What is the linear velocity if angular velocity $\vec{\omega} = 3 \hat{i} - 4 \hat{j} + \hat{k}$ and position vector $\vec{r} = 5 \hat{i} - 6 \hat{j} + 6 \hat{k}$?

Two masses $m$ and $2m$ are connected by a light string passing over a pulley (disc) of mass $M = 30m$ and radius $r = 0.1 \ m$. The pulley is mounted in a vertical plane and is free to rotate about its axis. The $2m$ mass is released from rest. Find its speed when it has descended through a height of $h = 3.6 \ m$. (Assume the string does not slip and $g = 10 \ m/s^2$)

In the following figure,a body of mass $m$ is tied at one end of a light string and this string is wrapped around a solid cylinder of mass $M$ and radius $R$. At the moment $t = 0$,the system starts moving. If the friction is negligible,the angular velocity at time $t$ would be

$A$ bob of mass $m$ is tied by a massless string whose other end is wound on a flywheel (disc) of radius $R$ and mass $m$. When released from rest,the bob starts falling vertically downwards. If the bob has covered a vertical distance $h$,then the angular speed of the wheel will be (There is no slipping between the string and the wheel,$g$ is the acceleration due to gravity).

$A$ binary star system consists of two stars,one of which has double the mass of the other. The stars rotate about their common centre of mass:

$A$ string is wrapped around a disc of mass $M$ and radius $R$ and the free end is fixed to the ceiling. The centre of mass falls down as the disc unwinds the string. The tension in the string is