From a square sheet of uniform density, a portion is removed shown as shaded region in the figure. Find the centre of mass of the remaining portion if the side of the square is $a$.

A uniform circular disc of radius '$a$' is taken. A circular portion of radius '$b$' has been removed from it as shown in figure. If the centre of 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 $O$ is given by:

Find the velocity of centre of mass if the triangle formed is equilateral.

Consider a semicircular ring of radius $R$. Its linear mass density varies as $\lambda ={\lambda }_0\sin \theta$. Locate its centre of mass.

The block wedge system shown in the figure is released from rest. All the surfaces are smooth. Find the speed of the wedge when the speed of the block w.r.t. wedge is $v$.

A block of mass $\mathrm{m}$ is moving with speed $v$ on a wedge of mass $M$ which is placed upon a horizontal surface. Track $AB$ is a circular track of radius $R$. What is the minimum speed $v$, so that the block reaches point $A$ (All surfaces are frictionless)?

Two blocks $A$ and $B$ are connected by a spring of stiffness $200 \mathrm{N} {\mathrm{m}}^{-1}$ and placed on a smooth horizontal surface. Initially the spring has its equilibrium length. The indicated velocities are imparted to $A$ and $B$. The maximum extension of the spring will be