A composite rod is made by joining a copper rod, end to end, with a second rod of different material but of the same area of cross section. At `25^(@)C`, the composite rod is `1 m` long and the copper rod is `30 cm` long. At `125^(@)C` the length of the composte rod increases by `1.91 mm`. When the composite rod is prevented from expanding by bolding it between two rigid walls, it is found that the constituent reds have remained unchanged in length in splite of rise of temperature. Find yong's modulus and the coefficient of linear expansion of the second red (Y of copper `=1.3xx10^(10) N//m^(2)` and `a` of copper `=17xx10^(-6)//K`).

Explanation

Since total expansion = expansion of the constituent rods `1.91xx10^(-3)=0.3xx17xx10^(-6)xx(125-25)+0.7xxaxx(125-25)`
or `1.91xx10^(-3)=5.1xx10^(-4)+70a`
or `70a=(1.91-0.51)xx10^(-3)=1.4xx10^(-3)`
When prevented from expanding thermal expansion is accompanied by elastic contration. Since the lengths remain unchanged, thermal expansion is equal to elastic contraction.
`therefore "strain"=(taDeltat)/(I)=aDeltar=17xx10^(-6)xx100=17xx10^(-4)`
and strees`=Yxx "strain"=1.3xx10^(10)xx17xx10^(-4)=22.1xx10^(6)`
For the second rod
Strain`=aDeltat=2xx10^(-5)xx100=2xx10^(-3)`
Stress `=Yxx"strain"=Yxx2xx10^(-3)`
But the same stress is effective throughout the composite rod.
`therefore Yxx2xx10^(-3)=22.1xx10^(6)` or `Y=11.1xx10^(9) N..m^(2)`