Question

A semicircular portion of radius $r=10\text{cm}$ is cut from a uniform rectangular plate as shown in the figure. The distance of center of mass $\text{C}$ of the remaining plate, from point $\text{O}$ is (COM of semicircle is at distance $\frac{4r}{3\pi }$ from the base)

• A. $7.75\text{cm}$
• B. $5\text{cm}$
• C. $17\text{cm}$
• D. $4.6\text{cm}$

Answer 1

The correct option is A $7.75\text{cm}$

Remaining area = Area of rectangle - Area of semicircular portion,

Area of rectangle $A_1=2r^2$
Area of semicircular portion $A_2=\frac{\pi r^2}{2}$

If $r_1$ and $r_2$ are the positions of centre of mass of the rectangle and the semicircular portion with respect to O.
$r_1=-\frac{r}{2}$ and $r_2=-\frac{4r}{3\pi }$
(left of origin is taken -ve)

Therefore, center of mass of remaining portion
$=\frac{A_1\times (-\frac{r}{2})-A_2\times (-\frac{4r}{3\pi })}{A_1-A_2}$

$=-\left(\frac{2r^2\times \left(\frac{r}{2}\right)-\frac{\pi r^2}{2}\times \left(\frac{4r}{3\pi }\right)}{2r^2-\frac{\pi r^2}{2}}\right)$
$=-\left(\frac{\frac{2r^3}{2}-\frac{4r^3}{6}}{2r^2-\frac{\pi r^2}{2}}\right)=\frac{-2r}{3(4-\pi )}$
(Putting $r=10\text{cm}$ (given))
$=\frac{-2\times 10}{3(4-\pi )}=-7.75\text{cm}$

So, the distance of center of mass $C$ of remaining plate, from point $O$ is $7.75\text{cm}$.