Question

A steel tape 1m long is correctly calibrated for a temperature of 27.0 °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 45.0 °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 °C ? Coefficient of linear expansion of steel = 1.20 × 10$^{-5}$ K$^{-1}$.

Answer 1

Length of the steel tape at temperature T = 27°C, l = 1 m = 100 cm
At temperature $T_1$ = 45°C, the length of the steel rod, $l_1$ = 63 cm
Coefficient of linear expansion of steel, α = 1.20 × 10$^{-5}$ K$^{-1}$
Let l2 be the actual length of the steel rod and l' be the length of the steel tape at 45°C.
l' = l + αl ($T_1$-T)
∴ l' = 100 + 1.20 x10$^{-5}$ x 100(45-27)
l' = 100.0216
Hence, the actual length of the steel rod measured by the steel tape at 45°C can be calculated as:

$l_2$ = $\frac{100.0216}{100}$ x 63 = 63.0136 cm

Therefore, the actual length of the rod at 45.0°C is 63.0136 cm. Its length at 27.0°C is 63.0 cm.