A body of mass m is moving in a circular orbit of radius R about a planet of mass M. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius `R/2`. And the other mass, in a circular orbit of radius `(3R)/(2)`. The difference between the final and initial total energies is :

Options

A. `+(Gm)/(6R)`
B. `-(GMm)/(2R)`
C. `-(GMm)/(6R)`
D. `(GMm)/(2R)`

Answer

C

Explanation

Total energy of body of mass m can be written as follows :
`T.E._(i) = -(GMm)/(2R)`
Total energy of bodies of equal masses of m/2 which are moving at radii R/2 and 3R/2.
`T.E._(f) = (-(GMm//2)/(2((R )/(2)))) + (-(GMm//2)/(2((3R)/(2)))) = -(GMm)/(2R) - (GMm)/(6R)`
`= -(4GMm)/(6R)`
`= -(2GMm)/(3R)`
`T.E._(f) - T.E._(i) = (-(2)/(3)(GMm)/(R )) - (-(1)/(2)(GMm)/(R ))`
`= -(GMm)/(6R)`
Hence option (c ) is correct.