Question

from a circular disc of radius R a square is cut with one of its radius as its diagonal the center of mass at a distance x from geometrical center of the disc find x.

Answer 1

Consider a circular sheet with radius r and mass $M$. A sqaure sheet with diagonal r and mass m is cut off from it.

Assuming centre of mass of circle to be origin i.e. at $0$.

Centre of mass of square sheet will be at distance of $\frac{r}{2}$ from it.

Side of square $=\frac{r}{\sqrt{2}}$

Area of square $=r^2/2$

Assuming metal sheet as a uniform density

Mass of square $m=M\times \frac{\frac{r^2}{2}}{\pi r^2}=\frac{M}{2\pi }$

Centre of a mass of remaining sheet =

$x=M\times 0-\frac{(\frac{M}{2\pi }\times \frac{r}{2})}{M-\frac{M}{2\pi }}$

$x=\frac{-r}{2}(2\pi -1)$

Hence, the center of mass of the remaining part lies at a distance $\frac{r}{2}(2\pi -1)$ towards right from center of the circle.