Three identical spherical shells,each of mass $m$ and radius $r$,are placed as shown in the figure. Consider an axis $XX'$ which touches two shells and passes through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the $XX'$ axis is:

$A$ disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. $A$ battery-operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the motor's thin shaft. Initially,both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega/n$,then the value of $n$ is. . . . .

$A$ simple pendulum of length $\ell$ is displaced so that its taut string is horizontal and then released. $A$ uniform bar of length $L$ pivoted at one end is simultaneously released from its horizontal position. If their motions are synchronous (i.e.,they have the same angular velocity at any angle $\theta$ below the horizontal), what is the length $L$ of the bar?

In the case of rotational dynamics,which one of the following statements is correct?
$[\vec{\omega} = \text{angular velocity}, \vec{v} = \text{linear velocity}, \vec{r} = \text{radius vector}, \vec{\alpha} = \text{angular acceleration}, \vec{a} = \text{linear acceleration}, \vec{L} = \text{angular momentum}, \vec{p} = \text{linear momentum}, \vec{\tau} = \text{torque}, \vec{f} = \text{force}]$

$A$ ring and a disc have the same mass and the same radius. The ratio of the moment of inertia of a ring about a tangent in its plane to that of the disc about its diameter is: (in $: 1$)

$A$ yo-yo is resting on a rough horizontal table. Forces $F_1, F_2$ and $F_3$ are applied separately as shown. The correct statement is