Elastic potential energy

Encyclopedia : E : EL : ELA : Elastic potential energy

The elastic potential energy stored in an elastic string or spring of natural length, l, and modulus of elasticity λ under an extension of x is given by:

[E = \frac]

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

[U_e = \int \, dx = \frac k x^2]

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)

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, [\Delta l]:

[U_e = \int }\, dl = \frac ^2} ]

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

[\frac = \frac ^2} = \frac Y ^2]

where [\varepsilon = \frac ] is the strain in the material.

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