The radii of two planets $A$ and $B$ are $R$ and $4 R$ and their densities are $\rho$ and $\rho / 3$ respectively. The ratio of acceleration due to gravity at their surfaces $\left(g_A: g_B\right)$ will be

Imagine earth to be a solid sphece of mass $M$ and radius $R$. If the value of acceleration due to gravity at a depth d below earth's surface is same as its value at a height $h$ above its surface and equal to $\frac{g}{4}$ (where $g$ is the value of acceleration due to gravity on the surface of earth), the ratio of $\frac{h}{d}$ will be

At what distance from the centre of the earth, the value of acceleration due to gravity $g$ will be half that on the surface ($R =$ radius of earth)

A man of $50\, kg$ mass is standing in a gravity free space at a height of $10\, m$ above the floor. He throws a stone of $0.5\, kg$ mass downwards with a speed $2\, m/s$. When the stone reaches the floor, the distance of the man above the floor will be …… $m$

If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately …….. minutes

(Take : $g =10 \,ms ^{-2},$ the radius of earth, $R =6400 \times 10^{3}\, m ,$ Take $\left.\pi=3.14\right)$