Two planets have the same mass, M. Each planet has a constant density, but the density of planet 2 is twice as high as that of planet 1. Identical objects of mass m are placed on the surfaces of the planets. What is the relationship of the gravitational potential energy, `U_(1)` on planet 1 to `U_(2)` on planet 2?

Options

A. `U_(1) = 2U_(2)`
B. `U_(1) = 8U_(2)`
C. `U_(1) = 2^(1//3)U_(2)`
D. `U_(1) = (U_(2))/((2)^(1//3))`

Answer

D

Explanation

`M = (4 pi)/3 R^(3) rho implies R^(3) rho = ` constant `" "`(i)
`:. U = ("G Mm")/(r)`
`U prop 1/r " "......(ii)`
from (i) & (ii), `U prop rho^(1//3)`