Question

Two straight metallic strips each of thickness $t$ and length $l$ are riveted together. Their coefficients of linear expansion are ${\alpha }_1$ and ${\alpha }_2$. If they are heated through a temperature $△t$, the bimetallic strip will bend to form an arc of radius :

• A. $\frac{t}{{\alpha }_1+{\alpha }_2}△t$
• B. $\frac{t}{({\alpha }_2-{\alpha }_1)△t}$
• C. $t({\alpha }_1+{\alpha }_2)△t$
• D. $t({\alpha }_2-{\alpha }_1)△t$

Answer 1

The correct option is B $\frac{t}{({\alpha }_2-{\alpha }_1)△t}$

Given,

$l=$ length of both the metallic stripes,

$t=$ thickness of both the metallic stripes,

let $l_1$ and $l_2$ be the final length of the metallic stripes after

expansion due to $\Delta T$ change.

$\Delta l=l_2-l_1=(l{\alpha }_2\Delta T-l{\alpha }_1\Delta T)$

We know,

$l=\theta r$

$\Delta l=\theta \Delta r$ Here, $\theta$ is constant as same arc angle is made by both the stripes.

and, $\Delta r=t$

then $\theta =\frac{l({\alpha }_2-{\alpha }_1)\Delta T}{t}=\frac{l}{r}$

$r=\frac{t}{({\alpha }_2-{\alpha }_1)\Delta T}$

Hence option $\text{B}$ is the correct answer.