Forces and elasticity: spring constant, k, in newtons per metre, N/m

The idea

The spring constant k is a measure of the stiffness of a spring or elastic object. It is defined by Hooke's law, F = k × e, where F is the force in newtons and e is the extension in metres. Rearranging gives k = F ÷ e, so k is the force needed to produce unit extension. Its unit is newtons per metre (N/m). A stiffer spring has a larger k and stretches less for the same force.

For example: If a force of 6 N produces an extension of 0.03 m, then k = 6 ÷ 0.03 = 200 N/m.

What earns the marks

• Defines the spring constant as the force per unit extension, given by k = F ÷ e, with unit N/m.
• Rearranges F = k × e to calculate k when force and extension are known.
• Determines k from the gradient of the straight-line section of a force–extension graph.
• Explains that a larger k indicates a stiffer spring that requires more force for the same extension.
• Recognises that k applies only up to the limit of proportionality, beyond which the relationship becomes non-linear.

Common mistakes and how to put them right

• Using extension in centimetres when calculating k.

  Put it right: Convert extension to metres so that k is in N/m.

• Confusing k with the gradient of a force–extension graph when axes are swapped.

  Put it right: Gradient = ΔF ÷ Δe gives k only when force is plotted on the y-axis and extension on the x-axis.

• Assuming k remains constant beyond the limit of proportionality.

  Put it right: k is constant only in the linear region, and the graph curves beyond that point.

Exam tips

• Write the unit N/m after any value of k and check that the arithmetic gives sensible units.
• When finding k from a graph, choose two points on the straight line and calculate gradient = change in y ÷ change in x.
• Convert all lengths to metres before dividing, and show the conversion step to avoid losing marks.