Question

A bimetallic strip is formed act of two identical strips one of copper and the other of brass. The coefficients of linear expansion of the two metals are ${\alpha }_c$ and ${\alpha }_{\beta }$. On heating, the temperature of the strip goes up by $\Delta T$ and the strip bends to form an arc of radius of curvature $R$. Then $R$ is:
(1) inversely proportional to $\Delta T$
(2) proportional to $\left|{\alpha }_{\beta }-{\alpha }_c\right|$
(3) proportional to $\Delta T$
(4) inversely proportional to $\left|{\alpha }_{\beta }-{\alpha }_C\right|$

• A. 1 and 2 are correct
• B. 2 and 3 are correct
• C. 1 and 4 and correct
• D. 1, 2 and 4 are correct

Answer 1

Answer: (C)

Let to be the initial length of each strip before heating.

Length after heating will be
$l_B=l_0\left(1+{\alpha }_{\beta }\Delta T\right)=(R+d)\theta$
and $R_c=l_0\left(1+{\alpha }_c\Delta T\right)=R\theta$
$\therefore \frac{R+d}{R}=\frac{1+{\alpha }_{\beta }\Delta T}{1+{\alpha }_c\Delta T}$
$\therefore 1+\frac{d}{R}=1+\left({\alpha }_{\beta }-{\alpha }_c\right)\Delta T$
[from binomial expansion]
$\therefore R=\frac{d}{\left({\alpha }_{\beta }-{\alpha }_c\right)\Delta T}$
Or $R\propto \frac{1}{\Delta T}$
and $\propto \frac{1}{|{\alpha }_{\beta }-{\alpha }_c|}$