11.1.2 Angular Acceleration Equations

Equations for Uniform Angular Acceleration

The kinematic equations of motion for uniform linear acceleration can also be re-written for rotational motion
The four kinematic equations for uniform linear acceleration are

$v = u + a t$

$s = u t + \frac{1}{2} a t^{2}$

$v^{2} = u^{2} + 2 a s$

$s = \frac{(u + v) t}{2}$

This leads to the four kinematic equations for uniform rotational acceleration

$ω_{2} = ω_{1} + α t$

$∆ θ = ω_{1} t + \frac{1}{2} α t^{2}$

${ω_{2}}^{2} = {ω_{1}}^{2} + 2 α ∆ θ$

$∆ θ = \frac{(ω_{1} + ω_{2}) t}{2}$

The five linear variables have been swapped for the rotational equivalents, as shown in the table below

Variable	Linear	Rotational
displacement	s	θ
initial velocity	u	ω₁
final velocity	v	ω₂
acceleration	a	α
time	t	t

Worked example

The turntable of a record player is spinning at an angular velocity of 45 RPM just before it is turned off. It then decelerates at a constant rate of 0.8 rad s⁻².

Determine the number of rotations the turntable completes before coming to a stop.

Answer:

Step 1: List the known quantities

Initial angular velocity, $ω_{1}$ = 45 RPM
Final angular velocity, $ω_{2}$ = 0
Angular acceleration, $α$ = −0.8 rad s⁻²
Angular displacement, $∆ θ$ = ?

Step 2: Convert the angular velocity from RPM to rad s⁻¹

One revolution corresponds to 2π radians, and RPM = revolutions per minute, so

$ω = 2 π f$ and $f = \frac{R P M}{60}$ (to convert to seconds)

$ω_{1} = \frac{2 π \times RPM}{60} = \frac{2 π \times 45}{60} = \frac{3 π}{2} rad s^{- 1}$

Step 3: Select the most appropriate kinematic equation

We know the values of $ω_{1}$ , $ω_{2}$ and $α$ , and we are looking for angular displacement $θ$ , so the best equation to use would be

${ω_{2}}^{2} = {ω_{1}}^{2} + 2 α ∆ θ$

Step 4: Rearrange and calculate the angular displacement $∆ θ$

$0 = {ω_{1}}^{2} + 2 α ∆ θ$

$- 2 α ∆ θ = {ω_{1}}^{2}$

$∆ θ = \frac{{ω_{1}}^{2}}{- 2 α} = \frac{{(\frac{3 π}{2})}^{2}}{- 2 \times - 0.8}$

Angular displacement, $∆ θ$ = 13.88 rad

Step 5: Determine the number of rotations in $∆ θ$

There are 2π radians in 1 rotation
Therefore, the number of rotations = $\frac{13.88}{2 π}$ = 2.2
This means the turntable spins 2.2 times before coming to a stop

AQA A Level Physics

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Equations for Uniform Angular Acceleration

Worked example

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Author: Ashika