A solid cylinder of mass $M$ and radius $R$ is pivoted at its centre and three particles of mass $m$ are fixed at the perimeter of the cylinder. Find the angular velocity of the cylinder after the system has moved through $90°$.

Figure shows a sphere from which a small sphere is excavated. Find the MI of this system about the centre of bigger circle.

Two particles of masses $m$ and $M$ are a distance $d$ apart and constitute a system. If $v$ is the frequency of revolution about the axis passing through the centre of mass and perpendicular to the line joining the masses, the rotational $K.E.$ of the system is,

Consider you are riding a bicycle. Case I:
You are pedalling the cycle and it is rolling on street.
Case II:
You have stopped pedalling and the cycle is still rolling on ground.

Find the product of the dot products of the unit vectors along the direction of friction on the two tyres in the two cases.

A disc of mass $M$ and radius $R$ is lying on the $x\text{-}y$ plane. The locus of all points on the $x\text{-}y$ plane about which the moment of inertia of the rod is same as that about $0$ will be,

The figure shows two systems in which the spring and sphere remain the same but the connection of spring changes.

Find the ratio of the friction forces in the two cases. (co-ordinate axes are as given)