$A$ particle is moving in a uniform circular motion. At which point in the plane of the circle will the angular momentum of the particle be conserved?

If the Earth shrinks to $1/8$ of its original volume, while maintaining the same mass, then the duration of the day will be (in $hrs$)

$A$ particle is moving in a circular orbit of radius $r_1$ with an angular velocity $\omega_1$. It jumps to another circular orbit of radius $r_2$ and attains an angular velocity $\omega_2$. If $r_2 = 0.5\, r_1$ and assuming that no external torque is applied to the system,then the angular velocity $\omega_2$ is given by:

An object of mass $m$ is tied to a thin string,which is passed through a hollow tube. The tube is held in one hand and the string in the other. The object is rotated in a circle of radius $R$ with a velocity $v$. If the string is pulled downwards to decrease the radius,which of the following quantities remains conserved?

Given below are two statements: One is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$.
Assertion $(A)$ : In a conical pendulum,angular momentum about its vertical axis remains constant.
Reason $(R)$ : Net torque about the vertical axis of a conical pendulum is not zero.
In the light of the above statements,choose the most appropriate answer from the options given below:

$A$ 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. Find the angular speed of the door just after the bullet embeds into it.