System of Particles and Rotational Motion

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