Question

A uniform circular sheet has a radius r. A square sheet of diagonal length equal to the radius of the plate is removed from one end of it. The distance of the centre of mass of the reminder plate from the centre of the original sheet is

• A. $\frac{r}{2(2\pi -1)}$
• B. $\frac{r}{(\pi -2)}$
• C. $\frac{2r}{(\pi -3)}$
• D. $\frac{r}{2(\pi -1)}$

Answer 1

Answer: (A)

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

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

Assume centre of mass of circle at O.

Center of a mass of square sheet will be at distance of $\frac{r}{2}$ from O.

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

Area Of square $=\frac{r^2}{2}$

Consider metal sheet as a uniform density,

$\therefore Massofm=\frac{M\times \frac{r^2}{2}}{\pi r^2}$

$=\frac{M}{2\pi }$

Now the centre of mass of remaining sheet $\left(X\right)$,

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

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

$\therefore$ centre of mass of remaining part at distance $\frac{r}{2(2\pi -1)}$ towards right from the centre