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Elastic potential energy is energy stored in a flexible object due to deformation caused by an external force. When you stretch a bow or compress a spring, the object stores this energy. Upon release, the object returns to its original shape, converting the stored energy back into kinetic energy or work. This energy depends on both the object's stiffness and the amount of deformation.

Using Hooke's law, elastic potential energy is calculated as U = 1/2 kx², where k is the spring constant and x is the displacement from the unstretched position. This formula shows that elastic potential energy depends on the square of the displacement, meaning doubling the stretch quadruples the stored energy. The spring constant k measures the stiffness of the spring.

The work done by a spring force equals the negative change in elastic potential energy. When an external force stretches or compresses a spring, work is done on the system, increasing elastic potential energy. Conversely, when the spring returns to its original position, it does work on the surroundings, decreasing elastic potential energy. This relationship follows from work done on a system by external force principles.

A mass-spring system exhibits both gravitational and elastic potential energy simultaneously. When a body hangs from a spring, the total potential energy U equals the sum of gravitational potential energy and elastic potential energy. As the mass moves, energy converts between these two forms. The equilibrium position represents a balance where both energy types contribute to the system's total potential energy.

Elastic potential energy depends on displacement squared because the restoring force itself increases linearly with displacement according to Hooke's law. As you stretch a spring further, the force required grows proportionally. The work done against this increasing force accumulates quadratically, resulting in the 1/2 kx² relationship. This squared relationship means small increases in displacement produce large increases in stored energy.

An ideal spring is perfectly elastic, meaning it returns completely to its original shape after deformation with no energy loss. It follows Hooke's law exactly, where the restoring force is proportional to and opposite the imposed displacement. Ideal springs are massless and exhibit elastic potential energy that depends only on the spring constant and displacement squares, independent of real-world factors like friction or material fatigue.

Elastic potential energy is a component of mechanical energy in conservative systems. In a mass-spring system, total mechanical energy remains constant as elastic potential energy converts to kinetic energy and gravitational potential energy. This relationship demonstrates conservation of mechanical energy, where the sum of all energy forms stays constant when only conservative forces act. Understanding elastic potential energy is essential for applying energy conservation principles.