In the figure shown,find the distance of the centre of mass of a system of a uniform circular plate of radius $3R$ from $O$,in which a hole of radius $R$ is cut whose centre is at a distance of $2R$ from the centre of the large circular plate.

$A$ uniform square plate is shown in the figure. Four identical small squares are removed from its corners. If squares $1, 2,$ and $3$ are removed,where will the center of mass $(C.M.)$ be located?

From a uniform circular disc of radius $2 \,cm$ (its centre of mass is at $O$), a circular portion of radius $1 \,cm$ is removed such that the shift in centre of mass is maximum. The disc is now rotated by an angle $\theta$ about an axis perpendicular to its plane and passing through $O$. If the magnitude of displacement of the new centre of mass is $\frac{1}{\sqrt{3}} \,cm$, then the $\theta$ is (in $^{\circ}$)

The $(x, y)$ coordinates (in $cm$) of the centre of mass of letter $E$ relative to the origin $O$,whose dimensions are shown in the figure are: (Take width of the letter $2 \ cm$ everywhere).

$A$ uniform square plate has a side of length $2R$. $A$ circular piece of maximum possible area is cut and removed from one of the quadrants of the plate as shown in the figure. Calculate the shift in the centre of mass of the plate.

$A$ uniform square plate of side $a$ has a circular hole of diameter $a$ cut out from it as shown in the figure. The center of the hole is at $(a/4, a/4)$ from the center of the square. The distance of the center of mass of the resulting plate from the center of the square is: