System of Particles and Rotational Motion

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is x5. The value of x is .

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reach upto maximum heights h1 and h2 respectively, above the initial level. The ratio h1 : h2 is n10. The value of n is .

In a system two particles of masses m1 = 3 kg and m2 = 2 kg are placed at certain distance from each other. The particle of mass m1 is moved towards the center of mass of the system through a distance 2 cm. In order to keep the center of mass of the system at the original position, the particle of mass m2 should move towards the center of mass by the distance cm.

Three identical spheres each of mass 2M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4 m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is 4 2x, where the value of x is .

A disc of radius R and mass M is rolling horizontally without slipping with speed v. It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is : [figure]

1. 34 v2g
2. v2g
3. 23 v2g
4. 12 v2g

A uniform rod AB of mass 2 kg and length 30 cm is at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be x s, where x= .

A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is J.

A body of mass 'm' is projected with a speed 'u' making an angle of 45 with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as 2mu3Xg. The value of 'X' is .

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is x20 kgm2, where the value of x is .

A particle of mass m is projected with a velocity 'u' making an angle of 30 with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :

1. mu3 2 g
2. zero
3. 32 mu2g
4. 316 mu3g