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An equilateral triangle ABC if formed by...

An equilateral triangle ABC if formed by two Cu rods AB and BC and one Al rod

it is heated in such a way that temperature of each rod increases by `Delta T.` Find change in the angle ABC. [Coeff. Of linear expansion for Cu. is `alpha_1,` Coeff. of linear expansion for Al is `alpha_2]`

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`cos theta = (l_1^2 + L_3^2 - l_2^2)/(2l_1 l_3)`
`2l_1 cos theta = l_1^2 + l_3^2 - l_2^2`
Differentiating it we have `2(l_1 dl_3 + l_3 dl_1) cos theta -2l_1 l_3 sin theta d theta =2l_1 dl_1 + 2l_3 dl_3 -2l_2 dl_2`
Here, `l_1 = l_2 = l_3 l, dl_1 = l alpha_1 Delta T, dl_2 = l alpha_2 Delta T and dl_3 =l alpha_1 Delta T`
`:. 2(lxxl alpha_1 Delta T + lxx l alpha_1 Delta T) cos theta -2lxxl sin theta d theta`
`=2 l xx lalpha_1 Delta T + 2l xx l alpha_1 Delta T - 2l xx l alpha_2 Delta T`
Dividing it by `2l^2` we get `2 alpha_1 Delta T cos theta - sin theta d theta = 2alpha_1 Delta T - (alpha_2 DeltaT)`
`sin theta d theta = 2 alpha_1 Delta T (cos theta -1) + alpha_2 Delta T`
Using `theta = 60^@` we have `(sqrt3)/(2) d theta = 2 alpha_1 Delta T ((1)/(2) -1) + alpha_2 Delta T = (alpha_2 - alpha_1) DeltaT :. d theta = (alpha_2 - alpha_1) Delta T xx (2)/(sqrt3)`
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