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As shown in the figure , an equilateral ...

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

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D is mid point condition to have CD constant
`AC^(2) - AD^(2) `constant `L_(2)^(2) - (L_(1)^(2))/(4)` = constant
Differentiating w.r.t t `2L_(2) (dL_(2))/(dt ) - 2 (L_(1))/(4) (dL_(1))/(dt) = 0 `
or `2L_(2) dL_(2) - 2 (L_(1))/(4) dL_(1) = 0`
`dL_(2) = L_(2) alpha_(2) Delta theta and dL_(1) = L_(1) alpha Delta theta`
` 2 L_(2) , L_(2) alpha_(2) Delta theta = (L_(1))/(2) L_(1) theta_(1) Delta theta`
but `L_(2) = L_(1)`
`therefore 2 alpha_(1) = (1)/(2) alpha_(1) " or " 4 alpha_(2) = alpha_(1)`
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