$A$ hemisphere of mass $3m$ and radius $R$ is free to slide with its base on a smooth horizontal table. $A$ particle of mass $m$ is placed on the top of the hemisphere. If the particle is displaced with a negligible velocity,find the angular velocity of the particle relative to the centre of the hemisphere at an angular displacement $\theta$,when the velocity of the hemisphere is $v$.

$A$ uniform circular disc of radius $R$, lying on a frictionless horizontal plane, is rotating with an angular velocity $\omega$ about its own axis. Another identical circular disc is gently placed on top of the first disc coaxially. The loss in rotational kinetic energy due to friction between the two discs, as they acquire a common angular velocity, is ($I$ is the moment of inertia of the disc).

Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same $(M)$ and the mass of the threads are negligible,the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at $\frac{r}{2}$ distance from the end $A$ of the handle will be ....... $Mr^2$?

$A$ particle of mass $m$ is moving along the side of a square of side '$a$',with a uniform speed $v$ in the $x-y$ plane as shown in the figure. Which of the following statements is false for the angular momentum $\vec L$ about the origin?

The position of an object having mass $0.1 \text{ kg}$ as a function of time $t$ is given as $\vec{r} = (10t^2\hat{i} + 5t^3\hat{j}) \text{ m}$. At $t = 1 \text{ s}$, which of the following statements are correct?
$A$. The linear momentum $\vec{p} = (2\hat{i} + 1.5\hat{j}) \text{ kg} \cdot \text{m/s}$.
$B$. The force acting on the object $\vec{F} = (2\hat{i} + 3\hat{j}) \text{ N}$.
$C$. The angular momentum of the object about its origin $\vec{L} = 15\hat{k} \text{ J} \cdot \text{s}$.
$D$. The torque acting on the object about its origin $\vec{\tau} = 20\hat{k} \text{ N} \cdot \text{m}$.
Choose the correct answer from the options given below:

State the equivalent physical quantities in linear motion corresponding to the moment of inertia and torque in rotational motion.