Materials: density, Hooke’s law, stress, strain and the Young modulus

Core ideas

Density and Hooke’s law

$\rho = \dfrac{m}{V}$ in kg m⁻³. To convert, $1\ \text{g cm}^{-3} = 1000\ \text{kg m}^{-3}$. For a wire, $V = AL = \pi r^2 L$.

Hooke’s law: the extension is directly proportional to the applied force, up to the limit of proportionality:

Here $k$ is the stiffness (spring constant or force constant) in N m⁻¹. It depends on the object’s dimensions as well as its material.

Two identical springs in series have stiffness $k/2$. In parallel they have $2k$.

Stress, strain and the Young modulus

• Tensile stress $\sigma = \dfrac{F}{A}$, in Pa (N m⁻²). $A$ is the cross-sectional area before stretching, $A = \dfrac{\pi d^2}{4}$.
• Tensile strain $\varepsilon = \dfrac{\Delta L}{L}$. It is a ratio, so it has no unit.
• Young modulus $E = \dfrac{\sigma}{\varepsilon} = \dfrac{FL}{A\Delta L}$, in Pa. It is the gradient of the linear part of the stress–strain graph.

The Young modulus is a property of the material. Stiffness belongs to the object. A thicker wire of the same steel is stiffer, but has the same $E$.

Stress–strain curves

For a ductile metal such as copper or mild steel:

1. Limit of proportionality: the end of the straight-line region, where Hooke’s law stops applying.
2. Elastic limit: the largest stress from which the material returns to its original length when the load is removed.
3. Yield point: the material suddenly extends with little or no increase in stress. Diagrams for mild steel often show an upper and a lower yield point.
4. Plastic region: the deformation is permanent because planes of atoms slip past each other.
5. Ultimate tensile stress (UTS): the maximum stress on the curve. After this the wire “necks”.
6. Breaking stress: the stress at fracture.

Brittle materials (glass, cast iron, ceramics) give a straight line all the way to fracture, with no plastic region. Polymers behave differently. Rubber gives a non-linear curve with very large strain and is elastic. Polythene stretches plastically a long way (its chains straighten and slide).

Loading and unloading:

• A metal loaded beyond its elastic limit unloads along a line parallel to the original line, leaving a permanent extension.
• Rubber unloads along a different curve (hysteresis). The area of the loop is the energy transferred to thermal energy in each cycle, so the band warms up.

Elastic strain energy

Work done stretching = energy stored = area under the force–extension graph. Within the limit of proportionality:

The area under a stress–strain graph is the energy stored per unit volume, $\tfrac{1}{2}\sigma\varepsilon$, in J m⁻³.

The Young modulus practical

• Use a long (about 2–3 m), thin wire. A long wire gives a large, measurable extension and a small percentage uncertainty in $\Delta L$.
• Measure the diameter with a micrometer at several points along the wire and in perpendicular directions. Check for zero error and take the mean.
• Measure $L$ from the clamp to the marker with a metre rule or tape. Record the extension from a marker against a scale, or use a vernier (Searle’s apparatus with a reference wire to cancel temperature and sagging effects).
• Add masses in equal steps, recording the extension on loading and again on unloading to check that you stayed below the elastic limit.
• Plot $\Delta L$ against $F$. Then $E = \dfrac{L}{A \times \text{gradient}}$. Alternatively plot stress against strain, where $E$ is the gradient.
• Safety: wear goggles (a wire can snap and whip) and put a sand tray or soft landing under the masses.
• The diameter usually dominates the uncertainty. Its percentage uncertainty is doubled in $A$.

Worked examples

Exam technique

• Convert everything to SI before substituting: mm to m ($\times 10^{-3}$), mm² to m² ($\times 10^{-6}$), and diameter to radius.
• Definitions are easy marks, so learn them word-perfectly. For example: the elastic limit is “the maximum stress (or force) beyond which the material does not return to its original length when the stress (or force) is removed”. Do not mix it up with the limit of proportionality.
• Ratio questions (a wire of twice the length and twice the diameter): write $\Delta L \propto \dfrac{FL}{d^2}$ and scale each factor. It takes 20 seconds and avoids arithmetic.
• Graph gradients: use a large triangle, read the axis units and powers of ten (such as “extension / mm”), and state which line you used.
• 6-mark practical answers: cover the measurements and instruments, the graph and how $E$ is calculated from its gradient, and the precautions (repeat diameters, check for zero error, reference wire, safety).

Common mistakes

Quick recap

• $F = k\Delta L$ up to the limit of proportionality; springs in series add as $1/k$, springs in parallel add as $k$.
• $\sigma = F/A$ (Pa), $\varepsilon = \Delta L/L$ (no unit), $E = \sigma/\varepsilon = FL/(A\Delta L)$.
• Energy stored = area under the $F$–$\Delta L$ graph = $\tfrac{1}{2}F\Delta L$ in the linear region; the area under $$– is energy per unit volume.
• Ductile: large plastic region. Brittle: fractures at the end of the linear region. Rubber: hysteresis loop, with energy dissipated as heat.
• Loading a metal beyond its elastic limit leaves a permanent extension; it unloads parallel to the loading line.
• Practical: long thin wire, micrometer at several points, plot $\Delta L$ against $F$, $E = L/(A \times \text{gradient})$.