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)q=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 the force acting on a test mass `m_(0)` placed at `O` is given by
`F=m_(0)E=m_(0)xx0=0`
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 axis. Since the spherical mass distribution behaves as if the whole mass is at its centre (for a point outside the sphere) and since all the points on the circle are equidistant from the centre of the sphere, the circle is a gravitational equipotential.