Question

A steel tape $1\, m$ 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 $63.0\, cm$ 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., $L=63\, cm$
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., $\Delta T= 45^{\circ} C -27^{\circ} C =18^{\circ} C =18\, K$
Clearly, $\Delta L=\alpha \cdot L \Delta T$
$=\left(1.20 \times 10^{-5} / K \right) \times(63\, cm ) \times 18\, K =0.0136\, cm$
Actual length of the rod at $45^{\circ} C$
$=63\, cm +0.0136\, cm =63.0136\, cm$