Question

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

• 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 $y^2+z^2=4$.

Answer 1

Answer: (A), (C), (D)

The correct options are
A The gravitational force due to this object at the origin 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 $y^2+z^2=4$.

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
$F=m_{0}E=m_0\times 0=0$
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 $x-$axis. Since the spherical mass distribution behaves as if the whole mass is at centre (for a point outside the sphere) and since the points on the circle are equilibrium from the centre of the sphere, the circle is a gravitational equipotential.