Two thin circular discs of mass $m$ and $4 m$, having radii of $a$ and $2 a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24} a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point ' $O$ ' is $\vec{L}$ (see the figure). Which of the following statement($s$) is(are) true?

($A$) The center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega / 5$

($B$) The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81 m a^2 \omega$

($C$) The magnitude of angular momentum of the assembly about its center of mass is $17 \mathrm{ma}^2 \mathrm{\omega} / 2$

($D$) The magnitude of the $z$-component of $\vec{L}$ is $55 \mathrm{ma}^2 \omega$

Find the components along the $x, y, z$ axes of the angular momentum $l$ of a particle. whose position vector is $r$ with components $x, y, z$ and momentum is $p$ with components $p_{ r }, p_{ y }$ and $p_{z} .$ Show that if the particle moves only in the $x -y$ plane the angular momentum has only a $z-$component.

Obtain the relation between torque of a system of particles and angular moment. 

Explain Angular momentum of a particle and show that it is the moment of linear momentum about the reference point. 

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