Question

A semicircular ring of mass $m$ and a semicircular disc of mass $2m$ of same radius are joined together as shown. Find the distance of COM from their geometric centre.

• A. $\frac{2R}{3\pi }$
• B. $\frac{2R}{9\pi }$
• C. $\frac{5R}{3\pi }$
• D. $0$

Answer 1

The correct option is B $\frac{2R}{9\pi }$

Let their geometric centre be at origin.
COM of semicircular disc, $y_1=\frac{4R}{3\pi }$
COM of semicircular ring, $y_2=\frac{-2R}{\pi }$
So, COM of the combined body $=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}$
$=\frac{\left(2m\right)\frac{4R}{3\pi }+m\frac{(-2R)}{\pi }}{m+2m}$
$=\frac{2R}{9\pi }$