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\(^{-1}\).
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\(^{-1}\).
Solution and Explanation
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.
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Concepts Used:
Thermal Expansion
Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. Temperature is a monotonic function of the average molecular kinetic energy of a substance.
The expansion of the solid material is taken to be the linear expansion coefficient, as the expansion takes place in terms of height, thickness and length. The gaseous and liquid expansion takes the volume expansion coefficient. Normally, if the material is fluid, we can explain the changes in terms of volume change.
The bonding force among the molecules and atoms differs from material to material. These characteristics of the compounds and elements are known as the expansion coefficient.
