In the above problem the angular velocity of the system after the particle sticks to it will be ....... $rad/s$

A rod $P$ of length $1\ m$ is hinged at one end $A$ and there is a ring attached to the other end by a light inextensible thread . Another long rod $Q$ is hinged at $B$ and it passes through the ring. The rod $P$ is rotated about an axis which is perpendicular to plane in which both the rods are present and the variation between the angles $\theta $ and $\phi $ are plotted as shown. The distance between the hinges $A$ and $B$ is ........ $m.$

The centre of mass of a body

The plank in the figure moves a distance $100\,mm$ to the right while the centre of mass of the sphere of radius $150\, mm$ moves a distance $75\,mm$ to the left. The angular displacement of the sphere (in radian) is  (there is no slipping anywhere) :-

A thin circular ring of mass $M$  and radius $R$  is rotating about its axis with a constant angular velocity $\omega .$ Two objects, each of mass $m,$  are attached gently to the opposite ends of a diameter of the ring. The ring rotates now with an angular velocity

Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$, the speed of centre of mass of the system is