A cavity of radius `R//2` is made inside a solid sphere of radius `R`. The centre of the cavity is located at a distance `R//2` from the centre of the sphere. The gravitational force on a particle of a mass `'m'` at a distance `R//2` from the centre of the sphere on the line joining both the centres of sphere and cavity is (opposite to the centre of cavity). [Here `g=GM//R^(2)`, where `M` is the mass of the solide sphere]

Options

A. `(mg)/(2)`
B. `(3mg)/(8)`
C. `(mg)/(16)`
D. None of these

Answer

B

Explanation

Gravitational field at mass m due to full solid sphere
`vec(E_(1)) = (rho vec(r))/(3 epsilon_(0)) = (rho R)/(6 epsilon_(0)) ....... [ epsilon_(0) = (1)/(4pi G) ] `
Gravitational field at mass m due to cavity `(-rho)`
`vec(E_(2)) = ((-rho) (R//2)^(3)vec(r))/(3 epsilon_(0) R^(2)) ........ [ "using " E = (rho a^(3))/(3 epsilon_(0)r^(2)) ] `
`= - ((-rho)R^(3))/(24 epsilon_(0) R^(2)) = (-rho R)/(24 epsilon_(0))`
Net gravitational field
`vec(E) = vec(E_(1)) + vec(E_(2)) = (rho R)/(6 epsilon_(0)) - (rho R)/(24 epsilon_(0))`
= `(rho R)/(8 epsilon_(0))`
Net force on m `rarr F = m vec(E) = (m rho R)/(8 epsilon_(0))`
Here, `rho = (M)/((4//3)pi R^(3)) & epsilon_(0) = (1)/(4 pi G) `then F = `(3mg)/(8)`
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