How can one verify that an object is undergoing rotation about a fixed axis?

$A$ spool of mass $M$ and radius $R$ is unwinding under gravity as a string is pulled. What is the acceleration of the spool?

The mass of the pulley is $m$ and its radius is $R$. Assume the pulley to be a uniform disc. The pulley is free to rotate about an axis passing through its centre and perpendicular to its plane. The string is massless and inextensible,and it is wrapped on the pulley. There is no slipping between the string and the pulley. The length of the string which is not wrapped on the pulley is $2R$. $A$ block of mass $m$ is released from the position as shown in the figure. The impulse exerted by the string on the pulley at the moment the string becomes taut is $J$. The value of $\frac{2m \sqrt{gR}}{J}$ is equal to:

$A$ tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega$. The force exerted by the liquid on the tube at the other end is

The velocity of the centre of mass of a rigid rod with respect to an observer $O$ is $\vec v_{cm} = (2\hat i + 3\hat j) \text{ m/s}$. The rod has an angular velocity about its centre of mass given by $\vec \omega = (3\hat j + 4\hat k) \text{ rad/s}$. Let $A$ be a point on the rod with position vector $\vec r = 2(\hat i + \hat k) \text{ m}$ with respect to the centre of mass. The velocity of the point $A$ with respect to $O$ is:

$A$ body rotates about a stationary axis. If the angular deceleration is proportional to the square root of the angular speed, then the mean angular speed of the body, given $\omega_0$ as the initial angular speed, is