Hooke's law, stress and strain
Pull twice as hard
- Pull a spring twice as hard and it stretches twice as far.
- That neat straight-line rule is Hooke's law — true up to a point.
- It lets us measure forces with springs and test materials.
Stretching and squeezing
- A tensile force stretches an object, giving an extension $x$.
- A compressive force squeezes it. The applied force is the load.
Hooke's law
- The extension is proportional to the load: $F = kx$.
- $k$ is the spring constant (stiffness), in $\dfrac{\text{N}}{\text{m}}$ — the gradient of an $F$–$x$ graph.
A load of $12\ \text{N}$ stretches a spring by $0.040\ \text{m}$. What is the spring constant?
$k = \dfrac{F}{x} = \dfrac{12}{0.040} = 300\ \dfrac{\text{N}}{\text{m}}$.
Within Hooke's law, doubling the load doubles the extension.
Yes — $F = kx$ is a straight-line (proportional) relationship, so $x$ scales with $F$.
The limit of proportionality
- Hooke's law only holds up to the limit of proportionality.
- Past it the $F$–$x$ line curves — the simple $F = kx$ no longer works.
The straight-line law $F = kx$ holds only up to the:
Beyond the limit of proportionality the $F$–$x$ graph curves and $F = kx$ no longer applies.
Springs together
- Series (end to end): $\dfrac{1}{k_{\text{total}}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$ — softer overall.
- Parallel (side by side): $k_{\text{total}} = k_1 + k_2$ — stiffer overall.
Two identical springs (each constant $k$) are joined in series. The combined spring constant is:
$\dfrac{1}{k_{\text{total}}} = \dfrac{1}{k} + \dfrac{1}{k} = \dfrac{2}{k}$, so $k_{\text{total}} = \tfrac{1}{2}k$ — the pair stretches more easily.
Stress and strain
- Stress $\sigma = \dfrac{F}{A}$ (force per area, in Pa).
- Strain $\varepsilon = \dfrac{x}{L_0}$ (extension ÷ original length — no unit).
Match each quantity to its formula.
Stress is force per area; strain is the fractional extension; the spring constant is load per extension.
The Young modulus
- $E = \dfrac{\sigma}{\varepsilon} = \dfrac{F L_0}{A x}$, in Pa (about $2 \times 10^{11}$ for steel).
- It is a property of the material only; the spring constant $k = \dfrac{EA}{L_0}$ also depends on size.
A material with Young modulus $2.0 \times 10^{11}\ \text{Pa}$ is under a stress of $1.0 \times 10^{8}\ \text{Pa}$. What is the strain?
$\varepsilon = \dfrac{\sigma}{E} = \dfrac{1.0 \times 10^{8}}{2.0 \times 10^{11}} = 5.0 \times 10^{-4}$.
The Young modulus of a wire depends on its length and thickness.
No — the Young modulus is a property of the material. The spring constant $k = EA/L_0$ is what depends on size.
Measuring the Young modulus
- Use a long, thin wire so the extension is big enough to measure.
- Find the area from the diameter (a micrometer), load it step by step, and plot $F$ against $x$: gradient $= \dfrac{EA}{L_0}$.
You've got it
- Hooke's law $F = kx$ holds up to the limit of proportionality
- stress $\sigma = \dfrac{F}{A}$, strain $\varepsilon = \dfrac{x}{L_0}$
- Young modulus $E = \dfrac{\sigma}{\varepsilon}$ — a property of the material, not its shape