$A$ particle of mass $m$ starts from rest at a distance $R$ from the centre and along the axis of a fixed ring of radius $R$ and mass $M$. Its velocity at the centre of the ring is:

Four particles,each of mass $M$,move along a circle of radius $R$ under the action of their mutual gravitational attraction as shown in the figure. The speed of each particle is:

Assume that a tunnel is dug along a chord of the Earth,at a perpendicular distance $(R / 2)$ from the Earth's centre,where $R$ is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel,it will execute a simple harmonic motion with a time period:

$A$ 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:

$A$ particle of mass $m_0$ is projected from the mid-point of the line joining two fixed particles each of mass $m.$ If the distance of separation between the fixed particles is $l,$ the minimum velocity of projection of the particle so as to escape to infinity is equal to

The gravitational force acting on a particle,due to a solid sphere of uniform density and radius $R$,at a distance of $3 R$ from the centre of the sphere is $F_1$. $A$ spherical hole of radius $(R / 2)$ is now made in the sphere as shown in the figure. The sphere with hole now exerts a force $F_2$ on the same particle. The ratio of $F_1$ and $F_2$ is