Question

A steel tape 1m long is correctly calibrated for a temperature of ${27.0}^{\circ }C$. The length of a steel rod measured by this tape is found to be 63.0cm on a hot day when the temperature is ${45.0}^{\circ }C$. What is the actual length of the steel rod on that day? What is the length of the same steel road on a day when the temperature is ${27.0}^{\circ }C$ ? Coefficient of linear expansion of steel $=1.20\times {10}^{-5}{K}^{-1}$.

Answer 1

Hint: Increase in length during linear thermal expansion is given by $l_2=l_1\left(1+\alpha \mathrm{\Delta }T\right)$, where $l_1$ and $l_2$ are initial and final lengths, $\mathrm{\Delta }T$ is change in temperature and $\alpha$ is coefficient of linear expansion.
Actual length of the rod is determined after calculating the length of steel tape at ${45.0}^{\circ }C$.

Complete step by step solution:
As given in the question,
Length of the steel tape at temperature $T={27}^{\circ }C$ is $l_1=1m=100cm$
At temperature $T_1={45}^{\circ }C$, the length of the steel rod, $l=63cm$
Coefficient of linear expansion of steel, $\alpha =1.2\times {10}^{-5}{K}^{-1}$
Let $l_1$ be the actual length of the steel tape and $l_2$ be the length of the steel tape at ${45}^{\circ }C$
Now, as we know that increase in length during linear thermal expansion is given by $l_2=l_1\left(1+\alpha \mathrm{\Delta }T\right)$ where, $l_1$ and $l_2$ are initial and final lengths, $\mathrm{\Delta }T$ is change in temperature and $\alpha$ is coefficient of linear expansion.
Here, $\mathrm{\Delta }T=T_1-T$ i.e. $\mathrm{\Delta }T={45}^{\circ }C-{27}^{\circ }C={18}^{\circ }C$.
Now, substituting the values of $l_1$, $\mathrm{\Delta }T$ and $\alpha$ in the equation $l_2=l_1\left(1+\alpha \mathrm{\Delta }T\right)$, we get,
$l_2=100\left(1+1.2\times {10}^{-5}\times 18\right)=100.0216cm$
Length of $1cm$ mark at ${27}^{\circ }C$ on this scale, at ${45}^{\circ }C=100.0216cm$
Length of $63cm$ measured by this tape at ${45}^{\circ }C={\displaystyle \frac{100.0216}{100}}\times 63=63.0136cm$
Hence, the actual length of the steel rod on that day was $63.0136cm$.

$\therefore$ The length of the same steel rod on a day when the temperature is ${27.0}^{\circ }C=63\times 1=63cm$, as the steel tape is correctly calibrated for a temperature of ${27.0}^{\circ }C$.

Note:
Be careful while calculating the temperature difference. Always subtract initial temperature from the final temperature to avoid the mistake of sign change.