Circular Motion

Overview

Circular Motion is the study of motion along a circular path. Although an object may move with constant speed, its velocity usually changes continuously because velocity is a vector quantity with both magnitude and direction.

Therefore, circular motion commonly involves:

• changing velocity
• acceleration toward the centre
• a resultant inward force

This topic connects ideas from:

• Kinematics
• Vectors
• Dynamics
• Forces
• Work, Energy and Power
• Gravitational Fields

For deeper treatment, see:

• Centripetal Acceleration and Force
• Circular Motion Force Analysis
• Vertical Circular Motion
• Orbital Motion in Gravity

Core Ideas

What Is Circular Motion?

An object undergoes circular motion when it moves along a circular path of radius $r$.

Examples:

• car turning around a bend
• stone tied to a string
• rotating fan blade tip
• satellite in orbit
• roller coaster loop

Circular motion may be:

• Uniform circular motion: speed remains constant
• Non-uniform circular motion: speed changes

Scalar vs Vector Distinction

This chapter requires careful distinction between scalars and vectors.

Scalars

• distance
• speed
• time
• radius
• period
• frequency

Vectors

• displacement
• velocity $\vec{v}$
• acceleration $\vec{a}$
• force $\vec{F}$

Important Reminder

A particle may move with constant speed but changing velocity because direction changes.

That is why uniform circular motion still has acceleration.

Angular Motion Quantities

Angular Displacement

Measured in radians.

For arc length $s$:

$$\theta =\frac{s}{r}$$

Angular Speed

Angular speed:

$$\omega =\frac{\Delta \theta }{\Delta t}$$

For one full revolution:

$$\theta =2\pi$$

Hence:

$$T=\frac{2\pi }{\omega }$$

and

$$f=\frac{1}{T}$$

So:

$$\omega =2\pi f$$

Where:

• $T$ = period
• $f$ = frequency

Speed and Tangential Velocity

The linear speed of an object moving in a circle is:

$$v=r\omega$$

Where:

• $v$ = speed (scalar)
• $r$ = radius
• $\omega$ = angular speed

If discussing the vector velocity $\vec{v}$:

• direction is always tangent to the circular path
• magnitude is $v$

Direction of Key Vectors

For uniform circular motion:

Velocity Vector

$$\vec{v}$$

• tangent to path
• perpendicular to radius

Centripetal Acceleration Vector

$${\vec{a}}_c$$

• points toward centre

Resultant Force Vector

$${\vec{F}}_{\text{net}}$$

• points toward centre when circular motion is maintained

Centripetal Acceleration Overview

Because direction of velocity changes, acceleration exists even if speed is constant.

Magnitude:

$$a_c=\frac{v^2}{r}$$

Also:

$$a_c={\omega }^{2}r$$

Direction:

• always toward centre of circular path

See Centripetal Acceleration and Force.

Centripetal Force Overview

Using Newton’s Second Law:

$$F_c=m{a}_c$$

So:

$$F_c=\frac{m{v}^2}{r}$$

or

$$F_c=m{\omega }^{2}r$$

Important Warning

“Centripetal force” is not an extra new force.

It is the resultant inward force provided by real forces such as:

• tension
• friction
• normal contact force
• gravitational force
• electric force

Basic Force Analysis Overview

In circular motion questions:

1. Identify the object
2. Draw all real forces
3. Choose radial inward direction
4. Resolve forces if needed
5. Apply:

$$\sum F_{\text{radial}}=\frac{m{v}^2}{r}$$

See Circular Motion Force Analysis.

Horizontal Circular Motion (Brief Overview)

Examples:

• car turning on flat road
• conical pendulum
• rotating platform
• banked track

Often, speed may remain constant while force direction changes continuously.

Possible inward forces:

• friction
• tension
• horizontal component of normal force
• horizontal component of tension

Non-Uniform Circular Motion (Brief Note)

If speed changes, there are two acceleration components:

Radial (centripetal)

Toward centre:

$$a_r=\frac{v^2}{r}$$

Tangential

Along tangent, due to changing speed.

Hence resultant acceleration is not purely inward.

Only brief awareness is needed at H2 level.

Energy Methods (Brief Note)

For vertical circular motion, speed often changes with height.

Use conservation of mechanical energy:

$$KE+PE=\text{constant}$$

Then combine with circular-force equations.

See:

Vertical Circular Motion

Worked Example 1: Angular Speed

A wheel rotates at frequency $5.0\text{ Hz}$.

Find angular speed.

Solution

$$\omega =2\pi f=2\pi (5.0)=10\pi {\text{ rad s}}^{-1}$$

Worked Example 2: Centripetal Acceleration

A car moves at $20{\text{ m s}}^{-1}$ around a bend of radius $50\text{ m}$.

Find centripetal acceleration.

Solution

$$a_c=\frac{v^2}{r}=\frac{20^2}{50}=8.0{\text{ m s}}^{-2}$$

Worked Example 3: Resultant Force

A $1000\text{ kg}$ car from Example 2 moves around the bend.

Find resultant inward force.

Solution

$$F=\frac{m{v}^2}{r}=1000\left(\frac{400}{50}\right)=8000\text{ N}$$

Toward the centre.

Formula Summary

Angular Motion

$$\omega =\frac{2\pi }{T}$$ $$f=\frac{1}{T}$$ $$\omega =2\pi f$$

Linear and Angular Relation

$$v=r\omega$$

Centripetal Acceleration

$$a_c=\frac{v^2}{r}$$ $$a_c={\omega }^{2}r$$

Centripetal Force

$$F_c=\frac{m{v}^2}{r}$$ $$F_c=m{\omega }^{2}r$$

Exam Relevance

1. Confusing Speed with Velocity

Speed can be constant while velocity changes.

2. Treating Centripetal Force as Extra Force

Wrong. It is the resultant inward force.

3. Wrong Direction of Velocity

Velocity is tangent to path, not toward centre.

4. Wrong Direction of Acceleration

For uniform circular motion, acceleration is toward centre.

5. Forgetting Force Resolution

Only inward radial components contribute to centripetal requirement.

6. Mixing Period and Frequency

$$f=\frac{1}{T}$$

Links

• Centripetal Acceleration and Force
• Circular Motion Force Analysis
• Vertical Circular Motion
• Orbital Motion in Gravity
• Dynamics
• Work, Energy and Power
• Gravitational Fields
• Vectors