$A$ particle of mass $m = 5$ is moving with a uniform speed $v = 3\sqrt{2}$ in the $XOY$ plane along the line $Y = X + 4$. The magnitude of the angular momentum of the particle about the origin is .......

Write the $SI$ unit of angular momentum and its dimensional formula.

$A$ 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 ........ $J-s$.

The position of a particle is given by $\vec r = (\hat i + 2\hat j - \hat k)$ and its momentum is $\vec p = (3\hat i + 4\hat j - 2\hat k)$. The angular momentum is perpendicular to:

$A$ particle moves along a circular path with decreasing speed. Hence,

$A$ point mass $m$ is attached to one end of a string passing through a cylindrical tube. The string is held in hand,and the point mass moves in a horizontal circle of radius $r_1 = 2 \ m$ with a speed of $v_1 = 4 \ m/s$. The string is then pulled down so that the radius is reduced to $r_2 = 1 \ m$. Calculate the new linear velocity,angular velocity,and the ratio of the final kinetic energy to the initial kinetic energy.