A metal rod A of 50 cm length expands by 0.10 cm when its temperature is raised from `0^(@)C" to "100^(@)C`. Another rod B of a different metal of length 60 cm expands by 0.06 cm for the same rise in temperature. A third rod C of 50 cm length is made up of pieces of rods A and B placed end to end expands by 0.03 cm on heating from `0^(@)C" to "50^(@)C`. Find the length of each portion of composite rod C.

Explanation

From the data for rod A, we have `DeltaL=alpha_(A)LDeltaT`
or `alpha_(A)=(DeltaL)/(LDeltaT)=(0.10)/(50xx100)=2xx10^(-5)""^(@)C^(-1)`
For rod B, we have `DeltaL=a_(B)LDeltaT`
or `alpha_(B)=(DeltaL)/(LDeltaT)=0.06/(60xx100)=10^(-5)""^(@)C^(-1)`
If rod C is made of segments of rod A and B of lengths `l_(1)andl_(2)` respectively then we have at `0^(@)C`.
`l_(1)+l_(2)=50cm" "...(a)`
At `T=50^(@)C" "l_(1).+l_(2).=50.03cm`
Thus `alpha_(A)l_(1)DeltaT+alpha_(B)l_(2)DeltaT=0.03cm`
or `2xx10^(-5)xxl_(1)xx50+10^(-5)xxl_(2)xx50=0.03cm`
or `2l_(1)+l_(2)=0.03/50xx10^(5)=60cm" "...(b)`
Solving (a) and (b) we get `l_(1)=10cmandl_(2)=40cm`