Three point masses,each of mass $m$,are kept at the corners of an equilateral triangle of side $L$. The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to

$A$ satellite of mass $5M$ orbits the Earth in a circular orbit. At one point in its orbit,the satellite explodes into two pieces,one of mass $M$ and the other of mass $4M$. After the explosion,the mass $M$ ends up travelling in the same circular orbit,but in the opposite direction. After the explosion,the mass $4M$ is in:

$A$ spherically symmetric gravitational system of particles has a mass density $\rho = \begin{cases} \rho_0 & \text{for } r \leq R \\ 0 & \text{for } r > R \end{cases}$ where $\rho_0$ is a constant. $A$ test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed $V$ as a function of distance $r$ $(0 < r < \infty)$ from the centre of the system is represented by:

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 $AG = BG = CG = 1 \; m$.

Select the correct choice$(s)$:

$A$ 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