Question

In the figure shown, there is a uniform circular plate of radius $3R$ and centre at $O$, from this plate a small circular part of radius $R$ is removed. The centre of the small circular part is at a distance of $2R$ from $O$.

Find the distance of the centre of mass of whole system from $O$.

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

Answer 1

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

The problem can be broken into two parts.
The CM of the plate after removing the smaller circular portion can be decomposed into two objects such as:
(i) CM of the circular plate of radius $3R$ without removing the circular portion of radius $R$
(ii) CM of the circular portion of radius $R$ which has a negative equivalent mass
Since the system symmetrical about the line joining the two centres, $CM$ will lie on this line.
Taking center of the circle of radius $3R$ as the origin, $CM$ can be calculated.
The co-ordinate of the CM of the large circle is the origin $O$.
The co-ordinate of the CM of the smaller circle is $3R-R=2R$;
$A_1=$ Area of the circle of radius $3R=\pi (3R{)}^2$,
$A_2=$Area of the circle of radius $R=\pi (R{)}^2$
The thickness and density of the circular plates are same every where. Hence, area can be taken instead of mass.
$X_{cm}=\frac{0\times A_1-2R\times A_2}{A_1-A_2}=\frac{-2R\times \pi R^2}{\pi \times 9R^2-\pi \times R^2}=-\frac{R}{4}$;
The $CM$ is to the left of the origin of the circle of radius $3R$.