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 solid 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
`vecE_(1)=(rhovecr)/(3epsilon_(0))=(rhoR)/(6epsilon_(0))............[epsilon_(0)=1/(4piG)]`
Gravitational field at mass `m` due to cavity `(-rho)`

`vecE_(2)=((-rho)(R//2)^(3))/(3epsilon_(0)R^(2))......[(rhoa^(3))/(3epsilon_(0)r^(2))]`
`=-((-rho)R^(3))/(24epsilon_(0)R^(2))=-(-rhoR)/(24epsilon_(0))`
Net gravitational field
`vecE=vecE_(1)+vecE_(2)=(rhoR)/(6epsilon_(0))-(rhoR)/(24epsilon_(0))=(rhoR)/(8epsilon_(0))`
Net force on `'m'toF=mvecE=(mrhoR)/(8epsilon_(0))`
Here, `rho=M/(4//3piR^(3)) & epsilon_(0)=1/(4piG)`
Then `F=(3mg)/8`