How many degrees of freedom does a rigid body have for a rotational motion about a fixed axis?

What is the direction of angular velocity for a rigid body?

$A$ ladder of length $L$ is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment,the speed of the end in contact with the horizontal floor is $v$ and the ladder makes an angle $\alpha = 30^o$ with the horizontal. If $dv/dt = 0$,then the angular acceleration of the ladder when $\alpha = 45^o$ is:

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 $400 \ g$ and $350 \ g$ are suspended from the ends of a light string passing over a heavy pulley of radius $2 \ cm$. When released from rest,the heavier mass is observed to fall $81 \ cm$ in $9 \ s$. The rotational inertia of the pulley is . . . . . . $kg \cdot m^2$. $(g = 9.8 \ m/s^2)$

$A$ 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: