The period `T_0` for a planet of mass 'm' above the sum in a circular orbit of radius R depends on m,R and G where G is the gravitational constant. Find expression for time period by dimensional methods.

Explanation

`T_0 alpha m^a R^b g^c , m to M , R to L , G to M^(-1) L^3 T^(-2) , T_0 to T therefore T^1 = K(M)^a (L)^b (M^(-1)L^3 T^(-2))^c`
comparing power of M , L and T on either sides
a-c =0 …..(1)
b-3c = 0 …..(2)
2c=1 `implies c = - 1/2`
`therefore b= 3/2 and a = - 1/2 therefore T_0 = K m^(-1/2) R^(3/2) G^(-1/2) , T_0 = K sqrt((R^3)/(GM))`