Two spherical planets P and Q have the same uniform density r, masses M_P and M_Q and surface areas A and 4A respectively. A spherical planet R also has uniform density r and its mass is (M_P + M_Q). The escape velocities from the planets P, Q and R are V_P, V_Q and V_R, respectively. Then

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $$\omega $$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $${\omega \over 2}$$. The ring and disc are separated by frictionless ball bearings. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of $$30^\circ $$ with the horizontal. Then with respect to the horizontal surface,

Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

Two identical discs of same radius R are rotating about their axes in opposite directions with the same constant angular speed $$\omega $$. The discs are in the same horizontal plane. At time t = 0, the points P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is v_r. In one time period (T) of rotation of the discs, v_r as a function of time is best represented by