A solid sphere of mass $500\,g$ and radius $5\,cm$ is rotated about one of its diameter with angular speed of $10\,rad \, s ^{-1}$. If the moment of inertia of the sphere about its tangent is $x \times 10^{-2}$ times its angular momentum about the diameter. Then the value of $x$ will be …………..

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 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 …….. $J-s$

$A$ particle of mass $0.5\, kg$ is rotating in a circular path of radius $2m$ and centrepetal force on it is $9$ Newtons. Its angular momentum (in $J·sec$) is: