A solid sphere of uniform density and radius 4 units is located with its centre at the origin O of coordinates. Two spheres of equal radii 1 unit with their centres at A(– 2, 0, 0) and B(2, 0, 0) respectively are taken out of the solid leaving behind spherical cavities as shown in figure

Difficult

A

The gravitational force due to this object at the origin is zero
B

The gravitational force at the point B (2, 0, 0) is zero
C

The gravitational potential is the same at all points of the circle $y^{2} + z^{2} = 36$
D

The gravitational potential is the same at all points on the circle $y^{2} + z^{2} = 4$

Solution

Since cavities are symmetrical w.r.t. O. So the gravitational force at the centre is zero.
The radius of the circle $z^{2} + y^{2} = 36$ is 6.
For all points for $r \geq 6,$ the body behaves as if whole of the mass is concentrated at the centre. So the gravitational potential is same.
Above is true for $z^{2} + y^{2} = 4$ as well.

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