A test particle is moving in a circular orbit in the gravitational field produced by a mass density `rho(r)=K/r^2`. Identify the correct relation between the radius R of the particle’s orbit and its period T:

Options

A. TR is a constant
B. `T//R^(2)` is a constant
C. `T//R` is a constant
D. `T^(2)//R^(3)` is a constant.

Answer

C

Explanation

`int_(0)^(R)(G/(R^(2))K/(r^(2))4 pir^(2)dr)m=m((2pi)/T)^(2)xxR`
`(GK4pi)/(R^(2))xxRxxm=m((2pi)/(T^(2)))xxR`
`(T^(2))/(R^(2))=` constant
`T/R=` constant