A steel tape `1 m` 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^(@)C`. Coefficient of linear expansion of steel `= 1.20 xx 10^(-5) //K`. what is the actual length of the steel rod on that day?

Options

A. `63.0136 cm`
B. `63.2134 cm`
C. `63.1526 cm`
D. `63.3136 cm`

Answer

A

Explanation

The steel tape gives correct reading only at the temperature `27^(@)C` at which it has been calibrated. At any other temperature `45^(@)C` the scale will expand and give less reading than the true value.
Hence length of the steel rod at `27^(@)C`, i.e., `L = 63 cm`
Let `DeltaL` be the increase in the length of the steel tape when temperature rises from `27^(@)C ` to `45^(@)C` , ie., `Delta T = 45^(@)C - 27^(@)C = 18^(@)C = 18K`.
Clearly , `Delta L = alpha L Delta T`
`=(1.20xx10^(-5)//K) xx (63cm) xx 18K = 0.0136 cm`
Actual length of the rod at `45^(@)C`
`=63 cm + 0.0136 cm = 63.0136 cm`.