System of Particles and Rotational Motion

A circular disk of moment of inertia It is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed i. Another disk of moment of inertia Ib is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed . The energy lost by the initially rotating disc to friction is

1. 12 Ib2(It + Ib) i2
2. 12 It2(It + Ib) i2
3. Ib - It(It + Ib) i2
4. 12 Ib It(It + Ib) i2

Two particles which are initially at rest, move towards each other under the action of their internal attraction. If their speeds are v and 2v at any instant, then the speed of centre of mass of the system will be :

1. 2v
2. zero
3. 1.5 v
4. v

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity . If two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity

1. MM+2m
2. (M+2m)M
3. MM+m
4. (M-2m)M+2m

If F is the force acting on a particle having position vector r and be the torque of this force about the origin, then

1. r > 0 and F < 0
2. r = 0 and F = 0
3. r = 0 and F 0
4. r 0 and F = 0

Four identical thin rods each of mass M and length l, form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is

1. 23Ml2
2. 133Ml2
3. 13Ml2
4. 43Ml2

Two bodies of mass 1 kg and 3 kg have position vectors i + 2 j + k and -3 i - 2 j + k, respectively. The centre of mass of this system has a position vector :

1. -2 i - j + k
2. 2 i - j - 2 k
3. - i + j + k
4. -2 i + 2 k

A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is 90 . The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is

1. ML26
2. 2 ML224
3. ML224
4. ML212

The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is

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

A particle of mass m moves in the XY plane with a velocity v along the straight line AB. If the angular momentum of the particle with respect to origin O is LA when it is at A and LB when it is at B, then [figure]

1. LA = LB
2. the relationship between LA and LB depends upon the slope of the line AB
3. LA < LB
4. LA > LB

A uniform rod AB of length l and mass m is free to rotate about point A. The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about A is ml23, the initial angular acceleration of the rod will be [figure]

1. mgl2
2. 32gl
3. 3g2l
4. 2g3l