Density, Hooke's law and stress-strainAQA A-Level Physics: Revision notes

Density is the mass per unit volume of a material:

$\rho=\dfrac{m}{V}$

Its SI unit is kg m⁻³. Convert carefully: 1 cm³ = 10⁻⁶ m³, 1 g = 10⁻³ kg, so 1 g cm⁻³ = 1000 kg m⁻³.

Example: a block 4.0 cm × 5.0 cm × 2.0 cm of mass 108 g has volume 40 cm³ = $4.0\times10^{-5}$ m³ and density $0.108\div4.0\times10^{-5}=2.7\times10^3$ kg m⁻³. For a hollow object, the volume of the material is $m/\rho$ and the cavity is the total volume minus this.

Hooke's law: the extension of a spring or wire is proportional to the applied force, provided the limit of proportionality is not exceeded:

$F=k\Delta L$

where $k$ is the spring constant (stiffness) in N m⁻¹, which is the force per unit extension. A stiffer spring has a larger $k$. For a spring where 4.0 N gives 0.080 m, $k=4.0\div0.080=50$ N m⁻¹, and a 3.0 N load gives $3.0\div50=0.060$ m.

The elastic limit is the greatest force (or stress) beyond which the material no longer returns to its original length when the force is removed. Beyond the elastic limit the material is permanently deformed.

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

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

measured in pascals (Pa = N m⁻²). Tensile strain is the extension per unit original length:

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

Strain is a ratio and has no unit. The breaking stress is the stress at which the material breaks.

Example: a steel wire of diameter 0.50 mm carries 80 N. $A=\pi(0.25\times10^{-3})^2=1.96\times10^{-7}$ m², so the stress is $80\div1.96\times10^{-7}=4.1\times10^{8}$ Pa. With extension 3.0 mm in 2.50 m, strain = $1.2\times10^{-3}$.

On a stress-strain graph of a ductile metal such as copper or steel, the first part is a straight line through the origin (Hooke's law). Beyond the elastic limit the material shows plastic deformation: the strain does not go back to zero when the stress is removed, and the material may extend a lot before it breaks.

A brittle material such as glass or a ceramic follows a straight line up to its breaking stress and then fractures suddenly at a small strain, with no plastic region and no warning.

The behaviour is read from a force-extension graph in the same way, with force and extension in place of stress and strain.

A lift cabin of mass 1800 kg hangs from a steel cable of diameter 12 mm and breaking stress $1.1\times10^9$ Pa.

1. Weight: $1800\times9.81=1.77\times10^4$ N.
2. Area: $\pi(12\times10^{-3})^2\div4=1.13\times10^{-4}$ m².
3. Stress: $1.77\times10^4\div1.13\times10^{-4}=1.6\times10^8$ Pa.
4. Factor of safety: $1.1\times10^9\div1.56\times10^8=7.0$.

A ductile cable stretches visibly before it fails, which gives a warning, so ductile metals are chosen for safety-critical cables rather than brittle ceramics.