A steel tape 1 m long is correctly cailbrated for 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 termperature is `45.0^(@)C`. What is the actual length of the steel rod on that day? What is the length of the same steel rood on a day when the temper ature is `27.0^(@)C` ? `alpha` of steel is `1.20xx10^(-5)""^(@)C^(-1)` ?

Explanation

Given length of the tape =1 m at `27^(@)C`
Length of the tape at `45.0^(@)C`
`l_(2)=1+alphal_(1)(theta_(2)-theta_(1))=1+0.000012xx1xx(45-27)`
`=1+0.0000216=1.0000216m`
Actual length of the steel rod `=(63)/(1.000216)`
`l_(2)=62.986cm`.
We know that
`alpha=(l_(2)-l_(1))/(l_(1)t_(2)-l_(2)t_(1))`
`(0.000012){l_(1)(45)-62.986xx27}=62.986-l_(1)`
`0.00054l_(1)-0.02040=62.986-l_(1)`
`63.0064=1.00054l_(1)`
`:.l_(1)=(63.0064)/(1.00054)=62.972cm`
Initial length of steel rod at `27^(@)C=62.972cm`.