Question

A rod has actual length at ${20}^{\circ }\text{C}$. Find the highest possible temperature at which the length should be measured, so that the error in measurement do not exceed $0.01\%$. The thermal coefficient of linear expansion is $2\times {10}^{-5}{/}^{\circ }\text{C}$

• A. ${20}^{\circ }\text{C}$
• B. ${25}^{\circ }\text{C}$
• C. ${30}^{\circ }\text{C}$
• D. ${35}^{\circ }\text{C}$

Answer 1

The correct option is B ${25}^{\circ }\text{C}$

Let the length of the rod be ${}^{\prime }{L}^{\prime }$
The maximum possible reading $=L+\frac{0.01}{100}L$
$=L+0.0001L$
$=1.0001L$
Using formula for total length
$L_{\text{net}}=L_0(1+\alpha \Delta T)$
$\Rightarrow 1.0001L=L[1+2\times {10}^{-5}(T-20\left)\right]$
$\Rightarrow 1.0001=1+2T\times {10}^{-5}-4\times {10}^{-4}$
$\Rightarrow 1.0001=1-0.0004+2T\times {10}^{-5}$
$\Rightarrow 1.0001=0.9996+2T\times {10}^{-5}$
$\Rightarrow 0.0005=2T\times {10}^{-5}$
$\Rightarrow 5\times {10}^{-4}=2T\times {10}^{-5}$
$\Rightarrow \frac{50}{2}=T$
$\Rightarrow T={25}^{\circ }\text{C}$ is the highest possible temperature at which the length should be measured