A wire is made by attaching two segments together end to end. One segment is made of aluminium and the other is steel. The effective coefficient of linear expansion of the two segment wire is `19xx10^(-6)//^(@)C`. The fraction length of aluminium is (linear coefficient of thermal expansion of aluminium and steel are `23xx10^(-6).^(@)C` and `12xx10^(-6)//^(@)C`,

Options

A. `(5)/(11)`
B. `(6)/(11)`
C. `(7)/(11)`
D. `(8)/(11)`

Answer

C

Explanation

The change in length of wire`=l_(Al)alpha_(Al)Deltatheta+l_(st)alpha_(st)Deltatheta` associated with temperature change of `Deltatheta`, where `alpha_(Al)` and `alpha_(St)` are the coefficient of linear expansion of aluminium and steel, respectively
`alpha_(Al)=23xx10^(-6)//^(@)C`
`alpha_(st)=12xx10^(-6)//^(@)C`
The effective coefficient of linear expansion of the two segments of wire`=19xx10^(-6)//^(@)C`
`l_1alpha_(Al)Deltatheta+l_2alpha_(st)Deltatheta=(l_1+l_2)alphaDeltatheta`
`(l_1)/((l_1+l_2))=(alpha-(l_2)/((l_1+l_2))alpha_(st))/(alpha_(Al))`
`[(l_1)/(l_1+l_2)=xl_1+l_2=l((l_2)/(l_1+l_2))=1-x]`
`x=(alpha-(1-x)alpha_(st))/(alpha_(Al))`
`x=(alpha-alpha_(st))/(alpha_(Al)-alpha_(st))=(19xx10^-6-12xx10^-6)/(23xx10^-6-12xx10^-6)=(7)/(11)`