$A$ uniform cylinder of radius $R$ is spun with angular velocity $\omega$ about its axis and then placed into a corner. The coefficient of friction between the cylinder and the planes is $\mu$. The number of turns taken by the cylinder before stopping is given by

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:

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery and sticks to it. The loss in kinetic energy due to the collision is ....... $J$.

$A$ 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?

$A$ rigid uniform bar $AB$ of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time,the angle made by the bar with the vertical is $\theta$. Which of the following statements about its motion is/are correct?
$[A]$ The midpoint of the bar will fall vertically downward
$[B]$ The trajectory of the point $A$ is a parabola
$[C]$ Instantaneous torque about the point in contact with the floor is proportional to $\sin \theta$
$[D]$ When the bar makes an angle $\theta$ with the vertical,the displacement of its midpoint from the initial position is proportional to $(1-\cos \theta)$

$A$ solid sphere $(A)$ of mass $5m$ and a spherical shell $(B)$ of mass $m$,both having the same radius $R$,are placed on a rough surface. When a force $F$ of the same magnitude is applied tangentially at the highest points of $A$ and $B$,they start rolling without slipping with accelerations $a_A$ and $a_B$,respectively. The ratio of $a_A$ to $a_B$ is . . . . . . .