$A$ particle of mass $100\,g$ is projected at time $t = 0$ with a speed $20\,ms^{-1}$ at an angle $45^{\circ}$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $t = 2\,s$ is found to be $\sqrt{K}\,kg\,m^2/s$. The value of $K$ is $............$ (Take $g = 10\,ms^{-2}$)

$A$ 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.

$A$ fan of moment of inertia $0.6\,kg \cdot m^2$ is turned up to a working speed of $0.5$ revolutions per second. The angular momentum of the fan is:

The units of angular momentum are

$A$ 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 $1 \, kg$)?

Two wheels are connected by a belt. The radius of the larger wheel is three times that of the smaller one. What is the ratio of the rotational inertia of the larger wheel to the smaller wheel,when both wheels have the same angular momentum?