Question

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 = 0 at r = `oo`, the potential at the centre of the cavity thus formed is:

(G = gravitational constant)

Options

• `(-GM)/(2R)`

• `(-GM)/(R)`

• `(-2GM)/(3R)`

• `(-2GM)/(R)`

Answer 1

`(-GM)/(R)`

Explanation:

Potential as a result of a solid sphere at point O,

V₁ = `(-GM)/(2R) [3 - (r/R)^2]`

Here, r = `R/2`

∴ V₁ = `(-GM)/(2R^3) [3R^2 - (R/2)^2]`

= `(-GM)/(2R^3) [3R^2 - (R^2/4)]`

= `(-GM)/(2R^3) xx (11R^2)/4`

= `(-11GM)/(8R)`

Spherical portion of `R/2` is removed, hence mass will be reduced by `1^(th)/8` of original mass.

Potential as a result of the cavity section at point O,

V₂ = `(-3)/2 (G(M/8))/(R/2) = (-3GM)/(8R)`

∴ Potential on point O because of the remaining component,

V = V₁ – V₂ = `(-11GM)/(8R) - ((-3GM)/(8R)) = (-GM)/R`