A solid sphere of uniform density and radius 4 units is located with its centre at the origin O of coordinates. Two sphere 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 if fig Then:

Options

A. the gravitational force due to this object at the origin is zero
B. the gravitational force at point B (2, 0, 0) is zero
C. the gravitational potential is the same at all points of circle `y^2+z^2=36`
D. the gravitational potential is the same at all points on the circle `y^2+z^2=4`

Answer

A, C, D

Explanation

The gravitational field intensity at point O is zero (as the cavities are symmetrical with respect to O).
Now, `y^2 + z^2 = 36` represents the equation of a circle with centre (0, 0, 0) and radius 6 units. The plane of the circle is perpendicular to the x-axis. Since the spherical mass distribution behaves as if the whole mass is at its centre (for a point outside sphere) and since all the points on the circle are equidistant from the centre of the sphere, the circle is a gravitational equipotential.