When a body of mass m is undergoing a circular motion of radius r around a body of mass M, - Physics - Gravitation

Question

When a body of mass m is undergoing a circular motion of
radius r around a body of mass M, the variation
of the force of gravitation (F) with distance (r)
is given as F = GMmrn
Then, the variation of the time period (T) with orbital
distance (R) for a circular orbit will be
(A) T∝R n (B)
T∝R –n (C)
T∝R (n + 1)/2 (D)
T∝R (n – 1)/2

Sanjay Upadhyay · Accepted Answer

Dear student
In the question the force is not clearly shown Since the force is
always inversely proportional to the distance 'r' then we will
assume the force as F=GMmrn
Now in the case of circular motion -
gravitational force = mass x centripetal acceleration ,
therefore
GMmrn=mv2ror, v2=GMr-(n-1)The time period of rotation T, of the circular path is given byT=2πrv=2πrGMr-(n-1)=2πGM r-(n-1)2-1 =2πGM r-(n+1)2Or, T α rn+12option C is correct
I hope it will help you
regards