Stretching Materials

When a force stretches a spring or wire, it produces an extension $\Delta x$ (the increase in length). Hooke's law states that, up to a point, the extension is directly proportional to the applied force:

$\Delta F = k\,\Delta x$

• $\Delta F$ is the applied (stretching) force in newtons.
• $\Delta x$ is the extension in metres.
• $k$ is the force constant (or spring constant / stiffness) in $\text{N m}^{-1}$. A larger $k$ means a stiffer material.

On a force–extension graph (force on the $y$-axis, extension on the $x$-axis), Hooke's law gives a straight line through the origin, and the gradient of that line is the force constant $k$.

The straight line does not continue forever. The point beyond which the graph is no longer linear is the limit of proportionality — beyond it, force and extension are no longer proportional and Hooke's law no longer holds.

If several identical springs are combined, the effective stiffness changes: springs in series give a smaller combined $k$ (each carries the full load and stretches), while springs in parallel share the load and give a larger combined $k$.

Deformation can be elastic or plastic:

• Elastic deformation — the material returns to its original length once the load is removed. Atoms are pulled slightly apart but spring back.
• Plastic deformation — the material does not return to its original length; some extension is permanent. Layers of atoms (or polymer chains) have slid past one another and cannot return.

Two limits are easy to confuse, and Edexcel examiners report this constantly:

• Limit of proportionality — the point up to which $\Delta F = k\,\Delta x$ holds (extension proportional to force). Just beyond it, the graph curves but the material may still be elastic.
• Elastic limit — the point beyond which deformation becomes plastic (permanent). It lies at or just beyond the limit of proportionality.

So a material can be stretched past its limit of proportionality and still spring back; only beyond the elastic limit does it stay deformed.

For some materials, especially rubber (a polymer), the loading and unloading curves do not coincide — the unloading curve lies below the loading curve, enclosing a loop. This is hysteresis. The area of the loop equals the energy converted to heat (internal energy) in one loading–unloading cycle, which is why a stretched and released rubber band warms up.

The force constant $k$ describes a particular sample (a specific wire of a given length and thickness). To compare the materials themselves we use stress, strain and the Young modulus.

Tensile stress is the force per unit cross-sectional area:

$\sigma = \frac{F}{A}$

with units of pascals ($\text{Pa} = \text{N m}^{-2}$). For a wire of diameter $d$ (radius $r$), the cross-sectional area is

$A = \pi r^2 = \frac{\pi d^2}{4}$

Tensile strain is the fractional increase in length — extension divided by original length:

$\varepsilon = \frac{\Delta L}{L}$

Strain has no units (metres ÷ metres); it is often quoted as a decimal or a percentage.

The Young modulus is the ratio of stress to strain in the region where the material obeys Hooke's law:

$E = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L/L} = \frac{FL}{A\,\Delta L}$

with units of pascals ($\text{Pa}$), the same as stress (because strain has no units). On a stress–strain graph, $E$ is the gradient of the straight (linear) region. A large Young modulus means a stiff material that resists stretching — steel ($\approx 2.0 \times 10^{11}\ \text{Pa}$) is much stiffer than copper ($\approx 1.2 \times 10^{11}\ \text{Pa}$).

Crucially, the Young modulus is a property of the material, not of the particular sample — it does not depend on the wire's length or thickness, whereas $k$ does.

A stress–strain graph (stress on the $y$-axis, strain on the $x$-axis) reveals how a material behaves as it is loaded to breaking. Key features for a ductile metal such as copper:

• Limit of proportionality (P) — up to here the graph is a straight line and $\sigma \propto \varepsilon$ (Hooke's law). The gradient here is the Young modulus.
• Elastic limit (E) — at or just beyond P; up to here deformation is elastic (recoverable), beyond it plastic.
• Yield point (Y) — the stress at which the material suddenly extends with little extra force; plastic flow begins.
• Ultimate tensile stress (UTS) — the maximum stress the material can withstand (the peak of the curve).
• Breaking (fracture) point (B) — where the sample actually snaps.

Different classes of material give characteristically different curves:

• Ductile (e.g. copper, mild steel) — a long plastic region; the material can be drawn into wires and undergoes large permanent extension before breaking.
• Brittle (e.g. glass, cast iron) — obeys Hooke's law right up to fracture, then snaps suddenly at the elastic limit with no plastic region. The graph is an almost straight line ending abruptly.
• Polymeric (e.g. rubber) — a low, curved stress–strain line (not obeying Hooke's law), capable of very large strains, often showing hysteresis between loading and unloading.

Stretching a material does work against the interatomic forces; this energy is stored as elastic strain energy (elastic potential energy). It equals the area under the force–extension graph.

For a material obeying Hooke's law (a straight line), the area is a triangle:

$E_{el} = \tfrac{1}{2}F\,\Delta x = \tfrac{1}{2}k\,\Delta x^2$

• Use $\tfrac{1}{2}F\Delta x$ when you know the final force and extension.
• Use $\tfrac{1}{2}k\Delta x^2$ when you know the force constant and extension.
• The two are equivalent because $F = k\,\Delta x$.

If the loading is non-linear (the graph curves — e.g. beyond the limit of proportionality, or for rubber), you cannot use $\tfrac{1}{2}F\Delta x$. Instead estimate the area under the curve, for example by counting squares on the grid or splitting it into trapezia. Each square represents (force per square) × (extension per square) joules.

When a stretched elastic material is released, the stored strain energy is returned. For a material undergoing plastic deformation or hysteresis, only part of the energy is recovered — the rest is dissipated as heat (the area between loading and unloading curves).

Core Practical 3 determines the Young modulus of a metal (e.g. copper) wire. A long thin wire is clamped at one end, run horizontally over a pulley, and loaded with slotted masses; a marker on the wire shows extension against a fixed scale.

Measurements and instruments:

• Original length $L$ from the clamp to the marker — measured with a metre rule (the longer the wire, the larger the extension, reducing percentage uncertainty).
• Diameter $d$ — measured with a micrometer screw gauge (resolution $0.01\ \text{mm}$). Take several readings at different points and orientations and average, because the wire may not be perfectly uniform. Then $A = \pi d^2/4$.
• Extension $\Delta L$ for each added load, read from the marker against the scale (force $F = mg$).

Processing and result:

• For each load, calculate stress $\sigma = F/A$ and strain $\varepsilon = \Delta L / L$.
• Plot stress (y) against strain (x) and draw the best-fit straight line through the linear region.
• The gradient of the straight line is the Young modulus $E$. Using a graph (rather than a single pair of readings) averages out random error.

Uncertainty in $E$: the diameter usually dominates because it is the smallest measurement and is squared in the area, so its percentage uncertainty is doubled when it propagates into stress. Add the percentage uncertainties in $F$, $d$ (×2), $L$ and $\Delta L$ to estimate the percentage uncertainty in $E$. Using a long thin wire and a sensitive micrometer keeps these percentages small.