Angular Velocity and Acceleration

We are already familiar with the concepts of linear velocity and linear acceleration, so angular velocity and angular acceleration are not too difficult to understand.

Angular velocity is, quite simply, how quickly an object is rotating.

Angular velocity is denoted by the Greek letter $\omega$.

The angular velocity measures how much an object has rotated in a given time. Thus, the unit of angular velocity is naturally radians per second.

Let us denote $\theta$ to be the angular displacement, and let $t$ be time.

We have the following result: $$\omega=\frac{d\theta}{dt}$$

Angular acceleration is just one step further. Naturally, as in linear motion, we are differentiating the angular velocity with respect to time to get the angular acceleration, denoted $\alpha$. The unit is radians per second squared.

We have the following: $$\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$$

Now, here is the trick that makes everything easier. Remember all of the kinematics equations for linear motion? Well, they all have simple analogous results for angular motion! The derivations are exactly the same, so you do not waste any time. All you have to do is to make the following substitutions: $$x\mapsto\theta$$ $$v\mapsto\omega$$ $$a\mapsto\alpha$$ For example, for constant angular acceleration, $$\theta_f=\theta_i+\omega_it+\frac{1}{2}\alpha t^2$$ With further study into angular motion, you will find even more nice substitutions for quantities such as mass and force; these will allow you to pull even more convenient equations from your linear motion repertoire!