Elastic strain energyAQA A-Level Physics: Revision notes

When a force stretches a wire or spring, it does work, and while the material stays elastic this work is stored as elastic strain energy. If the force is released, the energy is transferred back, for example to the kinetic energy of a launched object.

The work done equals the area under a force–extension graph. For a material obeying Hooke's law the graph is a straight line through the origin, so the area is a triangle.

For a straight-line graph, the area under it is $\frac{1}{2} F \Delta L$, where $F$ is the final force and $\Delta L$ is the extension, so $E = \frac{1}{2} F \Delta L$.

Using Hooke's law, $F = k \Delta L$, this becomes $E = \frac{1}{2} k \Delta L^2$. Doubling the extension therefore quadruples the stored energy.

Worked example: a spring of $k = 200$ N m⁻¹ is extended by 0.050 m. $F = 200 \times 0.050 = 10$ N, so $E = \frac{1}{2} \times 10 \times 0.050 = 0.25$ J.

A stretched or compressed spring can transfer its elastic strain energy to other stores. Ignoring losses, elastic strain energy = kinetic energy for a horizontal launch, and elastic strain energy = kinetic energy + gravitational potential energy if the object rises.

For a ball launched vertically from a spring compressed by $\Delta L$, which rises a height $h$ above the point where it leaves the spring: $\frac{1}{2} k \Delta L^2 = m g (h + \Delta L)$, because the ball also rises through the compression distance while it is pushed.

In real systems some energy is transferred to thermal energy through friction and air resistance, so the observed speed or height is less than predicted.

Beyond the elastic limit, a metal deforms plastically: layers of atoms slip past each other and the material does not return to its original length.

On unloading, the line on the graph falls parallel to the original straight section but ends at a permanent extension. The area between the loading and unloading lines is the energy transferred to the material as thermal energy, which is why a metal wire or paper clip warms when bent back and forth.

Engineers use elastic strain energy to recover energy that would otherwise be wasted, for example in a regenerative braking system that stores kinetic energy in a spring or flywheel and returns it on acceleration.

An evaluation weighs the benefits against the costs:

• Energy recovered reduces fuel or electricity use, so emissions and air pollution fall
• Less friction braking reduces brake wear and particulates
• The added mass of the store increases the energy needed to accelerate
• Manufacturing and disposal use energy and resources (whole-life cost)
• No system is 100% efficient, so some energy is always transferred to thermal energy