Question

In the figure shown, find out centre of mass of a system of a uniform circular plate of radius $3R$ from $\text{O}$ in which a hole of radius $R$ is cut whose centre is at $2R$ distance from the centre of large circular plate.

• A. $\frac{R}{2}$
• B. $\frac{R}{5}$
• C. $\frac{R}{4}$
• D. None of these

Answer 1

The correct option is C $\frac{R}{4}$

Let us consider the given system is consists of two masses $m_1$ and $m_2$ respectively.

Where,
$m_1=$ mass of the hole having radius $R$ and

$m_2=$ mass of the remaining part of the disc having radius $3R$

Hence, total mass $M=m_1+m_2$

$m_1=\sigma A=\pi R^2\sigma$ and

$m_2=M-m_1=\pi (3R{)}^2\sigma -\pi R^2\sigma =8\pi R^2\sigma$

As we know, COM of the system $M$ will be at its geometrical centre at $\text{O}.$

Using the relation of COM,

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

So, COM will be at the origin $\text{O}.$ Hence, we get,

$m_1{x}_1+m_2{x}_2=0$

$m_1{x}_1=-m_2{x}_2$

Given, COM of the $m_1$ is at $2R$ from the origin $\text{O}$ we get,

$\pi R^2\sigma \times 2R=-8\pi R^2\sigma \times x_2$

$\therefore x_2=-\frac{R}{4}$

Thus, COM of the remaining portion will be at $\frac{R}{4}$ distance on the left of the origin $\text{O}$.

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