AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 5: Torque & Rotational DynamicsRotational KinematicsRotational Kinematics Equations

Rotational Kinematics Equations (College Board AP® Physics 1: Algebra-Based): Study Guide

Written by: Katie M

Reviewed by: Caroline Carroll

Updated on 9 October 2024

Rotational kinematics equations

Rotating systems in a state of constant angular acceleration can be described by three rotational kinematic equations
These are analogous to the linear kinematic equations
- Each of the three rotational kinematics equations are listed on the equation sheet and will be provided in the exams
For all of these equations, the following conditions apply:
- angular acceleration is constant
- motion is relative to an axis of rotation
For all these equations:
- Time interval, $∆ t = t$ (i.e. the timer is assumed to start from zero, $t_{0} = 0$ )
- Change in angular velocity, $∆ ω = ω - ω_{0}$
- Angular displacement, $∆ θ = θ - θ_{0}$

Rotational kinematic equation 1

This equation is used when angular displacement is not required

$ω = ω_{0} + α t$

Where:
- $ω$ = final angular velocity, in $rad / s$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $t$ = time interval, in $s$

Rotational kinematic equation 2

This equation is used when final angular velocity is not required

$θ = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$

Where:
- $θ$ = final angular position, in $rad$
- $θ_{0}$ = initial angular position, in $rad$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $t$ = time interval, measured in $s$

Rotational kinematic equation 3

This equation is used when time is not required

$ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})$

Where:
- $ω$ = final angular velocity, in $rad / s$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $θ - θ_{0} = ∆ θ$ = angular displacement, in $rad$

Other helpful equations in rotational kinematics

Angular displacement can be calculated using angular velocity and time when angular acceleration is not required

$∆ θ = \frac{1}{2} (ω_{0} + ω) t$

Where:
- $∆ θ$ = angular displacement, in $rad$
- $ω_{0}$ = initial angular velocity, in $rad / s$
- $ω$ = final angular velocity, in $rad / s$
- $t$ = time interval, in $s$
Final angular position can be calculated when initial angular velocity is not required using the following equation:

$θ = θ_{0} + ω t - \frac{1}{2} α t^{2}$

Where:
- $θ$ = final angular position, in $rad$
- $θ_{0}$ = initial angular position, in $rad$
- $ω$ = final angular velocity, in $rad / s$
- $α$ = angular acceleration, in $rad / s^{2}$
- $t$ = time interval, in $s$

Table of rotational kinematics equations

Linear equation	Rotational equation	Quantity not required
$v_{x} = v_{x 0} + a_{x} t$	$ω = ω_{0} + α t$	$∆ θ$
$x = x_{0} + v_{x 0} t + \frac{1}{2} a_{x} t^{2}$	$θ = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$	$ω$
$v_{x}^{2} = v_{x 0}^{2} + 2 a_{x} (x - x_{0})$	$ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})$	$t$
$∆ x = \frac{1}{2} (v_{x 0} + v_{x}) t$	$∆ θ = \frac{1}{2} (ω_{0} + ω) t$	$α$
$x = x_{0} + v_{x} t - \frac{1}{2} a_{x} t^{2}$	$θ = θ_{0} + ω t - \frac{1}{2} α t^{2}$	$ω_{0}$

Worked Example

The turntable of a record player spins at an angular velocity of $45 rpm$ just before it is turned off. Its rotation then decelerates at a rate of $0.8 rad / s^{2}$ .

Determine the number of rotations the turntable completes before it comes to rest.

Answer:

Step 1: List the known quantities

Taking the initial direction of rotation as positive
Final angular velocity, $ω = 0$
Initial angular velocity, $ω_{0} = 45 rpm$
Angular acceleration, $α = - 0.8 rad / s^{2}$

Step 2: Convert the angular velocity from rpm to rad/s

One revolution corresponds to a rotation angle of $2 π$ radians
Therefore, the initial angular velocity is:

$ω_{0} = 45 rpm \times \frac{2 π}{60} = \frac{3 π}{2} rad / s$

Step 3: Choose the relevant rotational kinematic equation

The question asks for the number of rotations completed, which is equal to the ratio

$\frac{angular displacement of the turntable}{angular displacement of one rotation}$

Therefore, the quantity we need to calculate is $∆ θ$
The quantities we know are $ω$ , $ω_{0}$ and $α$
The quantity not required in this calculation is $t$

$ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})$

Step 4: Rearrange the equation

$0 = ω_{0}^{2} + 2 α (θ - θ_{0})$

$- 2 α (θ - θ_{0}) = ω_{0}^{2}$

$θ - θ_{0} = - \frac{ω_{0}^{2}}{2 α}$

Since angular displacement is $∆ θ = θ - θ_{0}$

$∆ θ = - \frac{ω_{0}^{2}}{2 α}$

Step 5: Substitute the known values and calculate the angular displacement

$∆ θ = - \frac{{(\frac{3 π}{2})}^{2}}{2 \times (- 0.8)} = 13.88 rad$

Step 6: Determine the number of rotations completed

There are $2 π$ radians in one rotation
Therefore, the number of rotations completed is

$\frac{angular displacement of the turntable}{angular displacement of one rotation} = \frac{13.88}{2 π} = 2.2$

This means the turntable spins 2.2 times before coming to rest

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent