Work Done and Elastic Potential Energy

• Identify the units for force (newtons, N), spring constant (newtons per metre, N/m), and extension or compression (metres, m).
• Explain that a force that stretches or compresses a spring performs work, resulting in elastic potential energy being stored in the spring.
• Recognize that the work done on a spring is equal to the elastic potential energy stored, provided the spring is not inelastically deformed.
• Describe the distinction between linear and non-linear relationships between force and extension.
• Calculate the spring constant in linear cases and interpret data from investigations evaluating the relationship between force and extension.
• Calculate the work done in stretching or compressing a spring (up to the limit of proportionality) using the elastic potential energy equation:

$E_e = \frac{1}{2} k e^2$

• Calculate relevant values of stored energy and track energy transfers.
• Describe and conduct a practical investigation to evaluate the relationship between force and extension for a spring.

Here, $E_e$ is elastic potential energy in joules (J), $k$ is the spring constant in newtons per metre (N/m), and $e$ is extension in metres (m). The same equation applies to compression, with $e$ representing how much shorter the spring has become.

Extension means the change in length, not the final length:

For compression, subtract the compressed length from the original length instead.

The spring constant describes stiffness. Within the proportional region, a stiffer spring needs a greater force for the same extension. It therefore stores more energy at that extension. For a given spring, energy depends on extension squared: doubling the extension stores four times as much energy, provided both extensions remain within the proportional region.

The limit of proportionality and inelastic deformation are not the same thing. The first concerns whether force and extension remain directly proportional; the second concerns whether the spring is permanently deformed. Use $E_e = \frac{1}{2}ke^2$ only up to the limit of proportionality.

From spring length to work done

Consider a spring with a spring constant of 250 N/m. Its unstretched length is 10.0 cm and its stretched length is 11.4 cm. Assume it remains within its limit of proportionality.

First find the extension:

Then substitute into the energy equation:

The spring stores approximately 0.025 J to two significant figures. Because it has not been inelastically deformed, the work done in stretching it from its unstretched length is also 0.025 J.

Finding a different unknown

The same relationship can be rearranged to calculate a spring constant or an extension:

For example, suppose 0.2 J of work stretches an initially unstretched spring by 4.5 cm within its proportional region. The stored energy is 0.2 J and the extension is 0.045 m, so:

This result describes the stiffness of the spring, not the force currently acting on it.

Following energy as a spring moves

Energy is conserved: stretching or compressing a spring transfers energy into its elastic potential store rather than creating energy. Releasing it transfers energy out again.

For a mass moving up and down on a vertical spring, energy can move between the spring’s elastic potential store, the mass’s kinetic store and the gravitational potential store of the mass–Earth system. At a turning point the mass is momentarily stationary, so its kinetic energy is zero, but energy remains in other stores.

To calculate a transfer, identify which store decreases and which stores increase. If all the energy leaving the spring’s elastic potential store becomes kinetic energy, the kinetic energy gained equals the elastic potential energy lost. If energy also goes into another store, it must be included in the energy balance; the kinetic energy gain alone is then smaller.

Use up to the limit of proportionality.