Question

A rod is measured to be of length $50\text{cm}$ at ${10}^{\circ }\text{C}$. Its length measured at ${50}^{\circ }\text{C}$ is $x$. Find $x$, if thermal coefficient of linear expansion of the rod is $2\times {10}^{-5}{/}^{\circ }\text{C}$ and of the scale is $3\times {10}^{-5}{/}^{\circ }\text{C}$.

• A. $49.98\text{cm}$
• B. $49.99\text{cm}$
• C. $50.01\text{cm}$
• D. $50.02\text{cm}$

Answer 1

The correct option is A $49.98\text{cm}$

Here $L_0=50\text{cm}$
${\alpha }_0=2\times {10}^{-5}{/}^{\circ }\text{C}$ [of rod]
${\alpha }_s=3\times {10}^{-5}{/}^{\circ }\text{C}$ [of scale]
$T_0={10}^{\circ }\text{C}$
$T={50}^{\circ }\text{C}$
Using formula,
Measured length of rod at temperature $T$ is :
$x=L_0[1+({\alpha }_0-{\alpha }_s\left)\right(T-T_0\left)\right]$
$x=50[1+(2\times {10}^{-5}-3\times {10}^{-5}\left)\right(50-10\left)\right]$
$x=50[1+(-{10}^{-5}\left)\right(40\left)\right]$
$x=50[1-4\times {10}^{-4}]$
$x=50[1-0.0004]$
$x=50\left(0.9996\right)$
$\therefore x=49.98\text{cm}$