Two small beads are initially at $A$ on an equilateral triangle made of uniform wires. The triangle is rotated about the vertical axis $AO$. The beads are released from rest simultaneously and slide down along $AB$ and $AC$ respectively (as shown in the figure). Which quantities are conserved as the beads slide down? (Friction is negligible)

The angular velocity of a body changes from $\omega_1$ to $\omega_2$ without the application of any external torque,but this change occurs due to a change in its moment of inertia. The ratio of the radii of gyration in the two cases will be:

If the mass of the Earth is kept constant and its radius is reduced to half, then the duration of the day will become ........ $hr$.

Due to the melting of ice at the poles of the Earth,the moment of inertia of the Earth ....,the angular velocity ....,and the duration of the day becomes ....

$A$ horizontal circular platform of radius $R = 0.5 \ m$ and mass $M = 0.45 \ kg$ is free to rotate about its axis. Two massless spring toy-guns,each carrying a steel ball of mass $m = 0.05 \ kg$,are attached to the platform at a distance $r = 0.25 \ m$ from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform,the balls have a horizontal speed of $v = 9 \ m/s$ with respect to the ground. The rotational speed of the platform in $rad/s$ after the balls leave the platform is:

$A$ uniform rod of length $6L$ and mass $8m$ is pivoted at its centre $C$. Two masses $m$ and $2m$ with speeds $2v$ and $v$ respectively,as shown in the figure,strike the rod and stick to it. Initially,the rod is at rest. If it rotates with angular velocity $\omega$ due to the impact,then $\omega$ will be: