Forces and elasticity: calculate a spring constant in linear cases

The idea

In the linear region of a force–extension graph, force and extension are directly proportional, so the spring constant k is the constant ratio F ÷ e. Rearranging F = k e gives k = F ÷ e. For example, if a 6.0 N force stretches a spring by 0.030 m, then k = 6.0 N ÷ 0.030 m = 200 N/m. The gradient of the straight-line section of a force–extension graph also equals k, so a steeper line means a stiffer spring. Convert extension to metres before dividing, and remember that extension is the increase in length, not the total length. The unit of k is newtons per metre (N/m). This calculation applies only while the graph is a straight line through the origin; beyond the limit of proportionality the relationship is non-linear and k is not constant.

What earns the marks

• State the relationship F = k e and rearrange it correctly to k = F ÷ e before substituting values.
• Identify the extension as the increase in length, calculated by subtracting the original length from the stretched length.
• Convert all lengths from centimetres or millimetres into metres before dividing force by extension.
• Substitute the force in newtons and the extension in metres, then evaluate k and give the unit N/m.
• Recognise that the gradient of the straight-line section of a force–extension graph is numerically equal to the spring constant.
• Check that the value of k is calculated only from data in the linear region, where force and extension are directly proportional.

Common mistakes and how to put them right

• Using the total stretched length instead of the extension.

  Put it right by subtracting the original length first.

• Forgetting to convert centimetres to metres.

  Put it right by dividing the length in cm by 100.

• Dividing extension by force instead of force by extension.

  Put it right by using k = F ÷ e.

Exam tips

• Write the equation, then the rearrangement, then the substitution with units, so the examiner can follow your method even if the final value is wrong.
• Check whether the question gives total length or extension; if it gives total length, calculate the extension before using k = F ÷ e.
• Give the unit N/m with your answer, and check that a stiffer spring has a larger value of k than a less stiff one.