Question

A semicircular portion of radius '$r$' is cut from a uniform rectangular plate as shown in the figure. The distance of center of mass $C$ of the remaining plate, from point $O$ 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

Answer: (D)

$\frac{2r}{3(4-\pi )}$

$x_r=r/2$ from O

$x_{sc}=4r/3\pi$ from O

$M_c{x}_c+m_{sc}{x}_{sc}=m_r{x}_r$

$m=\rho Ad$, $\rho$ and $d$ are constant

$A_c{x}_c+A_{sc}{x}_{sc}=A_r{x}_r$

$(2r^2-0.5\pi r^2)x_c+\left(4r/3\pi \right)\left(0.5\pi r^2\right)=0.5r\left(2r^2\right)$

$(4-\pi )x_c=2(r-\frac{2r}{3})$

$x_c=\frac{2r}{3(4-\pi )}$