Question

Three rods of equal lengths are joined to form an equilateral triangle ABC. D is the of AB. The coefficient of linear expansion is ${\alpha }_1$ for material of rod AB and ${\alpha }_2$ for n rods AC and BC. It the distance DC remains constant for small changes in temperature.

Answer 1

$D{C}^2=A{C}^2-A{D}^2$

$={\ell }^2-{\left(\frac{\ell }{2}\right)}^2$

$={\left[\ell \left(1+{\propto }_{2}t\right)\right]}^2-{\left[\frac{\ell }{2}\left(1+{\propto }_{1}t\right)\right]}^2$

${\ell }^2-\frac{{\ell }^2}{4}={\ell }^2\left(1+{\propto }_2^2{t}^2+2{\propto }_{2}t\right)-\frac{{\ell }^2}{4}\left(1+{\propto }_1^2{t}^2+2{\propto }_{2}t\right)$

$0={\ell }^2\left(2{\propto }_{2}t\right)-\frac{{\ell }^2}{4}\left(2{\propto }_{1}t\right)$

$2{\propto }_2=2\frac{{\propto }_1}{4},{\propto }_1=4{\propto }_2$