The masses and the radii of earth and moon are $M_1, R_1$ and $M_2, R_2$ respectively. Their centres are at distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres to escape to infinity is

A uniform ring of mass $m$ and radius $r$ is placed directly above a uniform hollow sphere of mass $M$ and of radius $R$. The centre of the ring is directly above the centre of sphere at a distance $\sqrt{3}\mathrm{r}$ as shown in the figure. The gravitational force exerted by the sphere on the ring will be:

A solid sphere of uniform density and mass $M$ has radius $4 \mathrm{m}.$ Its centre is at the origin of the coordinate system. Two spheres of radii $1 \mathrm{m}$ are taken out so that their centres are at $P(0,-2,0)$ and $Q(0,2,0),$ respectively. This leaves two spherical cavities. What is the gravitational field at the origin of the coordinate axes?

A solid sphere of uniform density and mass $M$ has radius $4 \mathrm{m}.$ Its centre is at the origin of the coordinate system. Two spheres of radii $1 \mathrm{m}$ are taken out so that their centres are at $P(0,-2,0)$ and $Q(0,2,0),$ respectively. This leaves two spherical cavities.