Given $V _{ CM }=2\; m / s , m =2\; kg , R =4\; m $

Find angular momentum of ring about origin if it is in pure rolling. $kgm ^{2} / s$

A particle of mass $m$ is moving with constant velocity $v$ parallel to the $x$-axis as shown in the figure. Its angular momentum about origin $O$ is ..........

A small particle of mass $m$ is projected at an angle $\theta $ with the $x-$ axis with an initial velocity $v_0$ in the $x-y$ plane as shown in the figure. At a time $t < \frac{{{v_0}\,\sin \,\theta }}{g}$, the angular momentum of the particle is

A particle is moving along a straight line parallel to $x$-axis with constant velocity. Find angular momentum about the origin in vector form

The position vector of $1\,kg$ object is $\overrightarrow{ r }=(3 \hat{ i }-\hat{ j })\,m$ and its velocity $\overrightarrow{ v }=(3 \hat{ j }+ k )\,ms ^{-1}$. The magnitude of its angular momentum is $\sqrt{ x } Nm$ where $x$ is

A pendulum consists of a bob of mass $m=0.1 kg$ and a massless inextensible string of length $L=1.0 m$. It is suspended from a fixed point at height $H=0.9 m$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse $P=0.2 kg - m / s$ is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J kg - m ^2 / s$. The kinetic energy of the pendulum just after the lift-off is $K$ Joules.

($1$) The value of $J$ is. . . . . .

($2$) The value of $K$ is. . . . .

Give the answers of the questions ($1$) and ($2$)