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 $63.0\text{cm}$ 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 rod 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

Step 1: Find length of steel tape at ${45}^{\circ }C$

Given, Length of the steel tape at temperature, $T={27}^{\circ }C$

$l=1m=100\text{cm}$

Coefficient of linear expansion of steel,

$\alpha =1.20\times {10}^{-5}{K}^{-1}$

Let $l_2$ be the length of the steel tape at ${45}^{\circ }C$, then from the formula $l$

Step 2: Find actual length of steel tape.

If actual length of steel rod at ${45}^{\circ }C\text{is}L$,

$L=\frac{100.0216}{100}\times 63$

$=63.0136\text{cm}$

Therefore, actual length of the rod at ${45}^{\circ }C\text{is}63.0316\text{cm}.$

Since the tape is calibrated at ${27}^{\circ }C$, therefore length of steel rod at ${27}^{\circ }C$ is same i.e., $63\text{cm}.$