Question

A semicircular portion of radius ${}^{\prime }{r}^{\prime }$ is cut from a uniform rectangular plate as shown in figure. The distance of centre of mass ${}^{\prime }{C}^{\prime }$ of remaining plate, from point ${}^{\prime }{O}^{\prime }$ 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 $\frac{2r}{3(4-\pi )}$

$Let$ $the$ $plate$ $density$ $be$ $\sigma$

$CM$ $will$ $shift$ $a$ $long$ $x$$-axis$

$X$ $=$ $\frac{-\frac{\pi r^2}{2}\times \sigma \times (-\frac{4r}{3\pi })+2r^2\sigma (-\frac{r}{2})}{2r\times r\times \sigma -\frac{\pi r^2}{2}\sigma }$ $=$ $\frac{\frac{2r}{3}-r}{2-\frac{\pi }{2}}$ $=$ $\frac{2r}{3(4-\pi )}$