Consider a large disk of radius R and two smaller disks, each of radius r = R / 50, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation Δθ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities ω and 2ω while the large disk is held stationary. The time τ at which the smaller disks are again in contact is:
[Use sin(Δθ)=Δθ and ignore gravity.]

A solid cylinder of radius R rolls without slipping with a center of mass speed v₀ = $\sqrt{\frac{gR}{3}}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, g is the acceleration due to gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

List-I shows four planar structures made of uniform solid rods each of mass $m$ and length $l$. In the List-II the possible moment of inertia of these structures about an axis $OCO'$, which lies in the plane of the structures, are given.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

List-I	List-II
(P)	(1) $$\frac{5}{4}ml^2$$
(Q)	(2) $$\frac{1}{6}ml^2$$
(R)	(3) $$\frac{1}{12}ml^2$$
(S)	(4) $$\frac{2}{3}ml^2$$
	(5) $$\frac{1}{3}ml^2$$

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$, inside a ring of radius $R>r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $T=\frac{2 \pi}{\omega}$. The correct expression for $\omega$ is ( $g$ is the acceleration due to gravity):