The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle $$\theta $$ without slipping and slipping down the incline without rolling is

A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions s^$$-$$2 is

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D₁ has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s^$$-$$1. Disc D₂ has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s^$$-$$1. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad s^$$-$$1) of the system is

The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centres and perpendicular to their planes are