A uniform metal bar of length \(10 \mathrm{~m}\) with a crack at its midpoint is clamped between two rigid…
A uniform metal bar of length \(10 \mathrm{~m}\) with a crack at its midpoint is clamped between two rigid supports. The bar buckles upward due to temperature rise of \(40^{\circ} \mathrm{C}\). If the coefficient of linear expansion of the metal is \(2.5 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\), the maximum displacement of the mid-point of the bar is
\(11.3 \mathrm{~cm}\)
\(22.3 \mathrm{~cm}\)
\(33.3 \mathrm{~cm}\)
\(44.3 \mathrm{~cm}\)
Solution
Given, length of bar \(L=10 \mathrm{~m}\), rise in temperature, \(\Delta T=40^{\circ} \mathrm{C}\) and coefficient of linear expansion, \(\alpha=25 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\)
As, linear expansion,
\(\begin{aligned}
\Delta L & =L \alpha \Delta t=10 \times 2.5 \times 10^{-6} \times 40 \\
\Rightarrow \Delta L & =0.1 \mathrm{~cm}
\end{aligned}\)
In the figure below, let the displacement of mid point of bar is \(x\),
Now, from \(\triangle O B A\),
\(\begin{aligned}
x & =\sqrt{(A B)^2-(O A)^2} \\
\Rightarrow x & =\sqrt{\left(\frac{10+0.001}{2}\right)^2-(5)^2} \\
\Rightarrow x & =0.223 \mathrm{~m}
\end{aligned}\)
\(\text{Hence, } x=22.3 \mathrm{~cm}\)
Hence, the correct option is (b).
