Forces and elasticity: force = spring constant × extension

The idea

This relationship describes Hooke's law for a spring or elastic object that returns to its original shape when the stretching force is removed. The force is the load applied, measured in newtons (N). The spring constant k measures stiffness in newtons per metre (N/m): a large k means a stiff spring needing a large force for a small extension. Extension e is the increase in length beyond the natural length, in metres (m), so e = stretched length − original length. For example, a spring of natural length 0.20 m stretches to 0.26 m under a 3.0 N load, so e = 0.06 m and k = F ÷ e = 3.0 N ÷ 0.06 m = 50 N/m. The law holds only up to the limit of proportionality; beyond that the graph of force against extension curves and the equation no longer applies.

What earns the marks

• State the equation as force = spring constant × extension and rearrange it as k = F ÷ e or e = F ÷ k when required.
• Identify force in newtons (N), spring constant in newtons per metre (N/m) and extension in metres (m), converting centimetres or millimetres to metres before substituting.
• Calculate extension as stretched length minus original length, not the total stretched length.
• Explain that the equation applies only while the spring obeys Hooke's law, up to the limit of proportionality, where force is directly proportional to extension.
• Interpret a straight-line force–extension graph through the origin: the gradient equals the spring constant.
• Compare springs using k, recognising that a steeper line means a larger spring constant and a stiffer spring.

Common mistakes and how to put them right

• Using the total stretched length as the extension.

  Put it right: Subtract the original length first, so a spring of natural length 0.20 m stretched to 0.26 m has e = 0.06 m.

• Substituting centimetres directly into F = k e.

  Put it right: Convert to metres, because 1 cm = 0.01 m, so 6 cm becomes 0.06 m.

• Assuming the equation works for any force.

  Put it right: Check that the spring is within its limit of proportionality, since beyond that point the force–extension graph curves and F = k e fails.

Exam tips

• Write the equation, substitute the known values with units, then rearrange only after substitution to reduce algebra errors.
• Check that your final unit matches the quantity: N for force, N/m for spring constant and m for extension.
• When a graph is given, read the gradient from a large triangle on the straight section rather than a single point, and state that the line must pass through the origin for Hooke's law.