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 :

Consider a hydrogen atom with $v_k, r_k,$ and $K_k$ denoting the velocity, orbital radius and kinetic energy of the electron in the $k^{{th}}$ orbit, respectively. The electron undergoes a transition from the $n^{{th}}$ orbit, emitting radiation corresponding to the Lyman series. Considering $h$ to be the Planck’s constant and $ ho_0$ the permittivity of the free space, the correct statement(s) is/are:

A particle is thrown with a speed v from a point O at an angle θ with the horizontal plane such that it passes through the point P at a height of 1 m and horizontal distance of 5 m from O, as shown in the figure. If acceleration due to gravity is g $\text{ms}^{-2}$, then the correct statement(s) is/are: