System of Particles and Rotational Motion

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m, would be: [figure]

1. 20 rad/s
2. 30 rad/s
3. 10 rad/s
4. 0 rad/s

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

1. 38Mr2
2. 2Mr2
3. 12Mr2
4. 32Mr2

[figure] Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At t=0, they are given an initial velocity VA=V0 AC, VB=V0 BA and VC=V0 CB. Here, AC, CB and BA are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is :

1. 12 a mV0
2. 3 a mV0
3. 32 a mV0
4. 32 a mV0

The center of mass of a thin rectangular plate (fig - x) with sides of length a and b, whose mass per unit area ( ) varies as = 0 xab (where 0 is a constant), would be . [figure]

1. ( 23a, 23b )
2. ( 23a, b2 )
3. ( 13a, b2 )
4. ( a2, b2 )

A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is

1. 290 g
2. 200 g
3. 190 g
4. 300 g

An object of mass 'm' is projected from origin in a vertical xy plane at an angle 45 with the x-axis with an initial velocity v0. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

1. mv032 2 g along negative z-axis
2. mv032 2 g along positive z-axis
3. mv034 2 g along positive z-axis
4. mv034 2 g along negative z-axis

A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45 . If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

1. 2 g
2. 1 2 g
3. 13 2 g
4. 2 g3

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be t1 and t2, respectively, then

1. t1 < t2
2. t1 = 2t2
3. t1 > t2
4. t1 = t2

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

1. 34
2. 43
3. 52
4. 25

A solid sphere of mass ' m ' and radius ' r ' is allowed to roll without slipping from the highest point of an inclined plane of length ' L ' and makes an angle 30 with the horizontal. The speed of the particle at the bottom of the plane is v1. If the angle of inclination is increased to 45 while keeping L constant. Then the new speed of the sphere at the bottom of the plane is v2. The ratio v12:v22 is

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

Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be

1. 1.5 cm
2. 2.0 cm
3. 0.5 cm
4. 1.0 cm

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that (t)=5t2-8t, where (t) is the angular position of the rotating disc as a function of time t. How much power is delivered by the applied torque, when t=2 s ?

1. 60 MR2
2. 72 MR2
3. 8 MR2
4. 108 MR2

A uniform circular disc of radius ' R ' and mass ' M ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above. [figure]

1. 1732MR2
2. 1332MR2
3. 932MR2
4. 732MR2

The torque due to the force (2 i+ j+2 k ) about the origin, acting on a particle whose position vector is ( i+ j+ k ), would be

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

A heavy iron bar, of weight W is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle with the horizontal. The weight experienced by the person is :

1. W
2. W
3. W2
4. W

A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity . If another disc of same dimensions but of mass M/2 is placed gently on the first disc co-axially, then the new angular velocity of the system is :

1. 45
2. 54
3. 32
4. 23

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is 8/x. The value of x is : [figure]

1. 34
2. 51
3. 67
4. 17

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 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

A heavy iron bar of weight 12 kg is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle 60 with the horizontal, the weight experienced by the man is :

1. 3 kg
2. 6 kg
3. 6 3 kg
4. 12 kg