A uniform sphere $A$ with radius $R$ exerts a force $F$ on a small particle $B$ situated at a distance $2 R$ from the centre of the sphere. A spherical portion of diameter $R$ is cut from the sphere $A$ as shown in the figure. If $F^{\prime}$ is the new gravitational force between the remaining part of the sphere $A$ and the particle $B$, then the correct relation between $F$ and $F^{\prime}$

A rocket is fired vertically with a speed of $4 \mathrm{~km} / \mathrm{s}$ from the earth's surface. How far from the earth does the rocket go before returning to the earth?

(Take, radius of earth $=6.4 \times 10^6 \mathrm{~m}$ and $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Three particles, each of mass $M$, situated at the vertices of an equilateral triangle of side length $l$. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original separation $l$. The initial speed that should be given to each particle is

A mass $M$ is split into two parts $m_0$ and $M-m_0$. These two masses are then separated by a distance $D$. If the gravitational force between the parts is maximum, then the ratio $\frac{m_0}{M}$ is