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 ( $θ$ ) is the angle through which the object rotates, measured in radians ( $rad$ ).

Tip

$360^{\circ} = 2 π$ radians.

Angular Velocity

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

Formula

$ω = \frac{d θ}{d t}$

Angular Acceleration

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

Alpha, not A!

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

Formula

$α = \frac{d ω}{d t}$

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

$a_{t} = \frac{d v}{d t}$

Radial Acceleration

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

Formula

$a_{r} = r ω^{2} = \frac{v^{2}}{r}$

Variable Key

$a_{r}$ is the radial acceleration, in metres per second squared.
$r$ is the radius of the rotating system, in metres.
$ω$ 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	$s$	$θ$
Initial Velocity	$u$	$ω_{0}$
Current Velocity	$v$	$ω$
Acceleration	$a$	$α$
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 $c = 2 π r$ , and if the circumference is directly proportional to the radius, then velocity, acceleration, etc. must be directly proportional to the radius as well.

Formulae

$\begin{aligned} s & = r θ \\ v & = r ω \\ a & = r α \end{aligned}$