Find the $\mathrm{CM}$ of the system of a L-shaped rod of mass $m$ and a particle of mass $m$ attached at the end of the rod as shown.

A ring and a particle of masses $2 m$ and $m$ are kept as shown in the figure, such that the $\mathrm{CM}$ of the combination lies at the origin. Find $x.$

Find $x$ and $y$ coordinates of the centre of mass of the plate shown in figure from which a square of side $2 \mathrm{m}$ is cut out.

Find centre of mass distance from point $O$ of the uniform circular arc shown in figure.

Determine th coordinate of centre of mass of a thin homogeneous plate having the form of a rectangle with sides $r$ and $2 r$ from which a semicircle with a radius $r$ is cut out.

The figure shows a square metal plate of side $l$ from which a square plate of side $a$ has been cut as shown in the figure. Find the ratio $\left(a/l\right)$ so that the centre of mass of the remaining L-shaped plate coincides with the point $A.$

Figure shows a fixed wedge on which two blocks of masses $2 \mathrm{kg}$ and $3 \mathrm{kg}$ are placed on its smooth inclined surfaces. When the two blocks are released from rest, find the acceleration of centre of mass of the two blocks.

There are two masses $m_1$ and $m_2$ placed at a distance $l$ apart, let the centre of mass of this system is at a point named $C.$ If $m_1$ is displaced by $l_1$ towards $C$ and $m_2$ is displaced by $l_2$ away from $C,$ find the distance from $C$where the new centre of mass will be located.

Let there are three equal masses situated at the vertices of an equilateral triangle, as shown in figure. Now particle $A$ starts with a velocity $v_1$ towards line $A B,$ particle $B$ starts with the velocity $v_2$ towards line$B C$ and particle $C$ starts with velocity $v_3$ towards line$C A.$ Find the displacement of the centre of mass of the three particles $A, B$ and $C$ after time $t .$ What would it be if $v_1=v_2=v_3$ ?