## Elastic Potential Energy

How much work is done when we stretch a spring a distance x from its equilibrium position?

First, we need to know about the general characteristics of a spring.

Experimentally, we find

This is known as Hooke's law

We might write this in equation form as F = k x. However, the force exerted

by the springis always in theopposite directionto the stretch (or compression) of the spring. Therefore, we write Hooke's law asF = - k xThis is the force exerted

by the spring. Theexternal forcewe exertonthe spring isF._{ext}= + k xThe proportionality constant k is known as the

spring constantand describes how stiff the spring is.Now we are again ready to ask

"How much work is done when we stretch a spring a distance x from its equilibrium position?"

We know how to handle a constant force. For a constant force, we know

W = F dOur spring force varies, but we can think of it as being (nearly) constant as we move through a (very) small distance,

The work done by a variable force is the

areaunder the "curve" on a Force - distance graph. For this Hooke's law force ofF, the work done_{ext}= k xtothe spring by the external force F_{ext}isW = (

^{1}/_{2}) k x^{2}This work done on the spring as it is stretched (or compressed) can be recovered. This is

stored workthat can be used to do workonsomething elsebythis spring. That means the stretched (or compressed) spring hasenergy--potential energy. This isspring potential energyorelastic potential energy.PE _{el}= (^{1}/_{2}) k x^{2}

^{or}U _{el}= (^{1}/_{2}) k x^{2}

Gravitational Potential EnergyEnergy ConservationReturn to ToC, Work and Energy(c) 2002, Doug Davis; all rights reserved