$A$ homogeneous disc of mass $2 \ kg$ and radius $15 \ cm$ is rotating about its axis (which is fixed) with an angular velocity $4 \ rad/s$. The linear momentum of the disc is

$A$ flywheel rotates about an axis. Due to friction at the axis,it experiences an angular retardation proportional to its angular velocity. If its angular velocity falls to half while it makes $n$ rotations,how many more rotations will it make before coming to rest?

$A$ rod of mass $m$ and length $l$ is hinged at one end to a horizontal floor and stands vertically. If it is allowed to fall,the velocity with which its upper end strikes the floor is:

$A$ uniform circular disc placed on a rough horizontal surface has initially a velocity $v_0$ and an angular velocity $\omega_0$ as shown in the figure. The disc comes to rest after moving some distance in the direction of motion. Then $\frac{v_0}{r\omega_0}$ is

Two thin circular discs of mass $m$ and $4m$,having radii of $a$ and $2a$,respectively,are rigidly fixed by a massless,rigid rod of length $l=\sqrt{24}a$ through their centers. This assembly is laid on a firm and flat surface,and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point $O$ is $\vec{L}$ (see the figure). Which of the following statement$(s)$ is(are) true?
$(A)$ The center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega/5$
$(B)$ The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81ma^2\omega$
$(C)$ The magnitude of angular momentum of the assembly about its center of mass is $17ma^2\omega/2$
$(D)$ The magnitude of the $z$-component of $\vec{L}$ is $55ma^2\omega$

$A$ non-uniform cylinder of mass $m$,length $l$,and radius $r$ has its center of mass at a distance $l/4$ from the geometric center,lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density $\rho$. The moment of inertia of the cylinder about its center of mass is $I$. The angular acceleration of the cylinder just after it is released from the horizontal position shown in the figure is: