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Potential energy is the ability of a body to do work due to its position relative to a reference point or datum. It equals the negative of the work done by a conservative force, such as gravity or an elastic restoring force, in bringing the body to that position. When a body is lifted, gravitational force does negative work, so potential energy increases. The relationship between potential energy and work is fundamental to understanding energy storage in mechanical systems.

Gravitational potential energy depends on a body's vertical distance from a reference datum where potential energy is zero. When a body moves upward, gravitational force does negative work, increasing potential energy. When moving downward, gravitational force does positive work, decreasing potential energy. The potential energy is expressed as the product of the body's weight and its height from the datum plane, becoming negative below the datum.

Elastic potential energy is the negative of the work done by the restoring force in a deformed elastic body. When a spring is compressed or stretched from its undeformed position, the applied force does positive work on the spring, increasing its elastic potential energy. This stored energy becomes available to do work on other bodies when the spring is released, making it essential for understanding energy storage in elastic systems.

The datum plane for zero potential energy is arbitrary because it does not affect the change in potential energy between two positions. Only differences in potential energy matter for calculating work and energy transfer. Whether you set the datum at ground level, a table surface, or any other reference point, the relative energy changes remain the same, making the absolute reference point a matter of convenience.

The elastic potential energy of a spring is equal to the total work done on the spring during compression or extension. For a compression distance x, the elastic potential energy is proportional to the square of the displacement, following the relationship between applied force and displacement. This energy represents the capacity of the spring to do work when it returns to its undeformed position.

When both gravitational and elastic forces act on a body, the total potential energy is the sum of gravitational potential energy and elastic potential energy. Each component is calculated independently based on the body's position relative to the datum and the deformation of elastic elements. This additive relationship allows engineers to analyze complex systems with multiple energy storage mechanisms.

Potential energy is fundamental to determining equilibrium positions and stability. Bodies naturally move toward configurations that minimize total potential energy. Understanding how potential energy changes with position helps identify stable and unstable equilibrium states. This relationship is essential for analyzing the potential energy criterion for equilibrium in mechanical systems and predicting system behavior under small perturbations.