How to Calculate Uniform Circular Motion

For a body going round a circle at constant speed, this finds the angular velocity, the period, the centripetal acceleration and the centripetal force. The speed may not change, but the direction does, so there is an acceleration pointing at the centre.

Going round a circle at a steady speed is uniform circular motion. The speed does not change, but the direction changes constantly.

a = \dfrac{v^2}{r}, \quad F = m\dfrac{v^2}{r}

$v$ is the speed, $r$ the radius, $a$ the acceleration towards the centre and $F$ the centripetal force. The angular velocity is $\omega = v \div r$ and the time for one lap is $T = 2\pi \div \omega$ .

Acceleration without a change of speed

Acceleration is a change of velocity, and velocity has a direction as well as a size. Going round a circle the direction changes all the time, so there is an acceleration even at constant speed. It always points at the centre of the circle.

Example

A 0.5 kg object goes round a circle of radius 2 m at 4 m/s. The angular velocity is $4 \div 2 = 2$ rad/s and the period is $2\pi \div 2 = 3.14$ seconds. The centripetal acceleration is $4^2 \div 2 = 8$ m/s² and the force is $0.5 \times 8 = 4$ N.

Not centrifugal force

The only force on the object points inwards. Whirling a weight on a string, the string pulls towards the centre. The sense of being flung outwards is an apparent force felt by whatever is going round, and from outside it is not there at all.

Notes

At the same radius, twice the speed needs four times the force. This is why taking a bend faster goes wrong so suddenly.