Potential Energy due to Gravitation

Gravitational potential energy for two masses separated by distance r is expressed as negative G times the product of the two masses divided by r, where G is the gravitational constant. This definition sets potential energy to zero at infinite distance, making it negative at finite separations. The potential energy increases as masses move farther apart, reaching its maximum value of zero at infinite distance.

Gravitational potential energy increases with distance because gravitational force is conservative, and work done against this attractive force increases potential energy. As objects separate, less gravitational attraction exists between them, requiring work input to maintain separation. This behavior is consistent with the conservation of energy principle, where kinetic energy decreases as potential energy increases.

The change in gravitational potential energy equals the negative work done by gravitational force during displacement. When an object moves from distance r1 to r2, the potential energy change is calculated by integrating the gravitational force over that displacement. This inverse relationship means stronger gravitational forces produce larger changes in potential energy over the same distance.

Gravitational potential energy is inversely proportional to distance because gravitational force depends on the inverse square of distance. Since potential energy is derived from integrating this force, it decreases with distance following an inverse relationship. The magnitude of potential energy therefore decreases as objects move apart, reflecting the weakening gravitational interaction.

A system is gravitationally bound when its total energy is negative, meaning the objects cannot escape each other's gravitational influence. Under gravity's influence, all masses fall from higher to lower potential energy while kinetic energies increase. If total energy is positive, the system is not gravitationally bound and objects can separate indefinitely.

Gravitational force is a conservative force, so the work done to move an object between two points depends only on initial and final positions, not the path taken. This property allows a gravitational potential energy function to be defined based solely on spatial coordinates. The path-independence makes gravitational potential energy a reliable state function for energy calculations.

Setting potential energy to zero at infinite distance establishes a reference point where Earth's gravitational field is negligible. This convention makes potential energy negative at finite distances, reflecting that work must be done to separate gravitationally bound objects to infinity. This reference choice simplifies calculations and aligns with the physical understanding that bound systems have negative total energy.