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 length60 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 by0.03 cm on heating from 0^(@)Cto 50^(@)C. Find the length of each portion of composite rod C.

A metal rod A of 50 cm length expands by 0.10 cm when its temperature

1.	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 length60 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 by0.03 cm on heating from 0^(@)Cto 50^(@)C. Find the length of each portion of composite rod C.
Answer» Solution :From the data for rod A, we have `DeltaL=alpha_(A) L DeltaT or alpha_(A)=(DeltaL)/(L DeltaT)=(0.10)/(50xx100)=2xx10^(-5)""^(@)C^(-1)` For ord B, we have `DeltaL = alpha_(B) L DeltaT (or) alpha_(B)=(DeltaL)/(L DeltaT)=(0.06)/(60xx100)=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)=50cm ""...(a)` At `T=50^(@)C, l_(1).+l_(2).=50.03 cm` Thus `alpha_(A)l_(1)DeltaT+alpha_(B) l_(2)DeltaT=0.03cm (or) 2xx10^(-5) XX l_(1)xx50 +10^(-5) xx l_(2)xx50 = 0.03 cm` (or) `2l_(1)+l_(2)=(0.03)/(50)xx 10^(-5) =60cm "" ...(b)` Solving (a) and (b) we get `l_(1)=10 cm and l_(2)=40cm`