7.4 Conservative forces and potential energy  (Page 2/10)

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

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 ${\text{PE}}_{\mathrm{s}}$ does not depend on the path taken; it depends only on the stretch or squeeze $x$ in the final configuration.