$A$ circular hole of radius $\left(\frac{a}{2}\right)$ is cut out of a circular disc of radius $'a'$ as shown in the figure. The centroid of the remaining circular portion with respect to point $'O'$ will be:

$A$ ring is formed by joining two uniform semi-circular rings $ABC$ and $ADC$. The mass of $ABC$ is thrice that of $ADC$. If the ring is hinged to a fixed support at $A$,it can rotate freely in a vertical plane. Find the value of $\tan \theta$,where $\theta$ is the angle made by the line $AC$ with the vertical in equilibrium.

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).

$A$ solid cone is placed on a horizontal surface has height $h$,radius $R$ and apex angle $\theta$ as shown. If the gravitational potential energy of the cone does not change as the position of the cone is changed from figure $(A)$ to figure $(B)$,then,

$A$ uniform circular disc of radius $a$ is taken. $A$ circular portion of radius $b$ has been removed from it as shown in the figure. If the centre of the hole is at a distance $c$ from the centre of the disc,the distance $x_2$ of the centre of mass of the remaining part from the initial centre of mass $O$ is given by

$A$ uniform square plate of side $a$ is placed in the $xy$-plane with its center at the origin. If a circular portion of diameter $a$ is removed from the plate as shown in the figure,what is the position of the center of mass of the remaining portion?