A metal rod (A) of `25 cm` length expands by `0.050 cm` when its temperature is raised from `0^@ C` to `100^@ C`. Another rod (B) of a different metal of length `40 cm` expands by `0.040 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 lengths of each portion of the composite rod.

Explanation

From the given data for rod A, we have
`DeltaL=alpha_(A)LDeltaT`
or `alpha_(A)=(DeltaL)/(L DeltaL)=0.05/(25xx100)`
`=2xx10^(-5).^(@)C^(-1)`
For rod B we have `DeltaL=alpha_(B)LDeltaT`
or `alpha_(B)=(DeltaL)/(LDeltaL)=0.04/(40xx100)`
`=10^(-5).^(@)C^(-1)`
If rod C is made of segments of rod A and B of lengths `l_(1)` and `l_(2)` respectively then we have at `.^(@)C`
`l_(1)+l_(2)=50 cm` ..........2
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 ` ..........1.22
Solving (1.21) and (1.22) we get `l_(1)=10 cm` and `l_(2)=40 cm`