An equilateral triangle ABC is formed by joining three rods of equal length and D is the mid-point of AB….

Q: An equilateral triangle ABC is formed by joining three rods of equal length and D is the mid-point of AB. The coefficient of linear expansion for AB is α1 and for AC and BC is α2. The relation between α1 and α2, if distance DC remains constant for small changes in temperature is

Numerical

(a) α1 = α2

(b) α1 = 4α2

(c) α2 = 4α1

(d) α1 = α2/2

Ans: (b)

Sol: Let CD = h , side of triangle = l

Since , CD2 = AC2 – AD2

$ \displaystyle h^2 = l^2 -\frac{l^2}{4} $

$ \displaystyle = [l(1+\alpha_2 \Delta \theta)]^2 -[\frac{l}{2}(1+\alpha_1\Delta \theta)]^2 $

$ \displaystyle = l^2(1 + 2\alpha_2 \Delta \theta)-\frac{l^2}{4}(1+2\alpha_1 \Delta \theta) $

$ \displaystyle l^2 -\frac{l^2}{4} = l^2 + l^2 . 2\alpha_2 \Delta \theta – \frac{l^2}{4} – \frac{l^2}{4}. 2\alpha_1 \Delta \theta $

$ \displaystyle \alpha_2 = \frac{\alpha_1}{4} $

$ \displaystyle \alpha_1 = 4 \alpha_2 $

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