Question

A semicircular portion of radius $r$ is cut from a uniform rectangular plate as shown in figure. The distance of centre of mass $\text{C}$ of remaining plate, from point $\text{O}$ is

• A. $\frac{2r}{(3-\pi )}$
• B. $\frac{3r}{2(4-\pi )}$
• C. $\frac{2r}{(4+\pi )}$
• D. $\frac{2r}{3(4-\pi )}$

Answer 1

The correct option is D $\frac{2r}{3(4-\pi )}$

Using the relation for COM,

$X_{COM}=\frac{A_1{x}_1+A_2{x}_2}{A_1+A_2}......\left(1\right)$

Here,

$A_1=2r^2;A_2=-\frac{\pi r^2}{2}$

And considering $O$ as a origin.

$\therefore x_1=-\frac{r}{2};x_2=-\frac{4r}{3\pi }$

Where,

$A_1$ = Area of complete rectangle

$A_2$ = Area of the semicircle (cavity)

$x_1=$ distance of COM of the complete rectangle from $O$

$x_2=$ distance of COM of the semicircle (cavity) from $O$

From equation $\left(1\right)$, we get,

$X_{COM}=\frac{[2r^2\times (-\frac{r}{2})]-[\frac{\pi r^2}{2}\times (-\frac{4r}{3\pi })]}{2r^2-\frac{\pi r^2}{2}}$

$X_{COM}=\frac{-r+\frac{2r}{3}}{\frac{(4-\pi )}{2}}=-\frac{2r}{3(4-\pi )}$

So, the distance between $\text{C}$ and $\text{O}$ is $\frac{2r}{3(4-\pi )}$.

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