Hooke's Law and Load–Extension Graphs: GCSE & IGCSE

Hooke’s law says that the force needed to stretch a spring is directly proportional to its extension, as long as you do not go past the limit of proportionality. In symbols, F = ke, where F is the force in newtons, e is the extension in metres and k is the spring constant in N/m. On a load–extension graph that is a straight line through the origin, and its gradient is k.

That is the whole idea, and it is worth a lot of marks across GCSE and IGCSE Physics. Most of the marks lost on this topic come from a handful of repeated slips, which this post walks through with numbers.

Hooke’s law and the spring constant calculation

The spring constant tells you how stiff a spring is. A large k means you need a large force for each metre of stretch. Rearranged, k = F ÷ e.

A spring has an original length of 12 cm. A 3.0 N load makes it 18 cm long. Many students write e = 18 here, and that is the commonest error on the topic. Extension is the change in length, so:

• Extension = 18 − 12 = 6 cm = 0.06 m
• Equation: k = F ÷ e
• Substitution: k = 3.0 ÷ 0.06
• Answer: k = 50 N/m

Three things earn the marks here: finding the extension first, converting to metres, and showing the substitution. The conversion matters because the unit is N/m. If you leave the extension in centimetres you get 0.5, which is correct only in N/cm. Some questions do accept N/cm, but only if you state the unit; mixing the two is a guaranteed lost mark.

Reading a load–extension graph

For a spring that obeys Hooke’s law, the graph of force (load) against extension is a straight line through the origin. Read the gradient from two well-separated points on the line, not from a single point and not from a point that sits on a curve.

Suppose the line passes through (0.02 m, 1.0 N) and (0.10 m, 5.0 N). The gradient is (5.0 − 1.0) ÷ (0.10 − 0.02) = 4.0 ÷ 0.08 = 50 N/m. Draw the triangle on the graph if the question lets you; it makes the method visible.

Pay attention to which way round the axes are. If force is on the vertical axis and extension on the horizontal, the gradient is k. Some questions plot extension against load instead, which is common in the IGCSE practical. Then the gradient is e ÷ F, which is 1 ÷ k, so k is the reciprocal of the gradient. Check the axis labels before you divide anything.

The limit of proportionality

Keep adding load and eventually the line stops being straight and begins to curve. The point where it leaves the straight line is the limit of proportionality. Beyond it, doubling the force no longer doubles the extension, so F = ke no longer applies.

That has an exam consequence. You can only use F = ke, or read k from the gradient, on the straight part. If a question gives you a force and extension that lie on the curved section, the calculation is not valid, and a question that asks you to explain why a student’s value of k is wrong is often testing exactly this.

A good answer is specific: “The graph is no longer a straight line, so force is no longer directly proportional to extension.” Writing only “the spring has stretched too much” describes the situation without explaining it.

Elastic and plastic deformation

A spring is elastic if it returns to its original length when you remove the load. If it stays stretched, the deformation is plastic, and the spring is permanently changed. The point beyond which a material no longer returns to its original shape is the elastic limit.

At this level the two limits are often close together, and not every specification asks you to separate them, so follow your own board’s wording. The safe distinction is this: the limit of proportionality is about the shape of the graph, while elastic versus plastic is about what happens when the load is removed. Do not use one phrase when the question asks for the other.

Elastic potential energy

Stretching a spring transfers energy to it. Within the limit of proportionality, the energy stored is E = ½ke². For the spring above, stretched by 0.06 m:

• E = ½ × 50 × 0.06²
• E = ½ × 50 × 0.0036 = 0.09 J

The same value is the area under the line on the force–extension graph, because the area of a triangle is ½ × base × height, which is ½ × e × F. Check it: ½ × 0.06 × 3.0 = 0.09 J. If a question gives you a graph and no value of k, the area is often the quicker route.

The trap is squaring the wrong thing. The extension is squared, not the force and not the whole product. Square the extension first, in metres, then multiply.

The required practical, step by step

Both GCSE and IGCSE courses expect you to know how to investigate the stretching of a spring, and the method is a common source of method marks. A clear answer covers these points:

• Clamp the spring vertically and fix a metre ruler beside it, as close to the spring as you can without touching it.
• Attach a marker, such as a pin or a thin strip of card, to the bottom of the spring so you have a clear point to read from.
• Measure the original length with no load, with your eye level with the marker to avoid parallax error.
• Add a mass in equal steps, for example 100 g at a time, and measure the new length each time. Remember that 100 g gives a weight of about 1 N.
• Work out extension for each load by subtracting the original length.
• Repeat the readings, ideally taking readings as you remove the masses too, and average them to improve reliability.
• Plot force against extension and draw a line of best fit through the straight part.

Keep the safety point in mind: stand clear, or use a tray or soft surface below the masses, because a mass can fall if the spring or clamp slips. If a question asks how to make the results more accurate, reading at eye level and using a marker are the points examiners look for.

Exam traps in one place

• Using length instead of extension. Always subtract the original length first.
• Mixing units. Convert cm to m before using N/m. Convert grams to kilograms before turning mass into weight.
• Reading k from the curve. Only the straight part gives the spring constant.
• Wrong axes. On an extension–load graph, k is the reciprocal of the gradient.
• Forgetting to show working. A bare answer of 50 earns nothing if the unit or the extension is wrong, while visible steps can still earn the method marks. We explain how that works in our post on method marks.

Practise it on your own board

Hooke’s law sits in a different place in each course, so go to the right notes for yours: AQA GCSE Forces and Elasticity, Edexcel GCSE Forces & Elasticity, Cambridge IGCSE Effects of Forces or Edexcel IGCSE Forces, Movement & Changing Shape. When you are ready to test yourself, try the elasticity worksheet on Exaim, where every written answer is marked against the board’s mark scheme, so you can see which method marks your working earned.