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

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

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

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