Three rods of equal lengths are joined to form an equilateral triangle ABC. D is the mid-point of AB. The coefficient of linear expansion is `alpha_1`, for material of rod AB and `alpha_2`, for material of rods of rods AC and BC. If the distance DC remains constant for small changes in temperature, then

Options

A. `alpha_1=2alpha_2`
B. `alpha_1=4alpha_2`
C. `alpha_1=8alpha_2`
D. `alpha_1=alpha_2`

Answer

B

Explanation

`(CD)^(2)=(AC)^(2) - (AD)^(2)`
`=l^(2) - [1/2]^(2)`
After temperature variation,
`(CD)^(2)=(AC)^(2) - (AD)^(2)`
`implies l^(2) - (l/2)^(2)=[l(1+alpha_(2)t)]^(2) - [1/2 l(1+ alpha_(2)t)]^(2)`
`therefore l^(2) -l^(2)/4 =l^(2)[1 + alpha_(2)^(2) t^(2) + 2alpha_(2)t]`
`= l^(2)/4 [1+ alpha_(1)^(2) l^(2) + 2alpha_(1)t]`
`alpha_(2)^(2)t^(2)` and `alpha_(1)^(2)t^(2)` are neglected since they are very small
`therefore l^(2)(2alpha_(2)t)=l^(2)/4 (2alpha_(1)t)`
`therefore alpha_(1)=4alpha_(2)`