$A$ satellite is orbiting the Earth at a height $h$ above the surface (where $R$ is the radius of the Earth and $h \ll R$). What is the minimum increase in the orbital velocity required for the satellite to escape the Earth's gravitational field? (Ignore the effect of the atmosphere.)

$A$ body situated on the surface of the earth becomes weightless at the equator when the rotational kinetic energy of the earth reaches a critical value '$K$'. The value of '$K$' is given by [$g$ = gravitational acceleration on earth's surface,$M$ = mass of the earth,and $R$ = radius of the earth].

Two particles of equal mass $m$ each move in a circle of radius $r$ under the action of their mutual gravitational attraction. The time period of each particle is

$A$ uniform sphere $A$ with radius $R$ exerts a force $F$ on a small particle $B$ situated at a distance $2R$ 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 find the correct relation between $F$ and $F^{\prime}$.

$A$ star $2.5$ times the mass of the sun and collapsed to a size of $12 \;km$ rotates with a speed of $1.2 \;rev/s$. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (Mass of the sun $= 2 \times 10^{30} \;kg$)

$A$ very long (length $L$) cylindrical galaxy is made of uniformly distributed mass and has radius $R$ $(R << L)$. $A$ star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of the star is $T$ and its distance from the galaxy's axis is $r$,then: