We have shown that a when a spring is stretched or compressed a distance x, it will exert forces with the same magnitude, but opposite directions. And the force is proportional to the displacement. Let's imagine compressing a spring. In this case, the force will increase as the spring compresses. Thus the instantaneous work will change as we compress the spring. We can define the average force for a spring much as we defined an average velocity for a constant acceleration

_av = ( + _o) =

where we have used that ₀ = 0, because at the equilibrium position x = 0 there is no force acting.

So the work done in compressing or stretching a spring is just

W = _av• = • = - kx $\cdot$ x $\cdot$ cos(180°) = k x²

Please note: this work done against the force of the spring can be thought of as "storing" energy in the spring. That is, when the spring is compressed or extended it stores potential energy.

This derivation works out much neater with the aid of calculus:

W = •d = k x dx = k x²

but has, of course, the same result.

© MultiMedia Physics, 1999