# Rotational Motion

## Basic Definitions

#### Angular Motion

Angular motion, or rotational motion, is the movement of an object which is rotating around an axis.

#### Angular Displacement

Angular displacement ( $\theta$ ) is the angle through which the object rotates, measured in radians ( $\text{rad}$ ).

Tip

${360}^{\circ }=2\pi$ radians.

#### Angular Velocity

Angular velocity ( $\omega$ ) is the rate of change of angular displacement, measured in radians per second ( ${\text{rad s}}^{-1}$ ).

Formula

$\omega =\frac{d\theta }{dt}$

#### Angular Acceleration

Angular acceleration ( $\alpha$ ) is the rate of change of angular velocity, measured in radians per second squared ( ${\text{rad s}}^{-2}$ ).

Alpha, not A!

Don't confuse $\alpha$ with $a$ (linear acceleration)! They look similar, but they are the Greek letter alpha and the letter A respectively.

Formula

$\alpha =\frac{d\omega }{dt}$

#### Linear Velocity

Linear velocity or tangential velocity ( $v$ ) is the instantaneous linear velocity of a rotating object - it is the velocity of an object revolving around an axis if the tether breaks.

#### Linear Acceleration

Linear acceleration or tangential acceleration ( ${a}_{t}$ ), like tangential velocity, is the instantaneous linear acceleration of a rotating object - it's the change in tangential velocity.

Formula

${a}_{t}=\frac{dv}{dt}$

Radial acceleration ( ${a}_{r}$ ) is the acceleration towards the centre of a rotating system, caused by centripetal force.

Formula

${a}_{r}=r{\omega }^{2}=\frac{{v}^{2}}{r}$

Variable Key

• ${a}_{r}$ is the radial acceleration, in metres per second squared.
• $r$ is the radius of the rotating system, in metres.
• $\omega$ is the angular velocity, in radians per second.
• $v$ is the linear velocity, in metres per second.

## Equations of Motion

The equations of motion can be applied to angular motion as well as linear motion, as long as all tangential/linear quantities are substituted for their angular counterparts.

Quantity Linear Angular
Displacement $s$ $\theta$
Initial Velocity $u$ ${\omega }_{0}$
Current Velocity $v$ $\omega$
Acceleration $a$ $\alpha$
Time (doesn't change!) $t$ $t$

You can convert between linear and angular motion by multiplying or dividing the radius as a factor. To think about this intuitively, this is because $c=2\pi r$, and if the circumference is directly proportional to the radius, then velocity, acceleration, etc. must be directly proportional to the radius as well.

Formulae

$\begin{array}{rl}s& =r\theta \\ v& =r\omega \\ a& =r\alpha \end{array}$