System of Particles and Rotational Motion

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

Two masses m and 2m are connected by a light string going over a pulley (disc) of mass 30m with radius r = 0.1 m. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2m mass is released from rest and its speed when it has descended through a height of 3.6 m is m/s. (Assume string does not slip and g = 10 m/s2)

A uniform rod of mass m and length l is suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is . (g acceleration due to gravity) [figure]

1. mg/3
2. mg/4
3. mg
4. mg/2

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is x12ML2 kg m2. The value of x is . [figure]

Two cars A and B each of mass 103 kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is J s.

1. 3.6 105
2. 105
3. 3 105
4. 2 105

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitude of the acceleration experienced by the blocks is (assume no slipping of string on pulley). [figure]

1. (M - m)g [ ( 83 )M + m ]
2. (M - m)g2M + m
3. (M - m)gM + m
4. (M - m)g [ ( 136 )M + m ]