Linear expansion of a rod

Text

The length of a metal rod at temperature $t$ (in $^\circ\mathrm{C}$) is $\ell(t)=\ell_{0}\left(1+1.2\cdot10^{-5}\,t\right),$ with length at $0\,^\circ\mathrm{C}$ equal to $\ell_{0}=2.5\ \mathrm{m}$. Find the change of length per degree, i.e. $\dfrac{d\ell}{dt}$.

Solution

$\ell(t)$ is linear in $t$, hence $\frac{d\ell}{dt}=\ell_{0}\cdot 1.2\cdot10^{-5}=2.5\cdot1.2\cdot10^{-5}=3.0\cdot10^{-5}\ \mathrm{m/^\circ C},$ that is $0.030\ \mathrm{mm}$ per degree. $\boxed{\frac{d\ell}{dt}=3.0\cdot10^{-5}\ \mathrm{m/^\circ C}}$