Hooke's Law

Hooke's law is the founding empirical statement of elasticity. Discovered by Robert Hooke (1660s), it states that for small deformations the strain produced is directly proportional to the stress applied.

Concept

Statement. Within the elastic limit, stress is directly proportional to strain:

$\sigma \propto \varepsilon \quad\Longrightarrow\quad \sigma = E\,\varepsilon$

The constant of proportionality $E$ is the modulus of elasticity and depends on the type of deformation:

• Young's modulus $Y$ for longitudinal stress and strain
• Shear modulus $G$ (or rigidity) for shear stress and strain
• Bulk modulus $B$ for volumetric stress and strain

For a spring, Hooke's law takes the familiar form:

$F = -k x$

where $k$ is the spring constant and $x$ is the displacement from the natural length. The minus sign indicates the restoring nature of the force.

Proportional limit. The point on the stress-strain curve up to which stress is strictly proportional to strain. Beyond this point, the curve may still be elastic (the body returns to original shape on removing load) but not linear.

Elastic limit. The maximum stress beyond which the body acquires a permanent (plastic) deformation.

Derivation

Consider a thin wire of length $L$, area $A$, stretched by a small force $F$, producing extension $\Delta L$.

Step 1 — Express stress and strain:

$\sigma = \frac{F}{A}, \quad \varepsilon = \frac{\Delta L}{L}$

Step 2 — Apply Hooke's law:

$\sigma = Y \varepsilon$

Step 3 — Substitute:

$\frac{F}{A} = Y \frac{\Delta L}{L}$

Step 4 — Rearrange to obtain the force-extension form:

$F = \left(\frac{YA}{L}\right) \Delta L = k\,\Delta L$

with the effective spring constant $k = YA/L$. Hence Hooke's law for a wire reduces to the linear-spring form when material constants are absorbed into $k$.

Step 5 — Validity. The law holds only for small strains (typically $\varepsilon < 10^{-3}$ for metals). Beyond the proportional limit, the curve deviates and eventually permanent set occurs.

Worked Example

A spring stretches by $4.0\,\text{cm}$ when a force of $20\,\text{N}$ is applied. (a) Find the spring constant. (b) Find the extension if a $35\,\text{N}$ force is applied, assuming Hooke's law still holds.

(a) $k = F/x = 20 / 0.04 = 500\,\text{N/m}$.

(b) $x = F/k = 35/500 = 0.07\,\text{m} = 7.0\,\text{cm}$.

Check: ratio of forces $35/20 = 1.75$, ratio of extensions $7/4 = 1.75$. Linear, as expected.

Common Confusions

• Hooke's law is not a universal law. It fails for large strains, brittle materials at high stress, and polymers (which are markedly nonlinear).
• Proportional limit ≠ elastic limit. A material can be elastic but nonlinear between these two points.
• The minus sign in $F = -kx$ for a spring denotes that the restoring force opposes displacement; it is not part of the magnitude.
• The modulus $E$ depends on the material, not on the size or shape of the sample (as long as it is isotropic and homogeneous).

Key Takeaways

• Hooke's law: stress is proportional to strain within the elastic limit.
• The proportionality constant is the modulus of elasticity, characteristic of the material.
• For a spring, the equivalent form is $F = -kx$.
• The law has a limited range of validity — small strains only.