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Two rods of equal cross-sections, one of...

Two rods of equal cross-sections, one of copper and the other of steel are joined to from a composite rod of length `2.0m` at `20^(@)C` the length of the copper rod is `0.5m`. When the tempertuare is raised to `120^(@)C`, the length of composite rod increases to `2.002m`. If the composite rod is fixed between two rigid walls and thus not allowed to expand, it is foundthat the length fo the component rod also do not change with increase in temperature. Calcualte the Young's modulus of steel. Given Young's modulus of copper `= 1.3xx10^(11) N//m^(2)` the coefficent of linear expansion of copper `alpha_(C) = 1.6xx10^(-5)//.^(@)c`

A

`2.6xx10^(11)Pa`

B

`1.6xx10^(10)Pa`

C

`1.3xx10^(10)Pa`

D

`0.9xx10^(10)Pa`

Text Solution

Verified by Experts

The correct Answer is:
A
The change in length of copper rod due to change in temprature from `20^(@)C` to `120^(@)C`.
`Delta l_(1) = l_(C) Delta T, 0.5 alpha_(C) (120-20) = 50alpha_(C)`
For steel,
`Delta l_(2) = l_(S) alpha_(S) Delta T = 1.5 alpha_(S) (120 - 20) = 150 alpha_(S)`
Total change in length,
`Delta l = Delta l_(1) + Delta l_(2) = 50 alpha_(C) + 150 alpha_(S)`
It is given that `Delta l = 0.002m`
`50alpha_(C) + 150 alpha_(C) = 0.002` or, `alpha_(S) = (0.002-50alpha_(C))/(150)`
`= (0.002-50xx1.6xx10^(-5))/(150) = 0.8xx10^(-5) /.^(@) c`
`F = YA alpha Delta Y, Y_(S) alpha_(S) = Y_(C) alpha_(C)`
`Y_(S) = 2.6xx10^(-11) N//m^(2)`
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