Question

Two semicircular rings of linear mass densities $\lambda$ and $2\lambda$ and of radius $3\pi \text{cm}$ each, are joined together to form a complete ring as shown in the figure given below.


Then, distance of the center of the mass of the complete ring from its centre will be:

• A. $\frac{9}{8}\text{cm}$
• B. $2\text{cm}$
• C. $\frac{4}{7}\text{cm}$
• D. $3\text{cm}$

Answer 1

The correct option is B $2\text{cm}$

Let the two half sings be placed on either side of the $y$ - axis as shown in the figure.
The system's $\text{COM}$ will lie in the $x-$ direction, due to symmetry of shape.


For half ring, the distance of $\text{COM}$ from the center of the ring is given by the formula:
$d=\frac{2r}{\pi }$; where $r=$ radius of the ring

Here, $d=\frac{2\times 3\pi }{\pi }=6\text{cm}$
$\therefore$ Centre of mass of the left ring = $(-6\text{cm},0)$
Centre of mass of right ring = $(6\text{cm},0)$
(taking origin at $\text{O}$)

Replacing rings by point masses placed at their respective $\text{COM}$ and applying formula for system's $\text{COM}$:

$x-$ coordinate of centre of mass of complete ring :
$x_{\text{CM}}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}=\frac{m(-6)+2m\left(6\right)}{3m}$

($\because (\lambda \propto m)$, hence ring with double density will have double mass.)

$\therefore x_{\text{CM}}=2\text{cm}$.