Two rods of length $$L2$$ and coefficient of linear expansion α$$2$$ are connected freely to a third rod of length $$L1$$ of coefficient of linear expansion α$$1$$ to form an isosceles triangle. The arrangement is supported on the knife edge at the midpoint of $$L1$$ which is horizontal. The apex of the isosceles triangle is to remain at a constant distance from the knife edge if

Question

Two rods of length $$L2$$ and coefficient of linear expansion α$$2$$ are connected freely to a third rod of length $$L1$$ of coefficient of linear expansion α$$1$$ to form an isosceles triangle. The arrangement is supported on the knife edge at the midpoint of $$L1$$ which is horizontal. The apex of the isosceles triangle is to remain at a constant distance from the knife edge if

Infinity Learn · Accepted Answer

The apex of the isosceles triangle to remain at a constant distance from the knife edge.  Thus, DC should remain constant before and after heating. Before expansion: In triangle $$ADC(DC)2=L22-L122$$          (i)After$$ expansion:$$(DC)2=$$[$$L2(1+$$α$$2t)]$$2-L12(1+$$α$$1t)2Equating$$ Eqs. (i) and (ii), we $$getL22-L122=$$[$$L2(1+$$α$$2t)$$]$$2-L12(1+$$α$$1t)2$$ $$L22-L124=L22+L22$$×$$2$$α$$2$$×$$t-L124-L124$$×$$2$$α$$1$$×t (Neglecting$$ higher $$terms)⇒$$L124(2$$α$$1t)=L22$$ (2$$α$$2t)      ⇒$$L1L2=2$$α$$2$$α$$1$$

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