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 $45^{\circ }C$. Coefficient of linear expansion of steel $=1.20\times 10^{-5}/K$. What is the actual length of the steel rod on that day?

• A. 63.0136 cm
• B. 63.2134 cm
• C. 63.1526 cm
• D. 63.3136 cm

Answer 1

Answer: (A)

The steel tape gives correct reading only at the temperature $27^{\circ }C$ at which it has been calibrated. At any other temperature $45^{\circ }C$ the scale will expand and give less reading than the true value.
Hence length of the steel rod at $27^{\circ }C$, i.e., $L=63cm$
Let $\Delta L$ be the increase in the length of the steel tape when temperature rises from $27^{\circ }C$ to $45^{\circ }C$,
i.e., $\Delta T=45^{\circ }C-27^{\circ }C=18^{\circ }C=18K$
Clearly, $\Delta L=\alpha \cdot L\Delta T$
$=\left(1.20\times 10^{-5}/K\right)\times (63cm)\times 18K=0.0136cm$
Actual length of the rod at $45^{\circ }C$
$=63cm+0.0136cm=63.0136cm$