An artifical satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. Using dimensional analysis show that the period of the satellite.
`T=k/Rsqrt(r^3/g)`
where k is a dimensionless constant and g is acceleration due to gravity.

Explanation

Given that
`T^2propr^3`or`Tpropr^(3//2)` Also T is a function of g and R
Let `Tpropr^(3//2)g^aR^b` where a, b are the dimensions of g and R.
(or) `T=kr^(3//2)g^aR^bto(1)`
where k is the dimensionless constant of proportionality
Form equation (1)
`[M^0L^0T^-1]=L^(3//2)(LT^-2)^a(L)^b=M^0L^(a+b+3/2)T^(-2a)`
Applying the principle of homogeneity of dimensions, we get
`a+b+3/2=0to(2) therefore-2a=1impliesa=1/2`
From eq (1), `(-1)/2+b+3/2=0impliesb=-1`
Substituting the values of `a` and `b` in eq.(1), we get
`T=kr^(3//2)g^(-1//2)R^-1`
`T=k/Rsqrt(r^3/g)`
This is the required relation.