7.2 Elastic energy

Potential energy and conservative forces

Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force    is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy     $(\text{PE})$ for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative . That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

Potential energy of a spring

First, let us obtain an expression for the potential energy stored in a spring ( ${\text{PE}}_{\mathrm{s}}$ ). 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 $\mathrm{\Delta }L$ are proportional, $F=k\mathrm{\Delta }L$ .) (See [link] .) For our spring, we will replace $\mathrm{\Delta }L$ (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 $W_{\mathrm{s}}=\text{Fd}=\left(\frac{\text{kx}}{2}\right)x=\frac{1}{2}{\text{kx}}^2$ . 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 [link] (c) we see that this area is also $\frac{1}{2}{\text{kx}}^2$ . We therefore define the potential energy of a spring    , ${\text{PE}}_{\mathrm{s}}$ , to be