Question

A non- homogeneous sphere of radius R has the following density variation :-

Find the gravitational field at a distance 2R from the centre of the sphere.
Please explain also.

Answer 1

First we have to find out the mass.
The mass of the inner most shell of the sphere is V1=43π(R3)3=0.049πR3
So mass of the inner most shell is m1=0.049πR3×ρ0.

For 2nd shell the volume is V2=43π[(3R4)3−(R3)3]=0.513πR3
so the mass of that shell is m1=0.513πR3×12ρ0=0.256πR3ρ0

The volume of the third shell is V3=43π[R3−(3R4)3]=0.771πR3
so mass of the outer most shell is m3=0.771πR3×18ρ0=0.096πR3ρ0

So the total mass is given by m=m1+m2+m3=0.049πR3ρ0+0.256πR3ρ0+0.096πR3ρ0=0.4010πR3ρ0

We know for any uniform sphere or uniform spherical shell the gravitational field out side the sphere is always like the mass is concentrated at the centre and we are calculating the field at the distance from the centre. In this case we have devided the whole sphere in three uniform sphere or spherical shell with same centre. So now we can consider that the mass is at the centre and so the field at the 2R distance is given by
Gm(2R)2=G×0.4010πR3ρ04R2=0.1πRGρ0