Suppose the gravitational potential due to a small system is k/r² at a distance r from it. What will be the gradational field? Can you think of any such system? What happens if there were negative masses?

A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let r be the distance of the body from the centre of the star and let its linear velocity be v, angular velocity ω, kinetic energy K, gravitational potential energy U, total energy E and angular momentum l. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.

Consider a planet moving in an elliptical orbit round the sun. The work done on the planet by the gravitational force of the sun

A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the Centre

A person brings a mass of 1 kg from the infinity to a point A. Initially the mass was at rest but it moves at a speed of 2ms^-1 as it reaches A. The work done by the person on the mass is -3 J. The potential at A is

If the acceleration due to gravity at the surface of the earth is g, the work done in slowly lifting a body of mass m from the earth’s surface to a height R equal to the radius of the earth is

The kinetic energy needed to project a body of mass m from the earth’s surface to infinity is

Show the nature of the following graph for a satellite orbiting the earth.

(a) KE vs orbital radius R

(b) PE vs orbital radius R

(c) TE vs orbital radius R.

Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the midpoint of the line joining the centers of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable?

Two stars each of one solar mass (= 2 × 10³⁰ kg) are approaching each other for a head-on collision. When they are a distance 10⁹ km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 10⁴ km. Assume the stars to remain undistorted until they collide. (Use the known value of G).

A small block of super dense material has a mass of 3 × 10²⁴ kg. It is situated at a height h (much smaller than the earth’s radius) from where it falls on the earth’s surface. Find its speed when its height from the earth’s surface has reduced to h/2. The mass of the earth is 6 × 10²⁴ kg.