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

Increase in length during linear thermal expansion is given by ${l_2} = {l_1}\left( {1 + \alpha \Delta T} \right)$, where ${l_1}$ and ${l_2}$ are initial and final lengths, $\Delta T$ 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 ${l_2} = {l_1}\left( {1 + \alpha \Delta T} \right)$ where, ${l_1}$ and ${l_2}$ are initial and final lengths, $\Delta T$ is change in temperature and $\alpha$ is coefficient of linear expansion.
Here, $\Delta T = {T_1} - T$ i.e. $\Delta T = 45^\circ C - 27^\circ C = 18^\circ C$.
Now, substituting the values of ${l_1}$, $\Delta T$ and $\alpha$ in the equation ${l_2} = {l_1}\left( {1 + \alpha \Delta T} \right)$, 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 $45^\circ C = \dfrac{{100.0216}}{{100}} \times 63 = 63.0136cm$
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.