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Two wire A and B of the same corss secti...

Two wire `A` and `B` of the same corss sectional area, young's modulli `Y_(1), Y_(2)` and coefficients of linear expansion `alpha_(1), alpha_(2)` respectively are joined together and fixed between rigid supports at either ends. The tension in the compound wire when the wire `A` is heated and Wire `B` is cooled at different temperature is same when wire `A` alone in cooled at same temperature as wire `B` earller. the correct option is

A

`(alpha_(1))/(alpha_(2)) gt (Y_(2))/(2Y_(1))`

B

`(alpha_(1))/(alpha_(2)) lt (Y_(2))/(2Y_(1))`

C

`(alpha_(1))/(alpha_(2)) gt (2Y_(2))/(Y_(1))`

D

`(alpha_(1))/(alpha_(2)) gt (Y_(2))/(Y_(1))`

Text Solution

Verified by Experts

The correct Answer is:
B
`T = Y Apropt, Y_(2)A_(2)alpha_(2)t_(2) - Y_(1)A_(1)alpha_(1)t_(1) = Y_(1)A_(1)alpha_(1)t_(2)`
`Y_(2)A_(2)alpha_(2)t_(2) = Y_(1)A_(1)alpha_(1)t_(2) = Y_(1)A_(1)alpha_(1)(t_(1) + t_(2))`
`(t_(1) + t_(2))/(t_(2)) = (Y_(2)A_(2)alpha_(2))/(Y_(1)A_(1)alpha_(1)) = (t_(1) - t_(2))/(t_(2)) = (Y_(2)A_(2)alpha_(2) - 2YA_(1)alpha_(1))/(Y_(1)alpha_(1)alpha_(1))`
`(Deltat)/(t_(2)) = (Y_(2)alpha_(2) - 2Y_(1)alpha_(1))/(Y_(1)alpha_(1))`
for `Deltat = +ve, Y_(2)alpha_(2) - 2Y_(1)alpha_(1) gt 0(Y_(2))/(2Y_(1)) gt (alpha_(1))/(alpha_(2))`
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