System of Particles and Rotational Motion

The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of F = i - j + k acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is .

[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 )

Two iron solid discs of negligible thickness have radii R1 and R2 and moment of inertia I1 and I2, respectively. For R2 = 2R1, the ratio of I1 and I2 would be 1/x, where x = .

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n = . Consider all the bodies have equal masses.

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