System of Particles and Rotational Motion

Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is kg m2. (g = 9.8 m/s2)

1. 8.3 10-3
2. 4.75 10-3
3. 1.86 10-2
4. 9.5 10-3

A thin uniform rod (X) of mass M and length L is pivoted at a height ( L3 ) as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is . (g = gravitational acceleration) [figure]

1. 32 gL
2. 3gL
3. 3 2 gL
4. 1 2 gL

A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to I1. A small concentric cylinder of length L/2 and radius R/3 carved from this cylinder has moment of inertia about its axis equal to I2. The ratio I1/I2 is .

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is :

1. 43 Lmd2
2. 32 Lmd2
3. 2L5md2
4. 2Lmd2

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (R < L) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M):

1. 34MR2 + 16ML2
2. 38MR2 + 712ML2
3. 34MR2 + 712ML2
4. 38MR2 + 16ML2

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by x256 M r2. The value of x is . [figure]

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is kg m2.

1. 0.18
2. 0.72
3. 0.36
4. 0.63

A circular disc has radius R1 and thickness T1. Another circular disc made of the same material has radius R2 and thickness T2. If the moments of inertia of both discs are the same and R1R2 = 2 then T1T2 = 1 . The value of is .

A uniform bar of length 12 cm and mass 20 m lies on a smooth horizontal table. Two point masses m and 2m are moving in opposite directions with same speed of v and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency . The ratio of v and is : [figure]

1. 33
2. 2 88
3. 32
4. 66

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are false
2. Statement I is false but Statement II is true
3. Statement I is true but Statement II is false
4. Both Statement I and Statement II are true