$A$ disc of mass $M$ with uniform surface mass density $\sigma$ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position $(\frac{x}{3} \frac{R}{\pi}, \frac{x}{3} \frac{R}{\pi})$,where $R$ is the radius of the disc and $x$ is ....... . (Round off to the nearest integer)

$A$ circular disc of radius $\alpha R$ has a smaller circular disc of radius $R$ removed from it such that their circumferences touch each other. The center of mass of the new disc is at a distance of $\alpha R$ from the center of the original large disc. What is the value of $\alpha$?

$A$ circular hole of radius $R/2$ is cut from a circular disc of mass $M$ and radius $R$ such that the circumference of the hole passes through the center of the disc. What is the moment of inertia of the remaining part about an axis perpendicular to the plane of the disc and passing through its center?

$A$ smaller cube with side $b$ (depicted by dashed lines) is excised from a bigger uniform cube with side $a$ as shown below,such that both cubes have a common vertex $P$. Let $X = a/b$. If the centre of mass of the remaining solid is at the vertex $O$ of the smaller cube,then $X$ satisfies:

The equilateral triangle $ABC$ is cut from a thin solid sheet of wood. $D, E$ and $F$ are the midpoints of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_0$. If the smaller triangle $DEF$ is removed from $ABC$,the moment of inertia of the remaining figure about the same axis is $I$. Then

In the cube of side $a$ shown in the figure,the vector from the central point $G$ of the face $ABOD$ to the central point $H$ of the face $BEFO$ is: