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 α_{1} and for AC and BC is α_{2}. The relation between α_{1} and α_{2}, if distance DC remains constant for small changes in temperature is

(a) α_{1} = α_{2}

(b) α_{1} = 4α_{2}

(c) α_{2} = 4α_{1}

(d) α_{1} = α_{2}/2

Ans: (b)

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

Since , CD^{2} = AC^{2} – AD^{2}

$ \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 $