For a particle in circular motion the centripetal acceleration is

The correct answer is: May be more or less than its tangential acceleration.

To understand why, let us analyze the two types of acceleration components that can act on a particle moving in a circular path of radius r:

1. Centripetal Acceleration (a_c):
This component is responsible for continuously changing the direction of the velocity vector to keep the particle in its circular path. It is always directed towards the center of rotation. Its magnitude is given by:
$a_c=\frac{v^2}{r}$
where v is the instantaneous speed of the particle and r is the radius of the circular path.

2. Tangential Acceleration (a_t):
This component is responsible for changing the magnitude of the velocity (i.e., the speed of the particle) and acts along the tangent to the circular path. Its magnitude is defined as:
$a_t=\frac{dv}{dt}$

Since the magnitude of centripetal acceleration depends on the speed of the particle, whereas the tangential acceleration depends on the rate at which the speed is changing, their relative values can vary:
• In uniform circular motion, the speed is constant. Therefore, the tangential acceleration is zero ($a_t=0$), while the centripetal acceleration is non-zero ($a_c>0$). In this case, $a_c>a_t$.
• If a particle starts moving from rest in a circular path, its initial speed is zero ($v=0$), which means the initial centripetal acceleration is zero ($a_c=0$). If it accelerates, its tangential acceleration is non-zero ($a_t>0$). In this case, $a_c<a_t$.
• As the speed increases, there will also be a moment when the magnitude of the centripetal acceleration becomes equal to the tangential acceleration ($a_c=a_t$).

Thus, the centripetal acceleration may be more, less, or equal to the tangential acceleration depending on the specific state of the particle's motion.