The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $\mathrm{V}(\mathrm{r})=\mathrm{kr}^2 / 2$, where $\mathrm{k}$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $\mathrm{R}$ about the point $\mathrm{O}$. If $\mathrm{v}$ is the speed of the particle and $\mathrm{L}$ is the magnitude of its angular momentum about $\mathrm{O}$, which of the following statements is (are) true?

$(A)$ $v=\sqrt{\frac{k}{2 m}} R$

$(B)$ $v=\sqrt{\frac{k}{m}} R$

$(C)$ $\mathrm{L}=\sqrt{\mathrm{mk}} \mathrm{R}^2$

$(D)$ $\mathrm{L}=\sqrt{\frac{\mathrm{mk}}{2}} \mathrm{R}^2$

$A$ uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^o$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above. What is the direction of the rod's angular momentum vector (calculated about its lower end)?

A ball of mass $1 \,kg$ is projected with a velocity of $20 \sqrt{2}\,m / s$ from the origin of an $x y$ co-ordinate axis system at an angle $45^{\circ}$ with $x$-axis (horizontal). The angular momentum [In $SI$ units] of the ball about the point of projection after $2 \,s$ of projection is [take $g=10 \,m / s ^2$ ] ( $y$-axis is taken as vertical)

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 …….

A disc of mass  $M$  and radius  $R$  is rolling with angular speed $\omega $ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is