Two rods $A$ and $B$ of identical dimensions are at temperature $30^{\circ} \mathrm{C}$. If $\mathrm{A}$ is…
Two rods $A$ and $B$ of identical dimensions are at temperature $30^{\circ} \mathrm{C}$. If $\mathrm{A}$ is heated upto $180^{\circ} \mathrm{C}$ and $\mathrm{B}$ upto $\mathrm{T}^{\circ} \mathrm{C},$ then the new lengths are the same. If the ratio of the coefficients of linear expansion of $\mathrm{A}$ and $\mathrm{B}$ is $4: 3,$ then the value of $T$ is
$230^{\circ} \mathrm{C}$
$270^{\circ} \mathrm{C}$
$200^{\circ} \mathrm{C}$
$250^{\circ} \mathrm{C}$
Solution
Change in length in both rods are same i.e.
$\Delta \ell_{1}=\Delta \ell_{2}$
$\ell \alpha_{1} \Delta \theta_{1}=\ell \alpha_{2} \Delta \theta_{2}$
$\begin{array}{l}
\frac{\alpha_{1}}{\alpha_{2}}=\frac{\Delta \theta_{2}}{\Delta \theta_{1}} \quad\left[\because \frac{\alpha_{1}}{\alpha_{2}}=\frac{4}{3}\right] \\
\frac{4}{3}=\frac{\theta-30}{180-30}
\end{array}$
$\theta=230^{\circ} \mathrm{C}$