14.10:
Potential Energy due to Gravitation
An object of mass m initially at a distance r1 from the Earth's center, is displaced to r2. Since the object moves in Earth's gravitational field, its potential energy changes from U1 to U2. The change in the potential energy is equal to the work done by the gravitational force to displace the object.
Here, the work done equals the negative integral of magnitudes of gravitational force and displacement. On substituting for gravitational force and integrating within the limits of r1 and r2, U2 minus U1 equals gravitational constant times the product of the two masses multiplied by one over r1 minus one over r2.
Considering the distance r2 as infinity, where the Earth's gravitational field is negligible, the potential energy U2 equals zero.
Therefore, the gravitational potential energy for any two masses separated by a distance r is expressed as minus G times the product of two masses, divided by r.
Typically, the potential energy increases as the masses move farther apart. Its maximum value is zero for masses at an infinite distance from each other.
14.10:
Potential Energy due to Gravitation
Since gravitational force is a conservative force, the amount of work done to move an object between two points in the gravitational field in which it resides is independent of the path taken. Thus, similar to the gravitational field, a gravitational potential energy function can be defined, which depends only on spatial coordinates.
Consider a mass gravitationally bound to another object. For example, the Earth is gravitationally bound to the Sun’s gravitational field. The potential energy of the Earth in the Sun’s gravitational field is defined such that its value is negative close to the Sun and increases to zero at large distances from the Sun.
Since the Earth and the Sun are not special cases, the result can be generalized for any two objects. Thus, under the influence of gravity, all masses fall from a higher to lower potential energy while their kinetic energies increase. Hence, the definition is consistent with the conservation of energy principle. If the total energy of a system is positive, it is not gravitationally bound.
The magnitude of the potential energy decreases with the distance between the two objects. It is inversely proportional to the distance because of the inverse-square dependence of the gravitational force on the distance.
This text is adapted from Openstax, University Physics Volume 1, Section 13.3: Gravitational Potential Energy and Total Energy.