Gravitation PYQs - Last 5 Years

A mass $M$ is broken into two parts of masses $m_1$ and $m_2$. How are $m_1$ and $m_2$ related, so that force of gravitational attraction between the two parts is maximum?

A body is projected vertically upwards from the surface of Earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be

A body which is initially at rest at a height $ R $ above the surface of the Earth of radius $ R $, falls freely towards the Earth, then its velocity on reaching the surface of the Earth is

A man of mass $m$ starts falling towards a planet of mass \(M\) and radius \(R\). As he reaches near to the surface, he realizes that will pass through a small role in the planet. As he enters the role, we sees that the planet is really mad...

What will be the acceleration due to gravity at a depth d, where g is acceleration due to gravity on the surface of earth?

If the earth stops rotating about its axis, then what will be the change in the value of g at a place in the equatorial plane? (Radius of earth = 6400 km)

From a solid sphere of mass M and radius R, a spherical portion of radius \({R \over 2}\) is removed as shown in the figure.

Taking gravitational potential V = 0 at r = \(\infty\), the potential at the centre of the cavity thus formed is