Question

Three identical circular plates of radius $R$ each are placed on a horizontal surface touching one another as shown in the figure. The y coordinate of the center of mass of the system is [Consider $\text{A}$ to be at the origin]

• A. $\frac{R}{3}$
• B. $\frac{R}{\sqrt{3}}$
• C. $\sqrt{3R}$
• D. $R$

Answer 1

The correct option is B $\frac{R}{\sqrt{3}}$


Here,
$P{C}^2=A{C}^2-A{P}^2$
$\Rightarrow PC=\sqrt{4R^2-R^2}$
$\Rightarrow PC=\sqrt{3}R$
Then, coordinates of $A$ is $(0,0)$
coordinates of $B$ is $(2R,0)$
coordinates of $C$ is $(R,\sqrt{3}R).$

$x_{com}$ of system $=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}$
$\Rightarrow x_{com}=\frac{m\times 0+m\times 2R+m\times R}{m+m+m}=R$

$y_{com}=\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3}{m_1+m_2+m_3}=\frac{m\times 0+m\times 0+m\times \sqrt{3}R}{m+m+m}$
$\Rightarrow y_{com}=\frac{\sqrt{3}mR}{3m}=\frac{R}{\sqrt{3}}$
Hence, the coordinates of the COM of the system are $(R,\frac{R}{\sqrt{3}})$