Kinematics of circular motion

What you'll learn

How the radian measures angle using arc length and radius.
How period and frequency describe repeated circular motion.
How to use $ω=2πT\omega = \frac{2\pi}{T}$ and $ω=2πf\omega = 2\pi f$ for angular velocity.
How to avoid the most common unit mistakes in circular-motion calculations.

1. Describing motion around a circle

Circular motion means motion along the circumference of a circle. The circumference is the distance all the way around the circle.

For an object moving in a circle:

the radius, $r$ , is the distance from the centre of the circle to the object;
the arc length, $s$ , is the distance travelled along part of the circumference;
one complete trip around the circle is called one revolution or one turn;
the object’s instantaneous velocity is tangential, meaning it points along the tangent to the circle at that instant.

The key idea is that circular motion is often easier to describe using angle turned through, rather than distance along the circle.

Diagram of circular motion showing radius, arc length, angle in radians, tangential velocity and angular velocity

2. The radian

At GCSE you probably used degrees. In A-Level circular motion, angles are usually measured in radians.

Definition

Radian

One radian is the angle at the centre of a circle when the arc length is equal to the radius. In symbols, $θ=sr\theta = \frac{s}{r}$ , where $θ\theta$ is the angle in radians, $s$ is the arc length in metres, and $r$ is the radius in metres.

The defining equation is:

θ=sr\theta = \frac{s}{r}

Because both $s$ and $r$ are lengths, the units cancel. Even so, you should usually write the angle unit as rad to show you are using radians.

For a full circle, the arc length is the circumference, $2πr2\pi r$ . So:

rad\theta = \frac{2\pi r}{r} = 2\pi \ \text{rad}

So:

one full turn is $rad2\pi \ \text{rad}$ ;
half a turn is $rad\pi \ \text{rad}$ ;
one quarter turn is $rad\frac{\pi}{2} \ \text{rad}$ .

Key Idea

Radians link distance and angle

Radians are useful because they directly connect distance around the circle to the angle swept out: $r\theta$ , as long as $θ\theta$ is in radians.

Example

Calculating an angle in radians

A bead moves along a circular track of radius 0.80 m. It travels an arc length of 0.60 m. Calculate the angle swept out in radians.

Use the radian definition because the question gives arc length and radius:
$θ=sr\theta = \frac{s}{r}$
Substitute the values, carrying the length units through:
$rad\theta = \frac{0.60\ \text{m}}{0.80\ \text{m}} = 0.75\ \text{rad}$
Check the size of the answer by comparing with a full circle:
$m2\pi r = 2\pi(0.80\ \text{m}) \approx 5.0\ \text{m}$
The bead travels much less than the full circumference, so an angle much less than $rad2\pi \ \text{rad}$ is sensible.

Common Mistake

Using degrees in radian equations

Equations such as $θ=sr\theta = \frac{s}{r}$ only work when $θ\theta$ is measured in radians. Do not substitute degrees into circular-motion formulae unless you have converted them to radians first.

3. Period and frequency

Circular motion is repetitive, so we often describe it using time per revolution or revolutions per second.

Definition

Period and frequency

The period, $T$ , is the time taken for one complete revolution, measured in seconds. The frequency, $f$ , is the number of revolutions per second, measured in hertz, Hz.

One hertz means one cycle per second. So frequency has the same base unit as per second, $s−1\text{s}^{-1}$ .

Period and frequency are reciprocals:

\frac{1}{T}

and

\frac{1}{f}

If something rotates quickly, it has a high frequency and a small period. If it rotates slowly, it has a low frequency and a large period.

Example

Finding frequency and period

A toy car completes 8.0 laps of a circular track in 20 s. Calculate its frequency and period.

Frequency is the number of complete revolutions per second:
$\frac{8.0}{20\ \text{s}} = 0.40\ \text{s}^{-1} = 0.40\ \text{Hz}$
Period is the time for one revolution, so use the reciprocal relationship:
$\frac{1}{f} = \frac{1}{0.40\ \text{Hz}} = 2.5\ \text{s}$
Check consistency by multiplying laps by time per lap:
$8.0 \times 2.5\ \text{s} = 20\ \text{s}$
This matches the total time, so the values are consistent.

Tip

Revolutions per minute

If a rotation rate is given in revolutions per minute, convert to revolutions per second before using it as frequency. Divide by 60 because 1 min = 60 s.

4. Angular velocity

The angular displacement of an object is the angle it has swept out, usually measured in radians.

Definition

Angular velocity

Angular velocity, $ω\omega$ , is the rate at which angular displacement changes. For uniform circular motion, it is the angle swept out per unit time, measured in radian per second, rad s $^{-1}$ .

For one complete revolution, the angle swept out is $rad2\pi \ \text{rad}$ , and the time taken is one period, $T$ .

So:

ω=2πT\omega = \frac{2\pi}{T}

Since $\frac{1}{T}$ , this can also be written as:

ω=2πf\omega = 2\pi f

These are the key equations for this part of the specification.

Key Idea

Frequency counts turns, angular velocity counts radians

Frequency tells you how many revolutions happen each second. Angular velocity tells you how many radians are swept out each second. Since one revolution is $rad2\pi \ \text{rad}$ , $ω=2πf\omega = 2\pi f$ .

Example

Calculating angular velocity

A rotor spins at $2.4 \times 10^{3}$ revolutions per minute. Calculate its angular velocity in rad s $^{-1}$ .

Convert revolutions per minute into frequency in hertz:
$\frac{2.4 \times 10^{3}}{60\ \text{s}} = 40\ \text{Hz}$
Use the angular velocity equation because frequency is now known:
$ω=2πf\omega = 2\pi f$ $s−1\omega = 2\pi(40\ \text{s}^{-1}) = 251\ \text{rad s}^{-1}$
Quote the answer to a sensible number of significant figures:
$s−1\omega \approx 2.5 \times 10^{2}\ \text{rad s}^{-1}$

Common Mistake

Confusing frequency and angular velocity

A frequency of 1.0 Hz does not mean an angular velocity of 1.0 rad s $^{-1}$ . One full revolution is $rad2\pi \ \text{rad}$ , so 1.0 Hz corresponds to $s−12\pi \ \text{rad s}^{-1}$ .

5. Choosing the right equation

For this sub-topic, your choice usually depends on what the question gives you.

If you know the period, use:

ω=2πT\omega = \frac{2\pi}{T}

If you know the frequency, use:

ω=2πf\omega = 2\pi f

If you know arc length and radius, use:

θ=sr\theta = \frac{s}{r}

If you need to move between period and frequency, use:

\frac{1}{T}

\frac{1}{f}

Tip

Same rotation, same angular velocity

For a rigid rotating object, such as a disc, all points complete one revolution in the same time. So they have the same $T$ , $f$ and $ω\omega$ , even though points farther from the centre travel a longer distance around the circle.

Exam technique

In the exam

Convert all times to seconds before using $T$ , $f$ or $ω\omega$ equations.
Check whether the question gives period or frequency, then choose either $ω=2πT\omega = \frac{2\pi}{T}$ or $ω=2πf\omega = 2\pi f$ .
Keep radians separate from degrees: circular-motion angle equations require radians.

Self review

Check yourself

A particle travels an arc length of 0.45 m around a circle of radius 0.30 m. What angle has it swept out in radians?
An object in circular motion has a period of 0.20 s. What are its frequency and angular velocity?
Why is an angular velocity in rad s $^{-1}$ not the same quantity as a frequency in Hz?

What you'll learn

How the radian measures angle using arc length and radius.
How period and frequency describe repeated circular motion.
How to use $ω=2πT\omega = \frac{2\pi}{T}$ and $ω=2πf\omega = 2\pi f$ for angular velocity.
How to avoid the most common unit mistakes in circular-motion calculations.

1. Describing motion around a circle

Circular motion means motion along the circumference of a circle. The circumference is the distance all the way around the circle.

For an object moving in a circle:

the radius, $r$ , is the distance from the centre of the circle to the object;
the arc length, $s$ , is the distance travelled along part of the circumference;
one complete trip around the circle is called one revolution or one turn;
the object’s instantaneous velocity is tangential, meaning it points along the tangent to the circle at that instant.

The key idea is that circular motion is often easier to describe using angle turned through, rather than distance along the circle.

Diagram of circular motion showing radius, arc length, angle in radians, tangential velocity and angular velocity

2. The radian

At GCSE you probably used degrees. In A-Level circular motion, angles are usually measured in radians.

Definition

Radian

The defining equation is:

θ=sr\theta = \frac{s}{r}

Because both $s$ and $r$ are lengths, the units cancel. Even so, you should usually write the angle unit as rad to show you are using radians.

For a full circle, the arc length is the circumference, $2πr2\pi r$ . So:

rad\theta = \frac{2\pi r}{r} = 2\pi \ \text{rad}

So:

one full turn is $rad2\pi \ \text{rad}$ ;
half a turn is $rad\pi \ \text{rad}$ ;
one quarter turn is $rad\frac{\pi}{2} \ \text{rad}$ .

Key Idea

Radians link distance and angle

Radians are useful because they directly connect distance around the circle to the angle swept out: $r\theta$ , as long as $θ\theta$ is in radians.

Example

Calculating an angle in radians

A bead moves along a circular track of radius 0.80 m. It travels an arc length of 0.60 m. Calculate the angle swept out in radians.

Use the radian definition because the question gives arc length and radius:
$θ=sr\theta = \frac{s}{r}$
Substitute the values, carrying the length units through:
$rad\theta = \frac{0.60\ \text{m}}{0.80\ \text{m}} = 0.75\ \text{rad}$
Check the size of the answer by comparing with a full circle:
$m2\pi r = 2\pi(0.80\ \text{m}) \approx 5.0\ \text{m}$
The bead travels much less than the full circumference, so an angle much less than $rad2\pi \ \text{rad}$ is sensible.

Common Mistake

Using degrees in radian equations

3. Period and frequency

Circular motion is repetitive, so we often describe it using time per revolution or revolutions per second.

Definition

Period and frequency

The period, $T$ , is the time taken for one complete revolution, measured in seconds. The frequency, $f$ , is the number of revolutions per second, measured in hertz, Hz.

One hertz means one cycle per second. So frequency has the same base unit as per second, $s−1\text{s}^{-1}$ .

Period and frequency are reciprocals:

\frac{1}{T}

and

\frac{1}{f}

If something rotates quickly, it has a high frequency and a small period. If it rotates slowly, it has a low frequency and a large period.

Example

Finding frequency and period

A toy car completes 8.0 laps of a circular track in 20 s. Calculate its frequency and period.

Frequency is the number of complete revolutions per second:
$\frac{8.0}{20\ \text{s}} = 0.40\ \text{s}^{-1} = 0.40\ \text{Hz}$
Period is the time for one revolution, so use the reciprocal relationship:
$\frac{1}{f} = \frac{1}{0.40\ \text{Hz}} = 2.5\ \text{s}$
Check consistency by multiplying laps by time per lap:
$8.0 \times 2.5\ \text{s} = 20\ \text{s}$
This matches the total time, so the values are consistent.

Tip

Revolutions per minute

If a rotation rate is given in revolutions per minute, convert to revolutions per second before using it as frequency. Divide by 60 because 1 min = 60 s.

4. Angular velocity

The angular displacement of an object is the angle it has swept out, usually measured in radians.

Definition

Angular velocity

Angular velocity, $ω\omega$ , is the rate at which angular displacement changes. For uniform circular motion, it is the angle swept out per unit time, measured in radian per second, rad s $^{-1}$ .

For one complete revolution, the angle swept out is $rad2\pi \ \text{rad}$ , and the time taken is one period, $T$ .

So:

ω=2πT\omega = \frac{2\pi}{T}

Since $\frac{1}{T}$ , this can also be written as:

ω=2πf\omega = 2\pi f

These are the key equations for this part of the specification.

Key Idea

Frequency counts turns, angular velocity counts radians

Example

Calculating angular velocity

A rotor spins at $2.4 \times 10^{3}$ revolutions per minute. Calculate its angular velocity in rad s $^{-1}$ .

Convert revolutions per minute into frequency in hertz:
$\frac{2.4 \times 10^{3}}{60\ \text{s}} = 40\ \text{Hz}$
Use the angular velocity equation because frequency is now known:
$ω=2πf\omega = 2\pi f$ $s−1\omega = 2\pi(40\ \text{s}^{-1}) = 251\ \text{rad s}^{-1}$
Quote the answer to a sensible number of significant figures:
$s−1\omega \approx 2.5 \times 10^{2}\ \text{rad s}^{-1}$

Common Mistake

Confusing frequency and angular velocity

A frequency of 1.0 Hz does not mean an angular velocity of 1.0 rad s $^{-1}$ . One full revolution is $rad2\pi \ \text{rad}$ , so 1.0 Hz corresponds to $s−12\pi \ \text{rad s}^{-1}$ .

5. Choosing the right equation

For this sub-topic, your choice usually depends on what the question gives you.

If you know the period, use:

ω=2πT\omega = \frac{2\pi}{T}

If you know the frequency, use:

ω=2πf\omega = 2\pi f

If you know arc length and radius, use:

θ=sr\theta = \frac{s}{r}

If you need to move between period and frequency, use:

\frac{1}{T}

\frac{1}{f}

Tip

Same rotation, same angular velocity

Exam technique

In the exam

Convert all times to seconds before using $T$ , $f$ or $ω\omega$ equations.
Check whether the question gives period or frequency, then choose either $ω=2πT\omega = \frac{2\pi}{T}$ or $ω=2πf\omega = 2\pi f$ .
Keep radians separate from degrees: circular-motion angle equations require radians.

Self review

Check yourself

A particle travels an arc length of 0.45 m around a circle of radius 0.30 m. What angle has it swept out in radians?
An object in circular motion has a period of 0.20 s. What are its frequency and angular velocity?
Why is an angular velocity in rad s $^{-1}$ not the same quantity as a frequency in Hz?

What you'll learn

1. Describing motion around a circle

2. The radian

3. Period and frequency

4. Angular velocity

5. Choosing the right equation

Review Kinematics of circular motion by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Kinematics of circular motion

What you'll learn

1. Describing motion around a circle

2. The radian

3. Period and frequency

4. Angular velocity

5. Choosing the right equation

Review Kinematics of circular motion by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Kinematics of circular motion

What you'll learn

1. Describing motion around a circle

2. The radian

3. Period and frequency

4. Angular velocity

5. Choosing the right equation

Review Kinematics of circular motion by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Kinematics of circular motion Revision Guide

Kinematics of circular motion

What you'll learn

1. Describing motion around a circle

2. The radian

3. Period and frequency

4. Angular velocity

5. Choosing the right equation

Review Kinematics of circular motion by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Kinematics of circular motion Revision Guide