Two wires A and B of the same cross sectional areas, Young's moduli $$Y1$$ and $$Y2$$ and coefficients of linear expansion α$$1$$ and α$$2$$ respectively are joined together and fixed between rigid supports at either ends. The tension in the compound wire when wire A is heated and wire B is cooled at different temperature is same when wire A alone is cooled at the same temperature as wire B earlier. Find correct option.

Question

Two wires A and B of the same cross sectional areas, Young's moduli $$Y1$$ and $$Y2$$ and coefficients of linear expansion α$$1$$ and α$$2$$ respectively are joined together and fixed between rigid supports at either ends. The tension in the compound wire when wire A is heated and wire B is cooled at different temperature is same when wire A alone is cooled at the same temperature as wire B earlier. Find correct option.

Infinity Learn · Accepted Answer

Tension in any wire, F $$=$$ YA αtGiven $$Y2A2$$α$$2t2-Y1A1$$α$$1t1$$ $$=$$ $$Y1A1$$α$$1t2$$         $$Y2A2$$α$$2t2$$ $$=$$ $$Y1A1$$α$$1(t1+t2)$$          $$(t1+t2)t2$$ $$=$$ $$Y2A2$$α$$2Y1A1$$α$$1or$$       $$t1-t2t2$$ $$=$$ $$Y2A2$$α$$2-2Y1A1$$α$$1Y1A1$$α$$1$$          ∆$$tt2$$ $$=$$ $$Y2$$α$$2-2Y1$$α$$1Y1$$α$$1$$   $$(since$$ $$A1$$ $$=$$ $$A2)For$$ ∆t > $$0Y2$$α$$2$$ > $$2Y1$$α$$1or$$ $$Y22Y1$$>α$$1$$α$$2$$

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