Circular Motion Basics

When a particle moves in a circle, it is constantly changing direction, which means it is accelerating even if its speed is constant. This centripetal acceleration is directed towards the centre of the circle and requires a centripetal force.

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1 Angular Speed

If a particle moves in a circle of radius r, its angular speed omega is:

omega = d theta/dt

where theta is the angle swept out. The SI unit is rad s^(-1).

For one complete revolution: theta = 2pi, time = T (period).

omega = 2pi/T = 2pi f

where f is the frequency (revolutions per second).

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2 Relationship Between Linear and Angular Speed

The linear speed v (tangential speed) is related to angular speed by:

v = r omega

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3 Centripetal Acceleration

A particle moving at constant speed v in a circle of radius r has acceleration directed towards the centre:

a = v^2/r = r omega^2

This is the centripetal acceleration. It changes the direction of the velocity but not its magnitude.

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4 Centripetal Force

By Newton's Second Law, the force required to maintain circular motion is:

F = mv^2/r = mr omega^2