A bullet of mass $10\, g$ and speed $500\, m/s$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0\, m$ wide and weighs $12\, kg$. It is hinged at one end and rotates about a vertical axis practically without friction . The angular speed of the door just after the bullet embeds into it will be

A particle of mass $1 kg$ is subjected to a force which depends on the position as $\vec{F}=-k(x \hat{i}+y \hat{j}) kgms ^{-2}$ with $k=1 kgs ^{-2}$. At time $t=0$, the particle's position $\vec{r}=\left(\frac{1}{\sqrt{2}} \hat{i}+\sqrt{2} \hat{j}\right) m$ and its velocity $\vec{v}=\left(-\sqrt{2} \hat{i}+\sqrt{2} \hat{j}+\frac{2}{\pi} \hat{k}\right) m s^{-1}$. Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z=0.5 m$, the value of $\left(x v_y-y v_x\right)$ is. . . . . $m^2 s^{-1}$

A particle of mass $'{m}'$ is moving in time $'t'$ on a trajectory given by

$\overrightarrow{{r}}=10 \alpha {t}^{2}\, \hat{{i}}+5 \beta({t}-5)\, \hat{{j}}$

Where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for ${t}=0$ at time ${t}=$ …..$seconds.$

Obtain the relation between angular momentum of a particle and torque acting on it. 

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 $55 \mathrm{ma}^2 \omega$