Forces and elasticity
When you apply a force to an object, it can stretch, compress or twist. If the deformation is reversible (the object returns to its original shape), it's elastic. If not, it's inelastic (or plastic).
Hooke's law
For elastic deformation, the extension is proportional to the force applied:
$F = ke$
- $F$ — force in newtons (N).
- $k$ — spring constant in N/m (a measure of stiffness).
- $e$ — extension in metres (m), measured from the natural length.
A "stiff" spring has high $k$.
Limit of proportionality
Hooke's law only works up to a certain point — the limit of proportionality. Beyond it, $F$ vs $e$ is no longer a straight line. Beyond the elastic limit, the spring no longer returns to its original shape.
Force-extension graph
Plot F (y-axis) vs e (x-axis):
- Straight line through origin while Hooke's law holds — gradient = $k$.
- Curve at higher F: deformation becomes inelastic.
Energy stored in a stretched spring
For elastic stretching:
$E_p = \frac{1}{2} k e^2$
This is the elastic potential energy stored. It comes from the work done by the stretching force, given that the average force during stretching is half the final force.
✦Worked example
A spring has $k = 50$ N/m. It's stretched by 0.20 m. Find:
(a) Force applied: $F = ke = 50 \times 0.20 = 10$ N. (b) Energy stored: $E_p = \frac{1}{2} k e^2 = 0.5 \times 50 \times 0.04 = 1.0$ J.
Required practical 6 — force and extension
Apparatus: clamp stand, spring, ruler, hangers and slotted masses.
Method:
- Hang spring from clamp; measure natural length.
- Hang a known mass; measure new length. Extension = new − natural.
- Repeat with increasing mass; record F (= mg) and e.
- Plot F vs e. The straight section gives $k$ (gradient).
Sources of error: ruler parallax; spring may be permanently deformed if you exceed elastic limit.
⚠Common mistakes
- Using new length, not extension. Extension is from natural length.
- Forgetting that beyond the limit of proportionality, $F = ke$ no longer applies.
- Confusing $k$ with stiffness verbally — high $k$ means stiff.
- Forgetting the ½ in elastic energy: $E_p = \frac{1}{2} k e^2$, NOT $k e^2$.
AI-generated · claude-opus-4-7 · v3-deep-physics