A bimetallic bar is made of two thin strips of dissimilar metals bonded together. As they are heated, the one with the larger average coefficient of expansion expands more than the other, forcing the bar into an arc, with the outer strip having both a larger radius and a larger circumference (Fig. P10.61). (a) Derive an expression for the angle of bending, $\theta$, as a function of the initial length of the strips, their average coefficients of linear expansion, the change in temperature, and the separation of the centers of the strips $\left(\Delta r=r_{2}-r_{1}\right) .$ (b) Show that the angle of bending goes to zero when $\Delta T$ goes to zero or when the two coefficients of expansion become equal. (c) What happens if the bar is cooled?