The density of a solid sphere of radius R is `P(r)=20(r^(2))/R^(2)` where, r is the distence from its centre. If the gravitational field due to this sphere at a distence 4 R from its centre is E and G is the gravitational constant, them the ratio of `(E)/(GR)` is

Options

A. `(pi)/(5)`
B. `3pi`
C. `(3pi)/(2)`
D. `pi`

Answer

D

Explanation

Volume of a spherical shell of thickness dr and of radius r is `dV=4pir^(2)dr`

So, mass of this shell is
`dm=pdV=(20r^(2))/(R^(2)). 4pi^(2).dr`
Total mass of complete solid sphere is
`M= int_(0)^(R)dM = int _(0)^(R)(80pir^(4))/(R^(2))dr`
`rArr M = (80piR^(5))/(5R^(2))=16piR^(3)`
Due to this mass M. gravitational field intensity at a distence 4R from its. centre is
`E = (GM)/((4R)^(2))`
`rArr E = (G.16piR^(3))/((4)^(2)R^(2))`
`rArr E = GpiRrArr(E)/(GR)=pi`