Question

A steel rod ($density=8000kg/m^3$) has a length of 2.0 m. It is bolted at both ends between immobile supports. Initially there is no tension in the rod because the rod just fits between the supports. If the tension is $100x\left(inN\right)$ that develops when the rod loses 3600J of heat, then find the value of x. Take specific heat capacity of steel as $450J/k{g}^{\circ }C$, coefficient of linear expansion $={10}^{-5}{/}^{\circ }C$, Young's modulus $=2\times {10}^{11}N/m^2$

Answer 1

Given : $Y=2\times {10}^{11}N/m^2$ $\alpha ={10}^{-5}{/}^{o}C$ $S=450J/k{g}^{o}C$ $L=2m$

Let the cross sectional area of the rod be $A$

Using $Q=mS\Delta T$ where $m=\rho AL$

$\therefore$ $Q=\left(\rho AL\right)S\Delta T$

OR $3600=8000\left(2A\right)\left(450\right)\Delta T$ $⟹A\Delta T=5\times {10}^{-4}$

Also $\Delta L=L\alpha \Delta T$

OR $\Delta L=\frac{2\left({10}^{-5}\right)\times 5\times {10}^{-4}}{A}$ $⟹A\Delta L={10}^{-8}$ m

Using $Y=\frac{TL}{A\Delta L}$

$\therefore$ $2\times {10}^{11}=\frac{T\times 2}{{10}^{-8}}$ $⟹T=1000$ N