Elasticity and materials

Revision notes

When a force stretches or compresses an object, the deformation can be elastic or plastic. Elastic deformation is reversible: when the force is removed the object returns to its original shape. Plastic deformation is permanent. For many springs and wires there is an initial region in which extension is directly proportional to applied force. In this proportional region Hooke's law can be written F = kx, where k is the spring constant and x is the extension from the natural length.

A larger spring constant means that a larger force is needed to produce the same extension, so the spring is stiffer. The limit of proportionality marks the end of the straight-line F-x behaviour; it should not automatically be described as the breaking point. A material may continue to behave elastically for some range beyond simple proportionality, depending on the context and model.

Work is done when a spring is stretched. For a linear spring obeying Hooke's law, the stored elastic energy is E = \(\dfrac{1}{2}kx^{2}\), which is also the area under a force-extension graph up to that extension. On a graph of force against extension, the gradient in the proportional region is k. Careful experiments use the extension, not the total spring length, and avoid exceeding safe loads.

Force and extension

When a force stretches a spring or elastic object, extension is the change in length, not the final length. Extension x = stretched length − original length. In the proportional region, Hooke's law applies: F = kx, where k is the spring constant. A larger k means a stiffer spring because more force is needed for the same extension.

The limit of proportionality is the point beyond which force and extension are no longer directly proportional. It is identified from the point where a force-extension graph stops being a straight line through the origin.

Elastic and plastic behaviour

Elastic deformation is reversible: the object returns to its original shape when the force is removed. Plastic deformation is permanent. Passing the limit of proportionality does not necessarily mean the object has already undergone permanent deformation, The elastic limit, beyond which deformation becomes permanent, can lie slightly beyond the limit of proportionality.

Energy stored in stretching

Work is done when a force stretches an object. The energy transferred to the elastic store equals the area under the force-extension graph. In the linear region this area is triangular, so \(E_{e}\) = \(\dfrac{1}{2}Fx\) = \(\dfrac{1}{2}kx^{2}\).

If a force-extension graph is not linear, do not use the triangle formula over the whole graph without justification; use the area under the actual graph.

Practical work

Measure the original length before loading. Add known forces gradually, measure extension at eye level and avoid exceeding safe loads. Repeating measurements and using a graph help identify anomalies and the proportional region. Loading and unloading measurements can reveal whether deformation is fully reversible.

Exam technique

Use metres for extension when k is in N/m. Distinguish 'length' from 'extension'. On graph questions, gradient of force against extension gives k; if axes are reversed, the gradient is \(\dfrac{1}{k}\) rather than k.