The weight of a body at the surface of earth is $18\,N$. The weight of the body at an altitude of $3200\,km$ above the earth's surface is $........\,N$ (given, radius of earth $R _{ e }=6400\,km$ )

Assuming the earth to be a sphere of uniform density the acceleration due to gravity

The mass and diameter of a planet have twice the value of the corresponding parameters of earth. Acceleration due to gravity on the surface of the planet is ........ $m/{\sec ^2}$.

A hemisspherical shell of mass $2M$ and radius $6R$ and a point mass $M$ are performing circular motion due to their mutual gravitational interaetion Their positions are shown in figure at any moment of time during motion. If $r_1$ and $r_2$ are the radii of circular path of hemispherical shell and point mass respectively and ${\omega _1}$ and  ${\omega _2}$ are the angular speeds of hemi-spherical shell and point mass respectively, then choose the correct option

When a body is taken from the equator to the poles, its weight

A certain planet completes one rotation about its axis in time $T$. The weight of an object placed at the equator on the planet's surface is a fraction $f(f$ is close to unity) of its weight recorded at a latitude of $60^{\circ}$. The density of the planet (assumed to be a uniform perfect sphere) is given by