Question

A steel scale measures the length of a copper wire as $80.0\text{cm}$, when both are at ${20}^{\circ }C$ (the calibration temperature for scale). What would be the scale reading for the length of wire, when both are at ${40}^{\circ }C$?
(Given ${\alpha }_{\text{steel}}=11\times {10}^{-6}$ per ${}^{\circ }C$ and ${\alpha }_{\text{copper}}=17\times {10}^{-6}$ per ${}^{\circ }C)$

• A. $80.0096\text{cm}$
• B. $80.0272\text{cm}$
• C. $1\text{cm}$
• D. $25.2\text{cm}$

Answer 1

The correct option is A $80.0096\text{cm}$

Linear expansion for copper
$L_c=L(1+{\alpha }_c\Delta T)...\left(1\right)$
for steel $L_s=L(1+{\alpha }_s\Delta T)...\left(2\right)$
Apparent change in length of the wire $\Delta L^{\prime }=L_c-L_s$
From equation (1) and (2),
$=L[{\alpha }_c-{\alpha }_s]\Delta T$
$=80(17\times {10}^{-6}-11\times {10}^{-6})\times 20=0.0096\text{cm}$
Hence, length of wire is measured as $80.0096\text{cm}$