In Fig. . (c), ∆ t Ž and the average acceleration becomes the instantaneous acceleration. It is directed towards the centre * .
Thus, we find that the acceleration of an object in uniform circular motion is always directed towards the centre of the circle. Let us now find the magnitude of the acceleration. The magnitude of a is, by definition, given by Let the angle between position vectors r and r ′ be ∆ θ . Since the velocity vectors v and v ′ are always perpendicular to the position vectors, the angle between them is also ∆ θ .
Therefore, the triangle CPP ′ formed by the position vectors and the triangle GHI formed by the velocity vectors v , v ′ and ∆ v are similar (Fig. .18a). Therefore, the ratio of the base-length to side-length for one of the triangles is equal to that of the other triangle. That is : = v Therefore, R If ∆ t is small, ∆θ will also be small and then arc PP ′ can be approximately taken to be| ∆ r |: r ≅ v t ≅ Therefore, the centripetal acceleration a c is : Fig.
. Velocity and acceleration of an object in uniform circular motion. The time interval ∆ t decreases from (a) to (c) where it is zero. The acceleration is directed, at each point of the path, towards the centre of the circle.
* In the limit ∆ t Ž , ∆ r becomes perpendicular to r . In this limit ∆ v → and is consequently also perpendicular to V . Therefore, the acceleration is directed towards the centre, at each point of the circular path. a c = v = v /R ( .
) Thus, the acceleration of an object moving with speed v in a circle of radius R has a magnitude /R and is always directed towards the centre . This is why this acceleration is called centripetal acceleration (a term proposed by Newton). A thorough analysis of centripetal acceleration was first published in