What you'll learn
Stretch a spring and it gets longer in a very predictable way — up to a point. That predictable behaviour is described by Hooke's law, and it lets us use springs to measure forces and store energy. For AQA GCSE Physics you need to understand elastic and inelastic deformation, apply Hooke's law, calculate the energy stored in a stretched spring, and interpret force–extension graphs. This guide covers the meaning of extension, the spring constant, the limit of proportionality, the energy stored, and the required-practical link on investigating force and extension. By the end you should be able to use the Hooke's law equation, read a force–extension graph, and explain the difference between elastic and inelastic deformation.
Key terms and definitions
Deformation — A change in shape caused by a force.
Elastic deformation — When an object returns to its original shape after the force is removed.
Inelastic deformation — When an object does not return to its original shape after the force is removed.
Extension — The increase in length of a spring or other object when a force is applied.
Hooke's law — Extension is directly proportional to the force, up to the limit of proportionality.
Spring constant — A measure of the stiffness of a spring; the force needed per metre of extension.
Limit of proportionality — The point beyond which extension is no longer proportional to force.
Elastic potential energy — The energy stored in a stretched or compressed spring.
Core concepts
Deforming an object
To change the shape of a stationary object, more than one force must act on it — a single force would just make it move. When two or more forces act, the object can be stretched, compressed or bent. Deformation can be elastic, where the object returns to its original shape when the forces are removed, or inelastic, where it stays permanently changed.
Hooke's law
Hooke's law states that the extension of an elastic object, such as a spring, is directly proportional to the force applied, provided the limit of proportionality is not exceeded. The equation is:
force = spring constant × extension (F = k e)
where force F is in newtons (N), the spring constant k is in newtons per metre (N/m), and the extension e is in metres (m). The spring constant measures how stiff the spring is — a larger spring constant means a stiffer spring that needs more force to stretch it by the same amount.
Extension, not length
It is important to use the extension, which is the increase in length, not the total length of the spring. If a spring is 5 cm long unstretched and 8 cm long when a force is applied, the extension is 8 − 5 = 3 cm. Using the total length instead of the extension is a common error.
The limit of proportionality
Hooke's law only holds up to the limit of proportionality. Below this point, a force–extension graph is a straight line through the origin, showing that extension is proportional to force. Beyond the limit of proportionality, the line curves, and the spring extends more for each extra newton — it no longer obeys Hooke's law. If stretched too far, the spring may be permanently deformed (inelastic deformation).
Force–extension graphs
A force–extension graph shows force on one axis and extension on the other:
- The straight-line part through the origin shows Hooke's law being obeyed; extension is proportional to force.
- The gradient of the straight-line section relates to the spring constant (a steeper line means a stiffer spring).
- The point where the line starts to curve is the limit of proportionality.
Energy stored in a spring
When a spring is stretched elastically, work is done and elastic potential energy is stored in it. This energy is released when the spring returns to its original shape. Provided the spring is not stretched beyond the limit of proportionality, the elastic potential energy stored is given by:
elastic potential energy = 0.5 × spring constant × (extension)² (Eₑ = ½ k e²)
This is why a stretched catapult or bow can transfer stored energy to an object, and why a spring can power a mechanism.
Required practical link
You can investigate the link between force and extension by hanging a spring, adding masses one at a time, and measuring the extension for each added weight. Plotting force (from the weights) against extension gives a straight line through the origin up to the limit of proportionality, confirming Hooke's law, and the gradient can be used to find the spring constant. To make it a fair test, use the same spring and measure from the same fixed point each time.
Working out the force from a mass
In the practical, the force stretching the spring is the weight of the masses hung on it, not the mass itself. Weight is calculated using weight = mass × gravitational field strength (W = m g), where g is about 9.8 N/kg on Earth. So a 100 g (0.1 kg) mass has a weight of 0.1 × 9.8 ≈ 1 N. This is the force you plot on your graph. Forgetting to convert mass into weight, or leaving the mass in grams, is a common source of error, so always convert to kilograms and multiply by g first.
Compression as well as extension
Springs can be compressed as well as stretched, and Hooke's law applies to compression too, provided the limit of proportionality is not exceeded. When a spring is compressed, it becomes shorter, and the same relationship holds: the change in length is proportional to the force. Elastic potential energy is stored in a compressed spring in the same way as a stretched one, which is why compressed springs are used in devices such as clothes pegs, retractable pens and vehicle suspension. Remembering that the physics works for both stretching and compressing helps with unfamiliar exam contexts.
Worked examples
Example 1: Using Hooke's law
A spring has a spring constant of 25 N/m. What force is needed to stretch it by 0.2 m? Force = spring constant × extension = 25 × 0.2 = 5 N.
Example 2: Finding the spring constant
A force of 12 N stretches a spring by 0.3 m. Calculate the spring constant. Rearranging F = k e gives k = F ÷ e = 12 ÷ 0.3 = 40 N/m.
Example 3: Finding the extension
A spring with a spring constant of 50 N/m has a 10 N force applied. Calculate the extension. e = F ÷ k = 10 ÷ 50 = 0.2 m.
Example 4: Energy stored
Calculate the elastic potential energy stored in a spring of spring constant 200 N/m stretched by 0.1 m. Eₑ = ½ k e² = 0.5 × 200 × (0.1)² = 0.5 × 200 × 0.01 = 1 J.
Common mistakes and how to avoid them
The most common error is using the total length of the spring instead of the extension. Always subtract the original length to find the extension before using the equation.
Students often forget to square the extension in the energy equation. It is ½ k e², so the extension must be squared — forgetting this gives a badly wrong answer.
Another mistake is using centimetres instead of metres. The spring constant is in N/m, so the extension must be in metres. Convert cm to m (divide by 100) first.
When reading a force–extension graph, do not describe the curved part as obeying Hooke's law. Hooke's law applies only to the straight-line region below the limit of proportionality.
Finally, do not confuse elastic and inelastic deformation. Elastic means it returns to its original shape; inelastic means it stays permanently deformed after stretching too far.
Exam technique for "Forces and elasticity (Hooke's Law)"
For calculations, write F = k e (or Eₑ = ½ k e²), rearrange if needed, and check that the extension is in metres. Show your rearrangement clearly to earn method marks.
Graph questions are common: identify the straight-line region as obeying Hooke's law, identify the limit of proportionality where the line curves, and use the gradient to find the spring constant. Practise reading values off the axes accurately.
When asked about the required practical, describe adding masses one at a time, measuring the extension each time, and plotting force against extension. Remember the difference between elastic and inelastic deformation, and be ready to explain the energy stored in terms of elastic potential energy.
Quick revision summary
- Deforming an object needs more than one force; deformation is elastic (returns to shape) or inelastic (stays deformed).
- Hooke's law: force = spring constant × extension (F = k e), valid up to the limit of proportionality.
- Use the extension (increase in length), in metres, not the total length.
- On a force–extension graph, the straight line through the origin obeys Hooke's law; it curves at the limit of proportionality.
- Elastic potential energy = ½ k e² — remember to square the extension.
- Investigate by adding masses to a spring and plotting force against extension.