The period of revolution of a satellite depends upon the radius of the orbit, the mass of the planet and the gravitational constant. Prove that the square of the period varies as the cube of the radius.

Explanation

The period of revolution of t of the satellite depends upon the radius of the orbit r the mass m of the planet and the grativational constant G
`t prop r^(x)m^(y)G^(z)`
Substituting the dimensional formula of t,r,m and G
`T=L^(x)M^(y)(M^(-1)L^(3)T^(-2)]^(z)`
`M^(0)L^(0)T=L^(x+3z)M^(y-z)T^(-2z)`
Using the principle of homogeneity of dimensions
`x+3z=0, y-z=0,y-z=0, -2z=1`
Solving `z=-1/2,y=-1/2,x=3/2`
`t prop r^(3//2)m^(-1//2)G^(-1//2)`
m and G are constants
`t^(2)propr^(3)`
i.e. the square of the period varies as the cube of the radius