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Length of a metal rod at temperature 27^...

Length of a metal rod at temperature `27^@ `C is 4.256m. Find the temperature at which the length of the same rod increases to 4.268m. (`alpha` for iron = `1.2 xx 10^(-5) K^(-1)`)

Text Solution

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Given
` T_1 = 27^@ C , l_1 =4.256 m,`
`l_2 =4.268 m , alpha =1.2 xx 10^(-5) K^(-1)`
we have
`alpha = (l_2 -l_1)/( l_1 (T_2-T_1)) = (l_2 -l_1 )/(l_1T_2-l_1 T_1)`
` therefore l_1 T_2 =l_1 T_1 +(l_2 -l_1 )/(alpha)`
` T_2 =(1)/(l_1) [ l_1 T_1 +(l_2 -l_1)/( alpha)]`
` = (1)/(4.256 ) [ ( 4.256 xx 27 ) + (4.268 -4.256 )/(1.2 xx 10^(-5))]`
`= (1)/(4.256 ) [ 114.912 + (0.012)/(1.2 xx 10^(-5))]`
` = ( 1)/(4.256 ) [ 114.912+1000]`
`=261.96^@ C`
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