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From a solid sphere of mass M and radius...

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 (G = gravitational constant)

A

`-(2GM)/(3R)`

B

`-(GM)/(R)`

C

`-(GM)/(2R)`

D

`(-2GM)/(R)`

Text Solution

Verified by Experts

The correct Answer is:
B
`M` is the mass of sphere, lat `rho` be the density of
sphere. So `M = V xx rho = (4)/(3) piR^(3) rho`
Similary if `M'` is the mass of cavity
`M' = (4)/(3) pi ((R )/(2))^(3) rho`
or `M' = (1)/(8)((4)/(3) pi R^(3) rho) = (M)/(8)`
Gravitational potential at a point lying inside the sphere at disatnce `r` from centre is
`V = -(GM)/(2R^(3)) (3R^(2) - r^(2))`
`:. V_(P) = - (GM)/(2R^(3)) [3R^(2) -((R )/(2))^(2)]`
`= - (Gm)/(2R^(3)) [(11R^(2))/(4)] = - (11Gm)/(8R)`
Gravitational potential due to cavity portion of sphere at its centre is `V_(P')`
`= - (3GM')/(2r) = (3)/(2) (G(M//8))/(R//2) = - (3GM)/(8R)`
Net gravitational pot. at `P`
`= - (11 GM)/(8R) - (-(3GM)/(8R)) = - (GM)/(R )`
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