Elastic Potential Energy

When you compress or stretch a spring from its natural length, you store elastic potential energy in it. This energy comes from the work you do against the spring force.

From Hooke's Law to Energy

Hooke's Law tells us the spring force varies with displacement:

where $k$ is the spring constant and $x$ is displacement from equilibrium. The spring force always points back toward equilibrium.

To compress or stretch the spring, you must apply an external force that balances the spring force at each position. Consider moving the spring slowly from equilibrium ($x = 0$) to displacement $x$. At each point, your applied force must match the spring force:

Since this force increases linearly with displacement, the work you do equals the area under the force-displacement graph—a triangle with base $x$ and height $kx$:

Work Changes the Potential Energy

The work you do compressing or stretching the spring does not increase the kinetic energy of the mass (we assume slow, controlled motion with negligible speed). Instead, this work changes the spring's stored energy. We define the change in elastic potential energy as equal to the work done:

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