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.]

Consider a circuit consisting of a capacitor of capacitance C and a coil with N turns per unit length, cross sectional area S and length d, where $d^2 \gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is L. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

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:

A double convex lens made of glass of refractive index 1.5 and radii of curvature of the curved surfaces 20 cm each is immersed in a liquid of refractive index n_L. The correct plot showing the variation of the power, in the units of diopter (D), as a function of n_L is :