$A$ uniform disc of mass $M$ and radius $R$ is rotating about a horizontal axis passing through its centre with angular velocity $\omega$. $A$ piece of mass $m$ breaks from the disc and flies off vertically upwards. The angular speed of the disc will be

If no external torque acts on a system,which of the following remains constant in both magnitude and direction?

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:

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery tangentially and sticks to it. Find the angular velocity of the system after the collision in $rad/s$.

$A$ circular disc has a moment of inertia $I_2$ about an axis passing through its center and perpendicular to its plane. Another disc with a moment of inertia $I_1$ is placed on it. If the system rotates about the same axis with an initial angular velocity $\omega$,what will be the final angular velocity of the combined system?

$A$ circular disc is rotating with an angular velocity $\omega$. If a child sits on the disc,which of the following quantities will be conserved?