System of Particles and Rotational Motion

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 position vectors of two 1 kg particles, (A) and (B), are given by rA= ( 1t2 i+ 2t j+ 3t k )m and rB= ( 1t i+ 2t2 j+ 3t k )m, respectively; ( 1=1 m/s2, 2=3n m/s, 3=2 m/s, 1=2 m/s, 2=-1 m/s2, 3=4p m/s), where t is time, n and p are constants. At t=1 s, | VA |= | VB | and velocities VA and VB of the particles are orthogonal to each other. At t=1 s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is L kg m2 s-1. The value of L is .

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 string is wrapped around the rim of a wheel of moment of inertia 0.40 kgm2 and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is x rad/s, where x is .

A circular disc reaches from top to bottom of an inclined plane of length l. When it slips down the plane, it takes t s. When it rolls down the plane then it takes ( 2 )1/2t s, where is .

A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of x and y coordinates of center of mass of plate is n9. The value of n is . [figure]

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

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be hours 30 minutes.

Three balls of masses 2 kg, 4 kg and 6 kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be kg m2.