Question

A rectangle $\text{OABC}$ of length $l$ and a thin ring of radius $\frac{l}{2}$, having equal masses, are placed in the cartesian coordinate system as shown in the figure. The coordinates of the centre of mass of the system is:

• A. $(\frac{l}{2},\frac{5l}{8})$
• B. $(\frac{l}{2},\frac{l}{8})$
• C. $(\frac{l}{2},\frac{l}{2})$
• D. $(\frac{l}{3},\frac{5l}{2})$

Answer 1

The correct option is A $(\frac{l}{2},\frac{5l}{8})$

We know that the $\text{COM}$ of a geometrically symmetric body lies at its geometrical centre.

Let the mass of both rectangle and circle be $m$.

Clearly, the coordinates of $\text{COM}$ of the rectangle is given by,

$x_{\text{COM}}=\frac{l}{2}$

$y_{\text{COM}}=\frac{l}{4}$

Now, the coordinates of $\text{COM}$ of the ring is given by,

$x_{\text{COM}}=\frac{l}{2}$

$y_{\text{COM}}=l$

So, the rectangle can be replaced by it's respective centre of mass, which is given by,

$C_1(x_1,y_1)=(\frac{l}{2},\frac{l}{4})$

So, the circle can be replaced by it's respective centre of mass, which is given by,

$C_2(x_2,y_2)=(\frac{l}{2},l)$

Now, the $\text{COM}$ of the combination is given by,

$x_{COM}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$

$x_{COM}=\frac{(\frac{ml}{2}+\frac{ml}{2})}{m+m}=\frac{l}{2}$

$y_{COM}=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}$

$y_{COM}=\frac{(\frac{ml}{4}+ml)}{m+m}=\frac{5l}{8}$

So, the coordinates of $\text{COM}=(\frac{l}{2},\frac{5l}{8})$

Hence, option $\left(A\right)$ is the correct answer.