A rod has \(L_0=2.00\,\mathrm{m}\), \(\alpha=1.5\times10^{-5}\,\mathrm{K^{-1}}\), and \(\Delta T=40\,\mathrm{K}\). Find \(\Delta L\).

A liquid has \(V_0=0.80\,\mathrm{m^3}\), \(\beta=9.0\times10^{-4}\,\mathrm{K^{-1}}\), and \(\Delta T=12\,\mathrm{K}\). Find \(\Delta V\).

A circular hole in a metal plate is heated uniformly. Does the hole diameter increase, decrease, or stay the same?

An aluminum strip is \(0.500\,\mathrm{m}\) long at \(18\,{}^\circ\mathrm{C}\). Find its length at \(68\,{}^\circ\mathrm{C}\), using \(\alpha=2.3\times10^{-5}\,\mathrm{K^{-1}}\).

A steel bridge span is \(45.0\,\mathrm{m}\) long. Estimate the expansion gap needed for a temperature increase of \(35\,\mathrm{K}\), using \(\alpha=1.2\times10^{-5}\,\mathrm{K^{-1}}\).

A solid expands isotropically with \(\alpha=1.0\times10^{-5}\,\mathrm{K^{-1}}\). Estimate its volume expansion coefficient \(\beta\).

A tank has volume \(2.0\times10^{-3}\,\mathrm{m^3}\) and is completely filled with liquid. The liquid has \(\beta_l=8.0\times10^{-4}\,\mathrm{K^{-1}}\), while the tank has \(\beta_t=3.6\times10^{-5}\,\mathrm{K^{-1}}\). Estimate the overflow volume for \(\Delta T=25\,\mathrm{K}\).

A rod would freely expand by \(0.90\,\mathrm{mm}\) when heated, but its ends are held fixed. Its cross-sectional area is \(1.5\times10^{-4}\,\mathrm{m^2}\), Young modulus is \(2.0\times10^{11}\,\mathrm{Pa}\), and initial length is \(3.0\,\mathrm{m}\). Find the compressive force.

A measuring tape made of steel is correct at \(20\,{}^\circ\mathrm{C}\). At \(35\,{}^\circ\mathrm{C}\), it is used to measure a table and reads \(1.500\,\mathrm{m}\). Is the true table length slightly larger or smaller than the reading? Estimate it using \(\alpha=1.2\times10^{-5}\,\mathrm{K^{-1}}\).

Derive \(\beta\approx3\alpha\) for an isotropic solid by starting from \(V=L^3\) and keeping only first-order terms in \(\alpha\Delta T\).

A composite bar is made from two materials of equal original length joined end to end. Their coefficients are \(\alpha_1\) and \(\alpha_2\). The total end-to-end length is free to change. Derive the effective linear expansion coefficient of the composite bar.

A rod of length \(L_0\), area \(A\), Young modulus \(Y\), and expansion coefficient \(\alpha\) is clamped between rigid walls at temperature \(T_0\). It is heated by \(\Delta T\). Derive the thermal stress and elastic energy stored in the rod, assuming linear elasticity.