Elasticity, springs and energy stored
Hooke's law
For a spring (and many other elastic materials) below the limit of proportionality:
F = k × e
- F = force in newtons (N).
- k = spring constant in N/m (a measure of stiffness).
- e = extension in metres (m) — the change in length, NOT the new length.
A 100 N/m spring stretches 1 cm for every 1 N of pull.
Linear vs non-linear behaviour
A force–extension graph for a spring is a straight line through the origin (linear) up to the limit of proportionality. Beyond that, the line curves — extension grows faster than expected. If stretched too far, the spring is permanently deformed (passes the elastic limit) and will not return to its original length.
| Region | Behaviour |
|---|---|
| Linear | F ∝ e — spring obeys Hooke's law |
| Beyond limit of proportionality | Curve — extension grows faster than F |
| Beyond elastic limit | Permanent (plastic) deformation |
Energy stored in a stretched spring
The work done in stretching a spring is stored as elastic potential energy:
E = ½ × k × e²
- E in joules (J), k in N/m, e in m.
This formula works while the spring obeys Hooke's law. Geometrically, the energy = area under the force–extension graph (a triangle for the linear region).
✦Worked example
A spring has spring constant k = 200 N/m. It is stretched by 5.0 cm.
- Convert: e = 0.05 m.
- F = k × e = 200 × 0.05 = 10 N.
- Energy stored = ½ × 200 × 0.05² = ½ × 200 × 0.0025 = 0.25 J.
Core Practical (Paper 2): force vs extension of a spring
Suspend a spring; measure unstretched length. Add 100 g (≈1 N) masses one at a time and measure the new length each time. Plot force (y) vs extension (x). Gradient = spring constant. The straight-line region shows Hooke's law; the curved region shows where it breaks down.
Edexcel exam tip
A common pitfall — using length instead of extension. Extension = current length minus natural length. Always check what the question gives you. Also: when calculating energy stored, square the extension, not just the force; ½ k e², not ½ k e.
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