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Three rods of equal of length are joined...

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,

A

`alpha_1 = alpha_2`

B

`alpha_1 =2 alpha_2`

C

`alpha_1 = 4 alpha_2`

D

`alpha_1 = 1/2 alpha_2`

Text Solution

Verified by Experts

The correct Answer is:
C
`DC^2 =AC^2 -AD^2 =l^2 - ((1)/(2))^2`
` DC^2 = [l (1+ alpha_2 t) ]^2 = [(1)/(2) (1+ alpha_t t)]^2`
`l^2 - (l^4)/(4) = l^2 =(1+ alpha _(2)^(2) t^2 + 2alpha _(2) t ) -(l^2)/(4) (1+ alpha_(1)^(2) t^2 + 2 alpha_1 t)`
Neelecting ` alpha_(2)^(2) t^2` and ` alpha_(1)^(2) t^2 , 0 = l^2 ( 2 alpha_1 t ) - (l^2)/(4) ( 2 alpha _1 t)`
` 2 alpha _2 = ( 2 alpha_1)/( 4) implies alpha_2`
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