4.5.3b - Elastic Potential Energy and Force-Extension Data

Stored energy in a spring

When a force stretches or compresses a spring, the force does work on the spring. Energy is transferred to the spring's elastic potential store.

If the spring is not inelastically deformed, the work done on the spring and the elastic potential energy stored are equal. That condition matters: if the spring is permanently damaged or does not return to its original shape, not all of the work done can be treated as recoverable elastic potential energy.

Stretching and compression are handled in the same way in this part of the specification. For a stretched spring, e means extension from the unstretched length. For a compressed spring, e means the size of the compression from the natural length.

extension = stretched length - unstretched length
compression = natural length - compressed length

For example, a spring with natural length 0.30 m that is compressed to 0.24 m has a compression of 0.06 m.

Hooke's law from data

For a spring in its linear region, the force is directly proportional to the extension. This relationship is Hooke's law.

The symbols and units are:

The spring constant tells you how stiff the spring is. A larger spring constant means a larger force is needed for the same extension or compression.

The same equation applies to compression of an elastic object. If a spring with k = 250 N/m is compressed by 0.040 m, then:

F = k e
F = 250 x 0.040
F = 10 N

To calculate a spring constant in a linear case, rearrange:

k = F / e

Example: a spring extends by 0.030 m when a force of 6.0 N is applied.

k = F / e
k = 6.0 / 0.030
k = 200 N/m

Linear and non-linear graphs

A force-extension graph usually has extension or compression on the horizontal axis and force on the vertical axis.

[DIAGRAM: asset_name: Force-Extension Data - diagram 1; asset_slug: 060_4_5_3b_elastic_potential_energy_and_force_extension_data_diagram1; file: diagram_assets/060_4_5_3b_elastic_potential_energy_and_force_extension_data_diagram1.png; recommended_method: deterministic_drawn; description: Exact assessed graph of force in N against extension or compression in m for a spring. It shows measured data points that form a straight line through the origin up to the limit of proportionality at e = 0.050 m and F = 10 N, then curve into a non-linear region. It labels the linear region, non-linear region, limit of proportionality, a gradient calculation for spring constant k = delta F / delta e = 200 N/m, and a shaded triangular area under the linear part to show work done equals elastic potential energy. Deterministic SVG-to-PNG drawing is used because axes, values, labels, data positions, gradient and shaded area are exact assessed information.]

In the linear region:

• the graph is a straight line through the origin
• F is directly proportional to e
• F / e is constant
• the gradient of a force-extension graph is the spring constant

The limit of proportionality is the point where the data first stops following a straight-line relationship. Beyond this point, the graph is non-linear. The force and extension still have a relationship, but they are no longer directly proportional, so one spring constant should not be calculated for the whole curved section.

This table matches the graph: