From a solid sphere of mass M and radius R, a spherical portion of radius R/2 is removed, as shown in the figure Taking gravitational potential `V =0at r = oo,` the potential at centre of cavity thus formed is: (G = gravitational constant)

Options

A. `(-2GM)/(3R)`
B. `(-2GM)/( R)`
C. `(-GM)/(2R)`
D. `(-GM)/(R )`

Answer

D

Explanation

(d) Potential at Point P (centre of cavity ) before removing the spherical portion,

`V_(1)=(-GM)/(2R^(3))(3TR^(2)-((R)/(2))^(2))`
`=(-GM)/(2R^(3))(3R^(2)-(R^(2))/(4))=(-11GM)/(8R)`
Mass of spherical portion to be removed,
`M'=(MV')/(V)=(M(4pi)/(3)((R)/(2))^(3))/((4pi)/(3)R^(3))=(M)/(8)`
Potential at point P due to spherical portion to be removed
`V_(2)=(-3GM')/(2R')=(-G(M//8))/(2(T//2))=(-3GM)/(8R)`
`:.` Potential at the centre of cavity formed
`V_(P)=V_(1)-V_(2)=(-11GM)/(8R)-((-3GM)/(8R))=(-GM)/(R)`