Three rods of equal length at 0^(@)C, are connected with one another to form an equilateral triangle ABC [Fig. 5.10]. Coefficient of linear expansion for the rod ABis alpha and that for the other two rods is beta. What will be the increment in the measure of the angle at C if the triangle is heated to t^(@)C?

Three rods of equal length at 0^(@)C, are connected with one another t

1.	Three rods of equal length at 0^(@)C, are connected with one another to form an equilateral triangle ABC [Fig. 5.10]. Coefficient of linear expansion for the rod ABis alpha and that for the other two rods is beta. What will be the increment in the measure of the angle at C if the triangle is heated to t^(@)C?
Answer» SOLUTION :At `0^(@)C, angleC=60^(@)=pi/3.` Suppose due to rise in temperature to `t^(@)C, angle C=2phi` where `phi=pi/6+theta` (assuming that `angle C` has increased by `2theta,theta` is very small) If L is the LENGTH of each rod at `0^(@)C`, then at `t^(@)C` the lengths of AC and BC become `l(l+betat)` and the length of AD becomes `1/2(1+alphat)`. So for `DeltaACD` in Fig. 5.10, `""(l//2(1+alphat))/sinphi=(l(1+betat))/(sin90^(@))` or, `"" sinphi=(l//2(1+alphat))/(l(1+betat))` or, `"" sin(pi/6+theta)=(1+alphat)/(2(1+betat))` or, `"" sin""pi/6costheta+cos""pi/6sintheta=(1+alphat)/(2(1+betat))` or, `"" 1/2costheta+sqrt(3)/2sintheta=1/2((1+alphat)/(1+betat))` or, `""1/2+sqrt(3)/2*theta=1/2((1+alphat)/(1+betat))` `""[therefore theta` is small, `sintheta=theta, costheta=1]` or, `"" sqrt(3)/2theta=1/2((1+alphat)/(1+betat)-1)` or, `"" theta=1/sqrt(3)((1+alphat-1-betat)/(1+betat))=((alpha-beta)t)/(sqrt(3)(1+betat))` `therefore` Increases in `angleC` is `2theta=(2(alpha-beta)t)/(sqrt(3)(1+betat)).`