Derive expression for the gravitational potential at a point in the gravitational field of the earth. How is gravitational P.E. related to gravitational potential ?

The gravitational potential at a point due to the earth is determined by the work done in bringing a body of mass m from infinity to a point at a distance r from the center of the earth. The work done in this process represents the gravitational potential energy of the body at that point.

At the surface of the earth, where the distance from the center is equal to the earth’s radius, the gravitational potential reflects the work done to bring the body from infinity to the surface.

The relationship between gravitational potential energy and gravitational potential is straightforward: gravitational potential energy is the product of gravitational potential and the mass of the body. This highlights how gravitational potential describes the energy per unit mass in the gravitational field.