$A$ force $\vec F = \alpha \hat i + 3\hat j + 6\hat k$ is acting at a point $\vec r = 2\hat i - 6\hat j - 12\hat k$. The value of $\alpha$ for which angular momentum about the origin is conserved is:

$A$ particle of mass $m$ moves with a velocity $v$ along the rim of a disc of radius $R$ in the counter-clockwise direction. The disc has a moment of inertia $I$ and is rotating in the clockwise direction with an angular velocity $\omega$. If the particle stops moving,what will be the angular velocity of the disc?

$A$ thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $\omega '$ equal to:

$A$ solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $\omega_1$ about its diameter. After some time,it starts losing mass at a uniform rate,with no change in its shape. The angular velocity of the sphere when its radius becomes $R/2$ is $x\omega_1$. The value of $x$ is . . . . . . .

$A$ person is standing at the edge of a circular plate that is rotating with a constant angular speed about an axis passing through its center and perpendicular to its plane. If the person starts walking along the radius towards the axis, the angular velocity of the system will:

$A$ child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$. The child now stretches his arms so that the moment of inertia of the system becomes double. The kinetic energy of the system now is