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(D) Let the rectangular plate have dimensions $2r 	imes r$. The area of the full rectangle is $A_1 = 2r 	imes r = 2r^2$. The centre of mass of the r

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) Let the rectangular plate have dimensions $2r 	imes r$. The area of the full rectangle is $A_1 = 2r 	imes r = 2r^2$. The centre of mass of the rectangle is at $x_1 = r/2$ from the edge $O$.The area of the removed semicircular portion is $A_2 = \frac{\pi r^2}{2}$. The centre of mass of a semicircle from its diameter is at a distance of $\frac{4r}{3\pi}$. Since the semicircle is cut from the rectangle,its centre of mass is at $x_2 = \frac{4r}{3\pi}$ from the edge $O$.The centre of mass of the remaining portion is given by:$x_{cm} = \frac{A_1 x_1 - A_2 x_2}{A_1 - A_2}$$x_{cm} = \frac{(2r^2)(r/2) - (\frac{\pi r^2}{2})(\frac{4r}{3\pi})}{(2r^2) - (\frac{\pi r^2}{2})}$$x_{cm} = \frac{r^3 - \frac{2r^3}{3}}{r^2(2 - \frac{\pi}{2})} = \frac{r^3(1 - 2/3)}{r^2(\frac{4 - \pi}{2})} = \frac{r/3}{(\frac{4 - \pi}{2})} = \frac{2r}{3(4 - \pi)}$

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

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