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The elastic potential energy is the potential energy of a string or spring that has elasticity. For a spring of natural length, l, and modulus of elasticity λ under an extension of x it can be calculated using the formula:

<math>E = \frac{\lambda x^2}{2l}</math>

This formula is obtained from the integral of Hooke's law:

<math>U_e = \int {k x}\, dx = \frac {1} {2} k x^2</math>

The equation is often used in calculations of positions of mechanical equilibrium.

Elastic Potential Energy is the kind of energy that is stored in a bow, or in a catapult, or in a spring.

The energy stored = the work done to stretch the bow, so:

Elastic Energy (joules) = Average Force (newtons) x Distance (metres)

[edit] Elastic Potential Energy in a Material

For a material of Young's modulus, Y (same as modulus of elasticity λ), cross sectional area, A₀, initial length, l₀, which is stretched by a length, <math>\Delta l</math>:

<math>U_e = \int {\frac{Y A_0 \Delta l} {l_0}}\, dl = \frac {Y A_0 {\Delta l}^2} {2 l_0}</math>

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

<math>\frac{U_e} {A_0 l_0} = \frac {Y {\Delta l}^2} {2 l_0^2} = \frac {1} {2} Y {\varepsilon}^2</math>

where <math>\varepsilon = \frac {\Delta l} {l_0}</math> is the strain in the material.