System of Particles and Rotational Motion

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

[figure] Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is J.

Two discs of moment of inertia I1=4 kg m2 and I2=2 kg m2, about their central axes & normal to their planes, rotating with angular speeds 10 rad/s & 4 rad/s respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is J.

A cylinder is rolling down on an inclined plane of inclination 60 . Its acceleration during rolling down will be x 3 m/s2, where x= (use g=10 m/s2).