Potential Energy of a Spring

Potential Energy of a Spring

First, let us obtain an expression for the potential energy stored in a spring (\({\text{PE}}_{s}\)). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force \(F\) on the spring and the resulting deformation \(\Delta L\) are proportional, \(F=k\Delta L\).) (See the figure below.) For our spring, we will replace \(\Delta L\) (the amount of deformation produced by a force \(F\)) by the distance \(x\) that the spring is stretched or compressed along its length.

So the force needed to stretch the spring has magnitude \(\text{F = kx}\), where \(k\) is the spring’s force constant. The force increases linearly from 0 at the start to \(\text{kx}\) in the fully stretched position. The average force is \(\text{kx}/2\). Thus the work done in stretching or compressing the spring is \({W}_{s}=\text{Fd}=\left(\cfrac{\text{kx}}{2}\right)x=\cfrac{1}{2}{\text{kx}}^{2}\). Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of \(F\) vs. \(x\) is the work done by the force. In figure (c) below(c) we see that this area is also \(\cfrac{1}{2}{\text{kx}}^{2}\). We therefore define the potential energy of a spring, \({\text{PE}}_{s}\), to be

where \(k\) is the spring’s force constant and \(x\) is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance \(x\). The potential energy of the spring \({\text{PE}}_{s}\) does not depend on the path taken; it depends only on the stretch or squeeze \(x\) in the final configuration.

(a) An undeformed spring has no \({\text{PE}}_{s}\) stored in it. (b) The force needed to stretch (or compress) the spring a distance \(x\) has a magnitude \(F=\text{kx}\) , and the work done to stretch (or compress) it is \(\cfrac{1}{2}{\text{kx}}^{2}\). Because the force is conservative, this work is stored as potential energy \(\left({\text{PE}}_{s}\right)\) in the spring, and it can be fully recovered. (c) A graph of \(F\) vs. \(x\) has a slope of \(k\), and the area under the graph is \(\cfrac{1}{2}{\text{kx}}^{2}\). Thus the work done or potential energy stored is \(\cfrac{1}{2}{\text{kx}}^{2}\).

The equation \({\text{PE}}_{s}=\cfrac{1}{2}{\text{kx}}^{2}\) has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy is energy due to position, shape, or configuration. For shape or position deformations, stored energy is \({\text{PE}}_{s}=\cfrac{1}{2}{\text{kx}}^{2}\), where \(k\) is the force constant of the particular system and \(x\) is its deformation. Another example is seen in the figure below for a guitar string.

Work is done to deform the guitar string, giving it potential energy. When released, the potential energy is converted to kinetic energy and back to potential as the string oscillates back and forth. A very small fraction is dissipated as sound energy, slowly removing energy from the string.