Elastic Potential Energy

When a spring or elastic string is stretched or compressed, it stores elastic potential energy (EPE). This energy can be converted to kinetic energy and vice versa, providing a rich source of mechanics problems.

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1 Hooke's Law

T = (lambda / l) * x = kx

where:

• T is the tension in the spring/string
• lambda is the modulus of elasticity (a property of the spring)
• l is the natural length
• x is the extension (or compression)
• k = lambda/l is the stiffness (spring constant)

Hooke's Law holds for extensions up to the elastic limit.

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2 Elastic Potential Energy Formula

The work done in stretching a spring from natural length by extension x is:

EPE = (lambda x^2) / (2l) = (1/2) k x^2

This is the area under the force-extension graph (which is a straight line through the origin for a Hooke's Law spring).

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3 Derivation

The tension at extension s is T = (lambda/l) s.