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 $3.77m$ 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

Answer: 7.5

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

$x^2={\ell }^2-{\ell }_0^2$
$={\ell }_0^2(1+\alpha \Delta T{)}^2-{\ell }_0^2$
$\simeq {\ell }_0^{2}2\alpha \Delta T$
$\text{ or }x={\ell }_0\sqrt{2\alpha \Delta T}$
$=\frac{3.77}{2}\sqrt{2\left(25\times 10^{-6}\times 32\right)}$
$=7.5\times 10^{-2}m$