Question

A bar with a crack at its centre buckles as a result of temperature rise of $32{ }^{\circ} C$. If the fixed distance $L_0$ is $3.77 \,m$ and the coefficient of linear expansion of the bar is $25 \times 10^{-6} /{ }^{\circ} C$ find the rise (in metre) of the centre. ___
$\left(7.5 \times 10^{-2}\right)$

Answer 1

Consider one half of the bar, its initial length $\ell_0=\frac{L_0}{2}$
Its length after increase in temperature $\Delta t$,
$\ell=\ell_0(1+\alpha \Delta T)$
By Pythogoras theorem

$x^2 =\ell^2-\ell_0^2 $
$ =\ell_0^2(1+\alpha \Delta T)^2-\ell_0^2 $
$ \simeq \ell_0^2 2 \alpha \Delta T $
$ \text { or } x =\ell_0 \sqrt{2 \alpha \Delta T} $
$= \frac{3.77}{2} \sqrt{2\left(25 \times 10^{-6} \times 32\right)} $
$= 7.5 \times 10^{-2} m $