1) Linear expansion (rods, lengths)
Here \(\alpha\) is the linear expansion coefficient (units \(1/^{\circ}\mathrm{C}\) or \(1/\mathrm{K}\)).
2) Area expansion (plates)
For an isotropic solid, area grows in two perpendicular directions, so typically:
\(\gamma = 2\alpha\).
3) Volume expansion (solids & liquids)
For an isotropic solid, volume grows in 3 directions, so a common approximation is
\(\beta \approx 3\alpha\).
For liquids, you usually use \(\beta\) directly.
4) Water anomaly (concept)
Water is unusual near \(0\text{–}4^{\circ}\mathrm{C}\): as it warms from \(0^{\circ}\mathrm{C}\) to \(4^{\circ}\mathrm{C}\),
its density increases (so volume decreases). That corresponds to an “effective” \(\beta < 0\) in that narrow range.
Outside that range, \(\beta\) becomes positive again.
The calculator’s anomaly option is a simple qualitative piecewise model for intuition; for high accuracy you’d use tabulated density vs temperature.
5) Worked example (matches the sample input)
Steel rod: \(L_0=1\,\mathrm{m}\), \(T_0=20^{\circ}\mathrm{C}\), \(T_1=100^{\circ}\mathrm{C}\), \(\alpha=12\times 10^{-6}/^{\circ}\mathrm{C}\).
6) University extension: bimetallic strips
If two bonded layers have different \(\alpha\), temperature change produces different free expansions.
Because they are bonded, the strip bends instead. This is the basis of thermostats and thermal switches.
