Consider a spring initially at its un-stretched length. To stretch the spring to an extension of *x*, we have to apply an external force to overcome the spring force.

If we choose to imagine the external force *F*_{ext} to be exactly equal to the spring force *F*_{spring}, then the elastic potential energy stored in the spring is simply equal to the amount of work done by *F*_{ext}.

If the spring obeys Hooke’s Law, then *F*_{spring} will increase according to the formula , where *k* is the force constant of the spring (aka spring constant). To match the spring force at every point of the stretch, *F*_{ext} that must also increase according to .

The work done by the external force is thus given by a triangular area under the *F-x* graph. The elastic potential energy of a spring (EPE) is thus given by the formula

Unlike GPE which increases linearly with height, EPE increases quadratically with extension.

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**Video Explanation**

Derivation of EPE Formula

**Concept Test**

0810

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