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 composite rod increases by `1.91 mm`. When the composite rod is prevented from expanding by holding it between two rigid walls, it is found that the constituent rods have remained unchanged in length inspite of rise of temperature. Find young's modulus and the coefficient of linear expansion of the second rod (Y of copper `=1.3xx10^(10) N//m^(2)` and `a` of copper `=17xx10^(-6)//K`).

Answer

`Y_(2) = 1.105 xx 10^(11) N//m^(2), alpha_(2) = 2 xx 10^(-5)//.^(@)C`

Explanation

`Deltal = 1.91 mm = Deltal_(1) +Deltal_(2)`
`rArr 0.191 cm = l_(1) alpha_(1) Delta theta +l_(1) alpha_(2) Delta theta`.
`rArr 0.191 = (30 xx 17 xx 10^(-6) +70 alpha_(2)) xx 100`
`alpha_(2) = 2 xx 10^(-5)`
`Y = (F)/(A alpha Delta theta)`
`F_(1) = F_(2)`
`Y_(1)A alpha_(1)Delta theta = Y_(2) A alpha_(2) Delta theta`.
`:. Y_(2) = (Y_(1)alpha_(1))/(alpha_(2)) = (1.3 xx 10^(11) xx 1.7 xx 10^(-5))/(2 xx 10^(-5))`
`= 1.105 xx 10^(11) N//m^(2)`