Spring Potential Energy

Hooke's Law, $$F=\left(-\left(kx\right)\right)$$, describes force exerted by a spring being deformed. Here, $F$ is the restoring force, $x$ is the displacement from equilibrium or deformation, and $k$ is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement.

In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a spring is ${\text{PE}}_{\text{el}}=\frac{1}{2}{\mathrm{kx}}^2$. Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence,

where ${\text{PE}}_{\text{el}}$ is the elastic potential energy stored in any deformed system that obeys Hooke’s law and has a displacement $x$ from equilibrium and a force constant $k$.

It is possible to find the work done in deforming a system in order to find the energy stored. This work is performed by an applied force $F_{\text{app}}$. The applied force is exactly opposite to the restoring force (action-reaction), and so $F_{\text{app}}=\text{kx}$. Figure 3.6.1 shows a graph of the applied force versus deformation $x$ for a system that can be described by Hooke’s law. Work done on the system is force multiplied by distance, which equals the area under the curve or $(1/2){\mathrm{kx}}^2$(Method A in the figure). Another way to determine the work is to note that the force increases linearly from 0 to $\mathrm{kx}$, so that the average force is $(1/2)\mathrm{kx}$, the distance moved is $x$, and thus $W=F_{\text{app}}d=[(1/2)\text{kx}](x)=(1/2){\mathrm{kx}}^2$ (Method B in the figure).