Question

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

Answer 1

Length of the steel tape at temperature $T={27}^{o}C,l=1m=100cm$

At temperature $T_1={45}^{o}C$, the length of the steel rod, $l_1$ = 63 cm

Coefficient of linear expansion of steel, $\alpha =1.2\times {10}^{-5}K$

Let $l$ be the actual length of the steel rod and l' be the length of the steel tape at ${45}^{o}C$.
$l^{\prime }-l=l\alpha (T_1-T)$

$l^{\prime }=100(1+1.20\times {10}^{-5}\times (45-27\left)\right)=100.0216cm$

Hence, the actual length of the steel rod measured by the steel tape at ${45}^{0}C$ can be calculated as:

$l_2=\left(100.0216/100\right)63=63.0136cm$
Therefore, the actual length of the rod at ${45}^{o}C$ is 63.0136 cm. Its length at ${27}^{o}C$ is $63.0cm.$