What is the gravitational force exerted by the sphere of mass $M$ and radius $a$ on a ring of mass $m$ and radius $a$,where the distance between the center of the ring and the center of the sphere is $r = 1.73a = \sqrt{3}a$?

Three equal masses of $m \; kg$ each are fixed at the vertices of an equilateral triangle $ABC.$
$(a)$ What is the force acting on a mass $2 \; m$ placed at the centroid $G$ of the triangle?
$(b)$ What is the force if the mass at the vertex $A$ is doubled?
Take $AG = BG = CG = 1 \; m$.

$A$ large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ (see figure). The point masses are connected by a rigid massless rod of length $\ell$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $M$ is at a distance $r = 3\ell$ from $M$,the tension in the rod is zero for $m = k\left(\frac{M}{288}\right)$. The value of $k$ is

$A$ particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a.$ The magnitude of the gravitational potential at a point situated at $a/2$ distance from the centre,will be

The solar constant on the surface of the earth is $S$. What will be its value on the surface of another planet which is about $5.3 \, AU$ away from the sun?

If a tunnel is cut at any orientation through the Earth,a ball released from one end will reach the other end in time ........ $\text{min}$ (neglect Earth's rotation).