If ar and at represents the magnitude of radial and tangential acceleration, the motion of the particle will be uniform circular motion if

The motion of a particle will be uniform circular motion if the tangential acceleration (at) is zero and the radial acceleration (ar) is not zero.

Here's why:

1. In uniform circular motion, the speed of the particle is constant, but its direction continuously changes towards the center of the circle. This change in direction indicates an acceleration.

2. This acceleration is the radial or centripetal acceleration (ar), which is always directed towards the center of the circle. It's responsible for changing the direction of the velocity vector.

3. The magnitude of the velocity (speed) remains constant in uniform circular motion. This means there's no acceleration in the direction of motion. In other words, the tangential acceleration (at) is zero.

So, for a particle to be in uniform circular motion, it must have a radial acceleration (ar) but no tangential acceleration (at).