Four particles $A, B, C$ and $D$ with masses $m_A=m, m_B=2m, m_C=3m$ and $m_D=4m$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is

Find the centre of mass of a uniform :

$(a)$ half-disc,$(b)$ quarter-disc. 

A narrow but tall cabin is falling freely near the earth's surface. Inside the cabin, two small stones $A$ and $B$ are released from rest (relative to the cabin). Initially $A$ is much above the centre of mass and $B$ much below the centre of mass of the cabin. A close observation of the motion of $A$ and $B$ will reveal that

$A$ slender uniform rod of length $\lambda$ is balanced vertically at a point $P$ on a horizontal surface having some friction. If the top of the rod is displaced slightly to the right, the position of its centre of mass at the time when the rod becomes horizontal :

In the figure shown a hole of radius $2\, cm$ is made in a semicircular disc of radius $6\pi$ at a distance $8 \,cm$ from the centre $C$ of the disc. The distance of the centre of mass of this system from point $C$ is  ......... $cm$.

Two semicircular rings of linear mass densities $\lambda $ and $3\lambda $ and of radius $R$ each are joining to form a complete ring. The distance of the centre of the mass of complete ring from its geometrical centre is