Question

Three uniform thin aluminum rods each of length $2m$ form an equilateral triangle $PQR$ as shown in the figure. The mid point of the rod $PQ$ is at the origin of the coordinate system. If the temperature of the system of rods increases by $50^{\circ }C$, the increase in $y$ -coordinate of the centre of mass of the system of the rods is ............ $mm$. (Coefficient of volume expansion of aluminium $=12\sqrt{3}\times 10^{-6}{K}^{-1}$ )

• A. 0.05
• B. 0.8
• C. 0.1
• D. 0.2

Answer 1

Answer: (D)

Initially,
$r_{COM}=\frac{m_1{r}_1+m_2{x}_2+m_3{r}_3}{\left(m_1+m_2+m_3\right)}$
$r_{COM}=\frac{1}{3}\left(\frac{\sqrt{3}}{2}\times 1\times 2\right)\hat{j}$
$\Rightarrow r_{COM}=\frac{1}{\sqrt{3}}\hat{j}$
Due to expansion each rod length becomes
$1_f=1_i(1+\alpha \Delta T)$
$1_f=1_i\left(1+\frac{\gamma }{3}\Delta T\right)$
$1_f=2\left(1+\frac{12\sqrt{3}\times 10^{-6}}{3}\times 50\right)$
$1_f=2\left(1+4\times 50\times \sqrt{3}\times 10^{-6}\right)$
$=2\left(1+2\sqrt{3}\times 10^{-4}\right)$
So, ${\left(r_{COM}\right)}_{\text{final }}=\frac{1}{3}\times \frac{\sqrt{3}}{2}\times \left(1+2\sqrt{3}\times 10^{-4}\right)\times 2\hat{j}$
$=\left(\frac{1}{\sqrt{3}}+2\times 10^{-4}\right)\hat{j}$
Hence, $\Delta y={\left(I_{COM}\right)}_{\text{final }}-{\left(I_{COM}\right)}_{\text{initial}}$
$=2\times 10^{-4}m=0.2mm$