Hooke's law: F = ke, straight line through the origin
Hooke's law: the extension of a spring is directly proportional to the applied force, up to the limit of proportionality. F = ke, with F in N, k (the stiffness) in N/m and e in m; extension = stretched length - natural length. On a force-extension graph, Hooke's law is the initial STRAIGHT line through the ORIGIN and its gradient equals k. Beyond the limit of proportionality the line stops being straight and F is no longer proportional to e.
Elastic behaviour: recovers shape when force removed
Elastic behaviour: the ability of a material to RECOVER its original shape once the forces causing the deformation are REMOVED. Both elements earn the marks: it recovers its original shape, and only after the deforming force is removed. This holds while the material stays within its ELASTIC LIMIT. Inelastic (plastic) behaviour: if the elastic limit is exceeded, the material is PERMANENTLY deformed and does not recover its shape when the force is removed.
Force-extension shapes: spring, wire, rubber band
Extension vs applied force for three materials. SPRING: a straight line through the origin (obeys Hooke's law) up to the limit of proportionality, then curves. METAL WIRE: a straight line through the origin for small forces (obeys Hooke's law), extending only slightly; if over-stretched it deforms plastically. RUBBER BAND: a curved line that does NOT obey Hooke's law — no straight region, so F is never proportional to e, and it follows different loading and unloading paths.
Drawn from real examiner reports.
"Proportional" is not "both increase"
"Directly proportional" means a graph that is a STRAIGHT LINE through the ORIGIN, so the ratio F/e is constant. Arguing that force and extension are proportional just because both increase is not credited — a curve, or a straight line that misses the origin, is not proportional. For Hooke's law it must be the straight line through the origin.
Jun 2024 1P Q10(c)(iv); Jun 2024 1PR Q7b(iv): "proportional" claimed only because both quantities increased — not credited; proportionality needs a straight line through the origin.
Extension is not total length
Extension is the CHANGE in length: e = stretched length - natural length. A spring of natural length 10 cm stretched to 16 cm has an extension of 6 cm (0.06 m), not 16 cm. Substituting the total length into F = ke, or forgetting to convert cm (or mm) to metres before finding k, are the two most common numerical errors on this topic.
Kink is the limit of proportionality
When asked to draw a graph for a material OBEYING Hooke's law, draw only the straight line through the origin — adding a curved section shows behaviour beyond Hooke's law and loses the mark. If a kink is shown, label it the LIMIT OF PROPORTIONALITY (where the line stops being straight), not the "elastic limit", which is not credited for the Hooke's-law point.
Jun 2024 1P Q5(a): a curved section added to a Hooke's-law graph lost the mark, and labelling the start of curvature "elastic limit" was not credited — only "limit of proportionality" was accepted.
Vague elastic definitions score zero
Answers such as "springy", "bounces back" or "bendy" earn nothing. The mark scheme needs "the material recovers its original shape" AND "when the deforming force is removed" — both elements. On a loading/unloading graph, elastic behaviour is shown by the line RETURNING TO THE ORIGIN once the load is removed; the proviso "when the load is removed" is needed for the second mark.
Jun 2024 1PR Q9b(ii): the proviso "when the load is removed" was required for the second mark on elastic behaviour.
Rubber band never obeys Hooke's law
A rubber band's force-extension graph is CURVED from the start with no straight region, so F is never proportional to e — it does not obey Hooke's law, unlike a spring or wire at small loads. A non-Hookean material also shows DIFFERENT loading and unloading lines. Do not draw a rubber band as a straight line.
Jun 2024 1PR Q9b(i): a non-Hookean material shows a line that is not straight, with different loading and unloading paths.
Gradient is ΔF/Δe, not the inverse
The spring constant k is the gradient of the straight region: k = ΔF/Δe (rise over run — change in force divided by change in extension). Inverting it to Δe/ΔF gives 1/k, not k. Read two widely spaced points off the straight line, keep extension in metres, and quote k in N/m.
Test for Hooke's law
To decide whether a material obeys Hooke's law, check TWO things about its force-extension graph: is it a straight line, and does it pass through the origin? Both are needed for "directly proportional". The gradient of the straight region is the spring constant k.
Convert extension to metres
Before finding k or using F = ke, convert the extension from cm or mm to metres (cm ÷ 100, mm ÷ 1000) and subtract the natural length first. Then k comes out in N/m. Mixing units is a frequent power-of-ten error.
Proportion shortcut in the linear region
Within the straight-line region you can scale directly: if force F1 gives extension e1, then F2 gives e2 = e1 × F2/F1. This avoids finding k. It only works inside the linear (Hooke's law) region — never beyond the limit of proportionality.
Draw the graph carefully
Label both axes with quantity AND unit (force / N, extension / m), start the line at the origin, and for a Hooke's-law material draw a straight line only. Use pencil so you can correct it. A correct, fully labelled graph can carry all the marks on its own.
Edexcel Science (Double Award) 4SD0 — spec statements 1.22–1.24. Covers the force–extension practical for springs, metal wires and rubber bands; Hooke's law and the linear region; and elastic behaviour.
| Quantity | Symbol | Relationship | Unit |
|---|---|---|---|
| Force (Hooke's law) | ( directly proportional to ) | newton, N | |
| Spring constant (stiffness) | (gradient of the linear region) | N/m | |
| Extension | metre, m |
Where = applied force (N), = spring constant / stiffness (N/m), = extension (m).
Hooke's law (statement 1.23): in the initial linear region of a force–extension graph, extension is directly proportional to the applied force. On the graph this is the straight line through the origin; beyond the limit of proportionality the line curves and the law no longer holds.
Elastic behaviour (statement 1.24): the ability of a material to recover its original shape after the deforming forces have been removed. If the elastic limit is exceeded the material is permanently (plastically) deformed and does not return to its original shape.
Define elastic behaviour.
A spring has a spring constant of 40 N/m. Within its linear region it is stretched so that the extension is 0.15 m.
Calculate the force applied to the spring. Give the unit of your answer.
