed is ω = v/r = 2π / T

These angular variables are vectors, just like their straight-line motion cousins. Which way do they point? If you take the fingers of your right hand and curl them the way an object is spinning, stick your thumb out and you get the direction of the angular velocity. The angular acceleration is in the same direction if the object is speeding up its spin, and in the opposite direction if its slowing down.

We'll often use clockwise and counterclockwise to indicate direction. The book (page 217) defines counterclockwise to be positive, but there's no good reason for that. As with straight-line motion, we can define the positive direction based on what's convenient in a particular case.

We will focus on rotation about a single axis of rotation, which is analogous to one-dimensional straight-line motion. Basically, if you understand 1-D motion you can do this - rotational motion is really just straight-line motion rolled up into a circle.

Displacement, velocity, and acceleration all have rotational equivalents. There are also rotational equivalents of mass, force, Newton's Laws, kinetic energy, momentum, etc. Any equation we used for straight-line motion has a rotational form that can be found by substituting the equivalent rotational variables.

For instance, how are angles, angular velocities, and angular accelerations related? The same way the linear variables are:

Angular velocity is the rate of change of angle

Instantaneous angular velocity: ω = dθ/dt

Average angular velocity = ω_avg = Δθ/Δt

Δθ = ∫ ω dt

Angular acceleration is the rate of change of angular velocity

Instantaneous angular acceleration: α = dω/dt

Average angular acceleration = α_avg = Δω/Δt

Δω = ∫ α dt