Three rods of equal lengths are joined to form an equilateral triangle A B C . D is the mid-point of A B . The coefficient of linear expansion is ₁ for material of rod A B and ₂ for material of rods A C and B C . If the distance D C remains constant for small changes in temperature, then

Step-by-step solution

C D^2=(A C)^2-(A D)^2=l^2- ( l 2 )^2 After small change in temperature C D^2=(A C)^2-(A D)^2= [l (1+ ₂ t ) ]^2- [ l 2 (1+ ₁ t ) ]^2 l^2- l^2 4 =l^2 [1+ ₂^2 t^2+2 ₂ t ]- l^2 4 [1+ ₁^2 t^2+2 ₁ t ] Neglecting ₂^2 t^2 and ₁^2 t^2 , (because very small quantity) 0=l^2 (2 ₂ t )- l^2 4 (2 ₁ t ) or ₁=4 ₂