What is the potential energy of a body of mass `m` relative of the surface of Earth of radius `R`, at a (a) height `h = R` above its surface (b) depth `d = R` below its surface.

Explanation

(a) Pot. Energy of the body at height `h (= R)` is equal to wirk done in taking a body from surafce of earth to height `R` is `= PE` of body at height `R - PE` of body on the surface of earth
`= - (GM m)/(2R) + (GM m)/(R ) = (GM m)/(2R) = (1)/(2) (g R^(2))/(R ) m`
`= (1)/(2) mg R`
(b) When body is at distance `x` from the centre of Earth where `x lt R`, then gravitational force on the body is,
`F = (G((4)/(3) pi x^(3) rho) m)/(x^(2)) = (4)/(3) pi G m rho x`
`P.E.` of body at depth `d (= R)` is equal to work done in taking the body from surface of Earth to centre of Earth. i.e.,
`P.E. = int_(R)^(0) (4)/(3) pi Gm rho x dx = (4)/(3) pi Gm rho ((x^(2))/(2))_(R)^(0)`
`= - (4)/(3) pi G rho m(R^(2))/(2) = - (1)/(2) mg R`
`[ :'g = (GM)/(R^(2)) = (G)/(R^(2)) xx (4)/(3) pi R^(3) rho = (4)/(3) pi G rho R]`