P5.3 Forces and Elasticity
Elastic and inelastic deformation
When a force deforms (changes the shape of) an object:
- Elastic deformation: The object returns to its original shape when the force is removed (e.g. a spring within its limit, a rubber band).
- Inelastic (plastic) deformation: The object does not return to its original shape (e.g. clay, a spring stretched beyond its limit).
Hooke's Law
For a spring (or elastic material) that has not been stretched beyond its elastic limit (limit of proportionality):
F = ke
F = force applied (N)
k = spring constant (N/m) — stiffness of the spring
e = extension (or compression) from natural length (m)
The extension is directly proportional to the applied force — the F–e graph is a straight line through the origin.
Spring constant k: the steeper the F–e graph, the stiffer the spring and the larger the value of k.
Elastic limit / limit of proportionality: Beyond this point, the spring stretches more than Hooke's Law predicts (graph curves) and may not return to its original shape.
✦Worked example
A spring has a natural length of 0.20 m. A 5 N force stretches it to 0.28 m.
Extension e = 0.28 − 0.20 = 0.08 m
k = F/e = 5/0.08 = 62.5 N/m
Elastic potential energy stored in a spring
When a spring is elastically deformed, it stores elastic potential energy:
Eₑ = ½ke²
Eₑ = elastic potential energy (J)
k = spring constant (N/m)
e = extension (m)
Note: Requires the deformation to be elastic (within Hooke's Law region).
Worked example: A spring with k = 500 N/m is compressed by 0.04 m.
Eₑ = ½ × 500 × (0.04)² = ½ × 500 × 0.0016 = 0.40 J
F–e graph interpretation
| Graph region | Meaning |
|---|---|
| Straight line through origin | Hooke's Law obeyed; proportional |
| Curve begins | Elastic limit / limit of proportionality exceeded |
| Permanent deformation when force removed | Inelastic deformation |
Required practical — Hooke's Law investigation
- Hang masses on a spring; record mass and spring length.
- Calculate extension: e = new length − natural length.
- Calculate force: F = mg (g = 9.8 N/kg).
- Plot F (y-axis) vs e (x-axis).
- Gradient of linear region = spring constant k.
- Note where graph departs from linearity = elastic limit.
Common exam errors
- Forgetting to subtract the natural length — extension ≠ total length.
- Squaring the extension in Eₑ = ½ke² but forgetting to square (writing ½ke instead).
- Using compression as a negative extension — the formula uses magnitude.
- Confusing elastic and inelastic deformation.
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