Question #01d18

Sep 26, 2017

I assume that "perfect circular motion" means "uniform circular motion"

Explanation:

We know that acceleration is rate of change of velocity with time. Also that velocity is a vector quantity.
Velocity can change in two ways: (1) change in magnitude (2) change in its direction.
Either change can occur due to some force acting on the object which produces acceleration. Perfect circular motion can be uniform or it can be non-uniform.
Uniform circular motion is at constant speed implies that magnitude of the velocity is constant, but direction is changing with time.

Lets us assume that acceleration makes an angle $\theta$ with the direction of motion. if we resolve the acceleration in two directions: perpendicular to and along the direction of motion we have

Component along the direction of motion$= \vec{a} \cos \theta$

The speed or magnitude of velocity is given to be constant

$\implies$ Acceleration along the direction of motion must be zero.

We have

$\vec{a} \cos \theta = 0$
$\implies \cos \theta = 0$
$\implies \theta = {90}^{\circ}$

This is possible only if acceleration is perpendicular to direction of motion.

Any component of acceleration along the direction of motion will change the magnitude of velocity.

We see that the velocity is always tangential to the circular path. From above figure we see that the acceleration is trying to change the direction of the velocity to maintain a circular path. That is force is pulling the object always towards the center of the circle.

For any other direction of acceleration we will have non-circular motion.