$A$ 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:

$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.
$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.

$A$ smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size,each of mass $m$,which can slide freely along the rod. Initially,the two beads are at the center of the rod and the system is rotating with angular velocity $\omega_0$ about its axis perpendicular to the rod and passing through its mid-point (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is:

An equilateral triangle $ABC$ is made of a uniform wire. Two identical beads are initially at $A$. The triangle is set to rotate about a vertical axis $AO$. The beads are then released simultaneously from rest to slide down along $AB$ and $AC$. Neglecting friction,which quantities are conserved as the beads slide down?

$A$ boy runs and sits on a rotating platform. Which of the following quantities will be conserved?

$A$ wheel is rotating freely with an angular speed $\omega$ on a shaft. The moment of inertia of the wheel is $I$ and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia $3I$ initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is: