System of Particles and Rotational Motion

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

A circular disc is to be made by using iron and aluminium so that it acquired maximum moment of inertia about geometrical axis. It is possible with

1. aluminium at interior and iron surround to it
2. iron at interior and aluminium surround to it
3. using iron and aluminium layers in alternate order
4. sheet of iron is used at both external surface and aluminium sheet as internal layers.

A disc is rolling, the velocity of its centre of mass is vcm. Which one will be correct?

1. the velocity of the highest point is 2vcm and of the point of contact is zero
2. the velocity of the highest point is vcm and of the point of contact is vcm
3. the velocity of the highest point is 2vcm and of the point of contact is vcm
4. the velocity of the highest point is 2vcm and of the point of contact is 2vcm.

For the adjoining diagram (a uniform right-angled triangular plate ABC), the correct relation between I1, I2 and I3 is (I – moment of inertia about the marked side) [figure]

1. I1 > I2
2. I2 > I1
3. I3 > I1
4. I3 > I2

For a hollow cylinder and a solid cylinder rolling without slipping on an inclined plane, then which of these reaches earlier

1. solid cylinder
2. hollow cylinder
3. both simultaneously
4. can't say anything