Springs: (b) Hooke’s law

The idea

Hooke’s law states that the extension of a spring or wire is directly proportional to the force applied, provided the limit of proportionality is not exceeded. Mathematically, F = kx, where F is the applied force in newtons, x is the extension in metres, and k is the spring constant in newtons per metre (N m⁻¹). The law also applies to compression, where x is the compression. For example, if a spring extends by 0.05 m when a force of 10 N is applied, then k = F/x = 10 / 0.05 = 200 N m⁻¹. A force–extension graph is a straight line through the origin up to the limit of proportionality; beyond that, the graph curves and Hooke’s law no longer applies. The spring constant is a measure of stiffness: a larger k means a stiffer spring.

What earns the marks

• Hooke’s law: the extension of a material is directly proportional to the applied force, provided the limit of proportionality is not exceeded.
• The equation is F = kx, where F is force in newtons, x is extension in metres, and k is the spring constant in N m⁻¹.
• The law applies to both extension and compression, with x representing the change in length from the natural length.
• A force–extension graph is a straight line through the origin for a material obeying Hooke’s law.
• The spring constant k is a measure of stiffness; a steeper graph indicates a larger k.
• The limit of proportionality is the point beyond which extension is no longer directly proportional to force.

Common mistakes and how to put them right

• Thinking Hooke’s law applies to all extensions.

  Put it right by checking whether the graph is linear through the origin.

• Confusing the spring constant k with the force constant of a different spring or with the elastic limit.

  Put it right by using k only as the gradient of the force–extension graph in the linear region.

• Using the total length instead of the extension in F = kx.

  Put it right by calculating extension as the increase from the natural length.

• Forgetting that k has units N m⁻¹, not N m.

  Put it right by remembering that k = F/x, so units are N divided by m.

Exam tips

• Always identify the extension from the natural length before substituting into F = kx.
• If a graph is given, calculate the gradient to find k, and check that the line passes through the origin.
• Remember that Hooke’s law applies to both stretching and compressing, but the sign of x may need a convention.
• In multiple-choice questions, look for the statement that extension is proportional to force, not to total length.