Q. A steel tape measures the length of a copper rod as 90.0 cm, when both are at 10$℃$, the calibration temperature, for the tape. What would be tape read for the length of the rod when both are at 30$℃$.Given, $\alpha\ $for steel$\ 1.2 \times 10^{- 5}℃^{- 1}\ $and$\ \alpha\ $for copper is$\ 1.7 \times 10^{- 5}℃^{- 1}.$
A) 90.01 cm
B) 89.90 cm
C) 90.22 cm
D) 89.80 cm
Explanation
(a) As the steel tape is calibrated at $10℃$, therefore, adjacent centimeter marks on the steel tape will be separated by a distance of
$l_{t} = l_{10}\left( 1 + \ \alpha_{s}\mathrm{\Delta}T \right) = (1 + \alpha_{s}20)$ cm
Length of copper rod at 30$℃$
= $90\left( 1 + \alpha_{c}20 \right)$cm
Therefore, number of centimeters read on the tape will be
= $\frac{90\left( 1 + \alpha_{c}20 \right)}{1\left( 1 + \alpha_{s}20 \right)} = \ \frac{90\left( 1 + 1.7 \times 10^{- 5} \times 20 \right)}{1\left( 1 + 1.2 \times 10^{- 5} \times 20 \right)}$
= $\frac{90 \times 1.00034}{1.00024} = 90.01$ cm
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.