Circular Motion

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1. Angular Speed and the Relation v = rω

What is circular motion?

When a particle (an object) moves along a circular path, we call this circular motion. Examples include a car going round a roundabout, a satellite orbiting Earth, or a ball on a string being swung in a circle.

Linear speed vs Angular speed

When something moves in a circle, we can describe how fast it's moving in two different ways:

• Linear speed (v): This is the actual distance traveled per second, measured in metres per second (m s⁻¹). It's the speed you would measure if you were sitting on the moving object.

• Angular speed (ω): This is how quickly the object is turning around the circle, measured in radians per second (rad s⁻¹). Think of it as how quickly the angle is changing as the object moves around the center of the circle.

What is a radian?

A radian is just another way to measure angles (instead of degrees). One complete circle = 360° = 2π radians. So:

• Half a circle = 180° = π radians
• Quarter circle = 90° = π/2 radians

The key relationship: v = rω

The linear speed (v) and angular speed (ω) are connected by a simple formula:

v = rω

Where:

• v = linear speed (m s⁻¹)
• r = radius of the circle (m)
• ω = angular speed (rad s⁻¹)

This makes sense: if you're further from the center (larger r), you have to travel faster (larger v) to complete the circle in the same time.

Example: A particle moves in a circle of radius 2 m with angular speed 3 rad s⁻¹.

• Linear speed v = rω = 2 × 3 = 6 m s⁻¹

Time period and frequency

• Time period (T): The time taken to complete one full circle. Since one complete circle is 2π radians: ω = 2π/T

• Frequency (f): The number of complete circles per second. Frequency = 1/T, so: ω = 2πf