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.
Given,
Length of the steel tape at temperature T = 27°C = 27 + 273.15 K = 300.15 K, L = 1 m
Coefficient of linear expansion of steel, α = 1.20 × 10-5 K-1
At temperature T1 = 45°C = 45 + 273.15 K = 318.15 K, the length of the steel rod,
L1 = 63 cm = 0.63m
Let L2 be the actual length of the steel rod and L2’ be the length measured by the steel tape at 45°C.
We know,
L2’ = L + αL(T1 - T)
∴ L2’ = 1 m + (1.20 × 10-5 K-1)(1 m)(318.15 K – 300.15 K)
⇒ L2’ = 1 + 0.000216
⇒ L2’ = 1.000216 m = 100.0216 cm
Hence, the actual length of the steel rod at 45°C is given by
L2 = (L2’/L)×L1
⇒ L2 = [(100.0216 cm)/(100 cm)](63 cm) = 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.