Question

Three metal rods of same length and area of cross-section are arranged to form a equilateral triangle as shown in figure. $S$ is the middle point of side $QR$. If $PS$ is independent of temperature, then
[${\alpha }_1$ is coefficient of linear expansion for rod $QR$ and ${\alpha }_2$ is that for $PQ$ and $PR$]

• A. ${\alpha }_1=2{\alpha }_2$
• B. ${\alpha }_1={\alpha }_2/2$
• C. ${\alpha }_1={\alpha }_2$
• D. ${\alpha }_1=4{\alpha }_2$

Answer 1

Answer: (D)

${\alpha }_1=4{\alpha }_2$

From the figure, we have
$P{S}^2=P{Q}^2-Q{S}^2$
$\Rightarrow I_0^2=I^2-\frac{I^2}{4}$
Differentiating with respect to time, we have
$2I_0\times \frac{d{I}_0}{dt}=2I\times {\left(\frac{dl}{dt}\right)}_{PQ}-\frac{1}{4}\times 2I\times {\left(\frac{dI}{dt}\right)}_{QR}$
Since, $\frac{d{I}_0}{dt}=0$
$\Rightarrow 2I{\left(\frac{dI}{dt}\right)}_{PQ}=\frac{1}{2}{\left(\frac{dI}{dt}\right)}_{QR}$
or $2{\alpha }_{PQ}=\frac{1}{2}\times {\alpha }_{QR}$
or $2{\alpha }_2=\frac{1}{2}\times {\alpha }_1$
or ${\alpha }_1=4{\alpha }_2$.