A mass m on the surface of the Earth is shifted to a target equal to the radius of the Earth. If R is the radius and M is the mass of the Earth, then work done in this process is

Options

A. `(mgR)/2`
B. mgR
C. 2mgR
D. `(mgR)/4`

Answer

A

Explanation

Gravitational potential energy of the mass m on the surface of the Earth is
`U_i=-(GMm)/R`
and that at a target at a height h (=R) form the surface of the Earth is
`U_f=-(GMm)/(R+h)=-(GMm)/(R+h)=-(GMm)/(2R)`
`:.` Work done in shifting the mass from the surface of the Earth to the target is
`W=U_f-U_i=-(GMm)/(2R)-(-(GMm)/R)`
`=-(GMm)/(2R)+(GMm)/R=(GMm)/R(1-1/2)`
`=-(GMm)/(2R)=(mgR)/2" "("as"g = (GM)/R^2)`