System of Particles and Rotational Motion

A mass m moves in a circle on a smooth horizontal plane with velocity v0 at a radius R0. The mass is attached to a string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally m moves in a circle of radius R02. [figure] The final value of the kinetic energy is

1. 2mv02
2. 12mv02
3. mv02
4. 14mv02

Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX' which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is [figure]

1. 165mr2
2. 4mr2
3. 115mr2
4. 3mr2

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

1. 25 N
2. 50 N
3. 78.5 N
4. 157 N

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

1. 5 : 7
2. 2 : 3
3. 2 : 5
4. 7 : 5

A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is [figure]

1. 2gL
2. 2g3L
3. 3g2L
4. gL

A small object of uniform density rolls up a curved surface with an initial velocity 'v'. It reaches upto a maximum height of 3v24g with respect to the initial position. The object is

1. hollow sphere
2. disc
3. ring
4. solid sphere

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

1. 1 : 2
2. 3 : 2
3. 2 : 1
4. 2 : 1

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

1. 60
2. 100
3. 120
4. 40

A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 m and its moment of inertia about the axle is 200 kg m2. It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 m s-1 relative to the ground. Time taken by the man to complete one revolution is

1. s
2. 3 2 s
3. 2 s
4. 2 s

The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through [figure]

1. B
2. C
3. D
4. A