7.3 Elastic Potential Energy

Elastic potential energy is the energy stored in an elastic material when it is temporarily deformed by an external force such as stretching, compressing, or twisting. This stored energy is equal to the work done to deform the object. When the external force is removed, the energy is released, allowing the object to return to its original shape, provided it has not been deformed beyond its elastic limit.

Work Done in Deforming a Material

To understand the energy stored in a material, consider a metallic wire being stretched by a pulling force $F$. As the force is applied, the wire's length increases by an amount $x$ (the extension). The work done by the applied force is stored in the wire as elastic potential energy.

The Force-Extension Graph

For an elastic material that obeys Hooke's Law, the force required to stretch it is directly proportional to its extension ($F\propto x$). A plot of force versus extension results in a straight line passing through the origin.

The work done in stretching the wire is represented by the area under the force-extension graph.

• Since the force increases linearly from 0 to a final value $F$, the average force applied is: $F_{avg}=\frac{0+F}{2}=\frac{1}{2}F$

• The work done ($W$) is the average force multiplied by the extension ($x$): $W=F_{avg}\times x=\frac{1}{2}Fx$

This work done is stored as elastic potential energy in the wire.

Formulas for Elastic Potential Energy

The elastic potential energy ($P.E{.}_{elastic}$) can be expressed in several ways:

a) In terms of Force and Extension:

As derived above, the stored energy is equal to the work done: $P.E{.}_{elastic}=\frac{1}{2}Fx$

b) In terms of the Spring Constant ($k$):

According to Hooke's Law, $F=kx$, where $k$ is the spring constant (a measure of stiffness). Substituting this into the first formula gives: $P.E{.}_{elastic}=\frac{1}{2}(kx)x=\frac{1}{2}k{x}^2$

c) In terms of Stress and Strain (Strain Energy Density):

A more general form, useful in materials science, expresses the energy stored per unit volume.

• Recall that Stress ($\sigma$) = $F/A$ and Strain ($\epsilon$) = $x/L$.

• Substituting $F=\sigma A$ and $x=\epsilon L$ into the energy formula: $P.E{.}_{elastic}=\frac{1}{2}(\sigma A)(\epsilon L)=\frac{1}{2}\times \sigma \times \epsilon \times (AL)$

• Since $AL$ is the volume of the wire, the strain energy per unit volume (or strain energy density) is: $\text{Energy Density}=\frac{P.E{.}_{elastic}}{\text{Volume}}=\frac{1}{2}\times \text{Stress}\times \text{Strain}$

Possible Questions and Answers

Q: Is elastic potential energy always recovered?

A: No. If a material is stretched beyond its elastic limit, it undergoes permanent (plastic) deformation. In this case, some of the work done is converted into heat, and not all of it is stored as recoverable elastic potential energy.

Q: What is a real-world example of elastic potential energy?

A: A stretched rubber band. The work you do to stretch it is stored as elastic potential energy. When you release it, this energy is converted into kinetic energy, causing the rubber band to fly. Other examples include a compressed spring, a drawn bow, and a bent diving board.