Body Moving in a Circular Path at Constant Speed

The magnitude of the velocity of the body is constant but the direction is constantly changing.

At any instant, the direction of the velocity is a tangent to the circular path.

At time t = 0, the body is at A (an arbitrarily chosen point), and the magnitude of its velocity in the direction A-O is zero.

At time t = Dt, the body has moved to B.

When at B, the magnitude of the component of the velocity in the direction B-O’ (which is parallel to A-O) is given by

So, there has been an acceleration in the direction A-O (or B-O’).

The magnitude of the acceleration is a = v/t which is equal to

For small angles, the sine of the angle is very nearly equal to the angle in radians.

In order to find the instantaneous value of the acceleration, we consider smaller and smaller time intervals. As t 0, 0 and so the acceleration is given by

which gives

also, the line B-O’ approaches the line A-O, so the direction of the acceleration is towards the centre of the circular path.

Remembering that v = r this gives us two useful expressions for calculating the magnitude of this centripetal acceleration.

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