Three rods of equal of length are joined to from an equilateral triangle ABC. `D` is the midpoint of AB. The coefficient of linear expansion is `alpha_(1)` for AB and `alpha_(2)` for `AC` and `BC` . If the distance `DC` remains constant for small changes in temperature,

Options

A. `alpha_2 3 alpha_1`
B. `alpha_2 = 4 alpha_1`
C. `alpha_1 = 3 alpha_2`
D. `alpha_1 = 4 alpha_2`

Answer

D

Explanation

`(OR)^(2) = (PR)^(2) - (OP)^(2)`
` = I^(2) - (1/2)^(2) = [l(1 + alpha_2t)]^(2)- [1/2 (1 + alpha_1 t)]^(2)`
`rArr l^(2) - (l^2)/(4) = l^(2)(1 + alpha_(2)^(2) t^(2) + 2alpha_(2) t) - (l^2)/(4) [ 1 + alpha_(1)^(2) t^(2) + 2alpha_(1)t)]`
`alpha_(2)^(2) t^(2),alpha_(1)^(2)t^(2) rarr` neglected
`0 l^(2)(2alpha_2 t) - (l^2)/(4) (2 alpha_1 t)`
`rArr 2 alpha_2 - (2 alpha_1)/(4) rArr alpha_1 - 4 alpha_2`