System of Particles and Rotational Motion

A thin circular ring of mass M and radius r is rotating about its axis with constant angular velocity . Two objects each of mass m are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with angular velocity given by

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

Which one of the following pairs of statements is correct ?

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

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

A wheel has angular acceleration of 3.0 rad/sec2 and an initial angular speed of 2.00 rad/sec. In a time of 2 sec it has rotated through an angle (in radian) of

1. 10
2. 12
3. 4
4. 6

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

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

The moment of inertia of a uniform circular disc of radius R and mass M about an axis touching the disc at its diameter and normal to the disc

1. 12MR2
2. MR2
3. 25MR2
4. 32MR2

The moment of inertia of a uniform circular disc of radius R and mass M about an axis passing through the edge of the disc and normal to the disc is

1. MR2
2. 12MR2
3. 32MR2
4. 72MR2

A drum of radius R and mass M, rolls down without slipping along an inclined plane of angle . The frictional force

1. dissipates energy as heat
2. decreases the rotational motion
3. decreases the rotational and translational motion
4. converts translational energy to rotational energy.

Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular velocity will be in the ratio

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

Three particles, each of mass m gram, are situated at the vertices of an equilateral triangle ABC of side l cm (as shown in the figure). The moment of inertia of the system about a line AX perpendicular to AB and in the plane of ABC, in gram-cm2 units will be [figure]

1. 34ml2
2. 2ml2
3. 54ml2
4. 32ml2

Consider a system of two particles having masses m1 and m2. If the particle of mass m1 is pushed towards the mass centre of particles through a distance d, by what distance would the particle of mass m2 move so as to keep the mass centre of particles at the original position?

1. m1m1 + m2d
2. m1m2d
3. d
4. m2m1d

A wheel having moment of inertia 2 kg m2 about its vertical axis, rotates at the rate of 60 rpm about this axis. The torque which can stop the wheel's rotation in one minute would be

1. 2 15 N m
2. 12 N m
3. 15 N m
4. 18 N m

A round disc of moment of inertia I2 about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia I1 rotating with an angular velocity about the same axis. The final angular velocity of the combination of discs is

1. I2 I1 + I2
3. I1 I1 + I2
4. (I1 + I2) I1

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is

1. 2 : 3
2. 2 : 1
3. 5 : 6
4. 1 : 2

A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is K. If radius of the ball be R, then the fraction of total energy associated with its rotational energy will be

1. K2 + R2R2
2. K2R2
3. K2K2 + R2
4. R2K2 + R2

A solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom ?

1. 2gh
2. 34gh
3. 43gh
4. 4gh

A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity . Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be :

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

A rod of length 3 m and its mass acting per unit length is directly proportional to distance x from one of its end then its centre of gravity from that end will be at :

1. 1.5 m
2. 2 m
3. 2.5 m
4. 3.0 m

A point P is considered at contact point of a wheel on ground which rolls on ground without slipping then value of displacement of point P when wheel completes half of rotation (if radius of wheel is 1 m)

1. 2 m
2. 2+4 m
3. m
4. 2+2 m

A solid sphere of radius R is placed on smooth horizontal surface. A horizontal force F is applied at height h from the lowest point. For the maximum acceleration of centre of mass, which is correct ?

1. h = R
2. h = 2R
3. h = 0
4. no relation between h and R.

A disc is rotating with angular speed . If a child sits on it, what is conserved

1. linear momentum
2. angular momentum
3. kinetic energy
4. potential energy