Centripetal Acceleration and Force

An object moving in a circle at constant speed is not in equilibrium. Its velocity is constantly changing direction, which means it is accelerating. This acceleration is always directed towards the centre of the circle and is called centripetal acceleration.

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Why There Must Be an Acceleration

Consider a satellite orbiting the Earth. At one instant it is moving east. A quarter of an orbit later, it is moving north. Its speed has not changed, but its velocity vector has rotated by 90°. A change in velocity — whether in magnitude or direction — means acceleration.

The direction of this acceleration can be found by considering the change in velocity vector, Δv. For a small time interval, the change in velocity points towards the centre of the circle. In the limit (infinitesimally small time interval), the acceleration is exactly directed towards the centre.

This is centripetal acceleration: always directed towards the centre of the circular path, perpendicular to the velocity at every point.

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Centripetal Acceleration Formulae

For an object moving in a circle of radius r at speed v with angular velocity ω:

$a = \frac{v^2}{r} = \omega^2 r = v\omega$a=rv2​=ω2r=vω