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_(1)=alpha_(2)`
B. `alpha_(1)=2alpha_(2)`
C. `alpha_(1)=4alpha_(2)`
D. `alpha_(1)=1//2alpha_(2)`

Answer

C

Explanation

`h^(2)=l^2-(l^(2))/(4)` (i)
At small change in temperature
`h^(2)=[l(1+alpha_(2)Deltatheta)]^(2)-[(l)/(2)(1+alpha_(1)Deltatheta)]^(2)`
`=l^(2)(1+2alpha_(2)Deltatheta)-(l^(2))/(4)(1+2alpha_(1)Deltatheta)`
`l^(2)-(l^(2))/(4)=l^(2)+2alpha_(2)l^(2)Deltatheta-(l^(2))/(4)-2alpha_(1)(l^(2))/(4)Deltatheta`
`alpha_(2)=(a_(1))/(4) impliesalpha_(1)=4alpha_(2)` .