As shown in the figure , an equilateral triangle ABC is formed by joining three rods of equal lengths and D is the midpoint of AB. Coefficient of linear expansion of the material of `AB is alpha_1` and that of AC and BC is `alpha_2` . If the length DC remains constant for small changes in temperature , then

Options

A. `alpha_1=alpha_2`
B. `alpha_1=4alpha_2`
C. `alpha_2=4 alpha_1`
D. `alpha_1=(alpha_2)/(2)`

Answer

B

Explanation

For a change of temperature by t, change in length DC is `Delta DC,` then `Delta C^2= Delta AC^2- Delta AD^2`
`= l(1+ alpha_2t)^(2)- (1/2(1+ alpha_2t))^2`
`= l^2(2 alpha_2t) - l^2/4 (2 alpha_1 t)`
(After neglecting terms `alpha_(2)^(2)t^(2)` and `alpha_(1)^(2)t^2`, being very small
As, `Delta DC=0 rArr l^2 2 alpha_(2)t - (l^2)/(4)(2 alpha_1 t)=0`
`rArr alpha_(1) =4 alpha_(2)`