Question

From a circular disk of radius ($R=50\text{cm}$), a square is cut with one of its radii as the diagonal of the square (as shown in figure). The distance of the center of mass of the remaining part from the geometrical center of the disc is

• A. $7.5\text{cm}$
• B. $9.4\text{cm}$
• C. $6.8\text{cm}$
• D. $4.73\text{cm}$

Answer 1

The correct option is D $4.73\text{cm}$


Consider a circular sheet with radius $R$ and mass $M$. A square sheet with diagonal $R$ and mass $m$ is cut from it.
Assuming center of mass of the disc to be the origin $(x_1,y_1)$ =$(0,0)$
Center of mass of square sheet will be $(x_2,y_2)=$ $(\frac{R}{2},0)$

Side of square = $\frac{R}{\sqrt{2}}$
$\Rightarrow$ Area of square = $\frac{R^2}{2}$
Assuming metal sheet is of uniform density,
Mass of square $m=\frac{M}{\pi R^2}\times \left(\frac{R^2}{2}\right)=\frac{M}{2\pi }$

Therefore, center of a mass of remaining sheet is given by
$X_{com}=\frac{\left(M\right)\times 0-\left(\frac{M}{2\pi }\right)\times \frac{R}{2}}{\left(M\right)-\left(\frac{M}{2\pi }\right)}$
i.e $X_{com}=-\frac{R}{2(2\pi -1)}$

Putting $R=50\text{cm}$ (given)
$x=-\frac{50}{2(2\pi -1)}=-4.73\text{cm}$

$Y_{com}$ will be zero due to symmetry
Hence, the center of mass of remaining part lies at a distance $4.73\text{cm}$ left of the center of the disc.