Question

A steel scale is correct at $0^{0}C.$ The length of a brass tube measured by it at 40${}^{0}C.$ is $5m$. The correct length of the tube at $0^{0}C$ is ( coefficients of linear expansion of steel and brass are $11\times {10}^{-6}{/}^{0}C$ and $19\times {10}^{-6}{/}^{0}C$)

Answer 1

Let correct length of brass rod be $L_o$ at $0^{o}C$.

Linear expansion coefficients:

Steel $={\propto }_1=11\times {10}^{-6}{K}^{-1}$

Brass $={\propto }_2=19\times {10}^{-6}{K}^{-1}$

at ${40}^{o}C$, length of brass road will be:

$L=L_o(1+{\propto }_2\Delta T)$ where $\Delta T=40K$

$\Rightarrow L=L_o(1+40{\propto }_2)$ ....$\left(i\right)$

also, the steel rod initially showing $5m$ reading at $0^{o}C$ will expand to give incorrect reading of $L$ at ${40}^{o}C$.

$L=5(1+{\propto }_1\Delta T)$ [Initial reading of $5$ expands]

$\Rightarrow L=5(1+40{\propto }_1)$ ...$\left(2\right)$

From $\left(1\right)$ & $\left(2\right)$

$L_o=\frac{5(1+40{\propto }_1)}{(1+40{\propto }_2)}=\frac{5(L+40\times 11\times {10}^{-6})}{(1+19\times {10}^{-6}\times 40)}$

$\Rightarrow L_0\simeq 4.9984m$