Find the magnitude and direction of centripetal acceleration in case of uniform circular motion

Hi,

Please find below the solution to the asked query:

Uniform circular motion is the motion of a body in a circle at a constant speed. As the body moves in circle therefore direction of a body changes at a constant rate, As, a result it, body experience a uniform acceleration.


In the diagram shown, direction of velocity of the body at two instants of time is shown. By vector addition, we find that change in velocity, Δv , points toward the center of the circle. As acceleration is the ratio of the change in velocity over time. Therefore, acceleration also points towards the centre of the circle. Further, as the body moves in the circle. Therefore, ratio of the change in velocity to velocity is same as the ratio of the change in position to position. We can write.
$$\begin{gathered} \frac{∆v}{v}=\frac{∆r}{r}=\frac{v∆t}{r} \\ \Rightarrow \frac{∆v}{∆t}=\frac{v^2}{r} \\ \Rightarrow \alpha =\frac{v^2}{r} \end{gathered}$$
This is the magnitude of the centripetal acceleration.