Question

how is body in uniform circular motion  is related to its time period 

Answer 1

It is because it moves at a uniform speed at the same period of time.

Answer 2

A body in uniform circular motion has a uniform velocity of motion,i.e;equal distance is covered in equal amounts of time .
A particle under UCM covers a constant distance in completing circular trajectory in one revolution, which is equal to the perimeter of the circle.
s=2πr
Further the particle covers the perimeter with constant speed. It means that the particle travels the circular trajectory in a constant time given by its time period as :
T=2πr/v.
Hope i helped u:)