A particle of mass $M=0.2 kg$ is initially at rest in the $x y$-plane at a point $( x =-l, y =-h)$, where $l=10 m$ and $h=1 m$. The particle is accelerated at time $t =0$ with a constant acceleration $a =10 m / s ^2$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec{L}$ and $\vec{\tau}$, respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x , y$ and $z$-directions, respectively. If $\hat{k}=\hat{i} \times \hat{j}$ then which of the following statement($s$) is(are) correct?

$(A)$ The particle arrives at the point $(x=l, y=-h)$ at time $t =2 s$.

$(B)$ $\vec{\tau}=2 \hat{ k }$ when the particle passes through the point $(x=l, y=-h)$

$(C)$ $\overrightarrow{ L }=4 \hat{ k }$ when the particle passes through the point $(x=l, y=-h)$

$(D)$ $\vec{\tau}=\hat{ k }$ when the particle passes through the point $(x=0, y=-h)$

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 is moving in a circular path of radius $a,$ with a constant velocity $v$ as shown in the figure.The centre of circle is marked by $'C'$. The angular momentum from the origin $O$ can be written as

A particle of mass $m$ moves along line $PC$ with velocity $v$ as shown.  What is the angular momentum of the particle about $O$ ​

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 ($2$)