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

In the rotational motion of a rigid body,all particles of the body have

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):-

$A$ rigid body is rotating with variable angular velocity $\omega = (a - bt)$ at any instant of time $t$. The total angle subtended by it before coming to rest will be ($a$ and $b$ are constants).

Let $\vec{A}$ be a unit vector along the axis of rotation of a purely rotating body and $\vec{B}$ be a unit vector along the velocity of a particle $P$ of the body away from the axis. The value of $\vec{A} \cdot \vec{B}$ is

$A$ disc at rest is subjected to a uniform angular acceleration about its axis. Let $\theta$ and $\theta^{\prime}$ be the angle made by the disc in the $2^{\text{nd}}$ and $3^{\text{rd}}$ second of its motion,respectively. The ratio $\frac{\theta}{\theta^{\prime}}$ is