An isosceles triangle is formed with a thin rod of length $l_{1}$ and coefficient of linear expansion $α_{1}$ , as the base and two thin rods each of lenglh /, and coefficient of linear expansion $α_{2}$ , as the two sides. The distance between the apex and the midpoint of the base remain unchanged as the temperature is varied. The ratio of lengths $\frac{l_{1}}{l_{2}}$ is

Question

Infinity Learn · Accepted Answer

The distance between the apex and the midpoint of the base, using Pythagoras theorem l $$=$$ (l2)2-(l12)2or$$ $$l2$$ $$=$$ $$(l2)2-(l12)2------------(i)Differentiating$$ $$(i) w.r.t. temperature $$0$$ $$=$$ $$2l2$$×$$dl2dT-2(l12).12$$×$$dl1dTl12$$×$$l1$$α$$1$$ $$=2l2$$×$$l2$$α$$2(l1l2)2$$ $$=$$ $$4$$α$$2$$α$$1$$    ⇒ $$l1l2$$ $$=$$ $$2$$α$$2$$α$$1$$

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