$A$ thin rod of mass $M$ and length $L$ is struck at one end by a ball of clay of mass $m$,moving with speed $v$ as shown in the figure. The ball sticks to the rod. After the collision,the angular momentum of the clay-rod system about $A$,the midpoint of the rod,is:

$A$ particle starts from the point $(0, 8) \, m$ and moves with a uniform velocity of $\vec{v} = 3 \hat{i} \, m/s$. What is the angular momentum of the particle about the origin after $5 \, s$ (mass of the particle is $1 \, kg$)?

$A$ particle of mass $m$ moves along a straight line $AB$ in the $XY$ plane. Let $L_A$ and $L_B$ be the angular momentum of the particle about the origin $O$ when it is at points $A$ and $B$ respectively. Then:

The moment of inertia of a uniform rod of mass $M$ and length $L$ about an axis through its centre and perpendicular to its length is given by $\frac{ML^2}{12}$. Now,consider one such rod pivoted at its centre,free to rotate in a vertical plane. The rod is at rest in the vertical position. $A$ bullet of mass $M$ moving horizontally at a speed $v$ strikes and embeds itself in one end of the rod. The angular velocity of the rod just after the collision will be

The angular momentum of a flywheel having a moment of inertia of $0.4\, kg\, m^2$ decreases from $30\, kg\, m^2/s$ to $20\, kg\, m^2/s$ in a period of $2\, s$. The average torque acting on the flywheel during this period is ......... $N\, m$.

$A$ solid cylinder of mass $2\ kg$ and radius $0.2\ m$ is rotating about its own axis without friction with angular velocity $3\ rad/s$. $A$ particle of mass $0.5\ kg$ and moving with a velocity $5\ m/s$ strikes the cylinder and sticks to it as shown in figure. The angular momentum of the cylinder before collision will be ........ $J-s$.