The centre of the mass of \(3\) particles, \(10~\text{kg},\)  \(20~\text{kg},\) and \(30~\text{kg},\) is at \((0,0,0).\) Where should a particle with a mass of \(40~\text{kg}\) be placed so that its combined centre of mass is \((3,3,3)?\)
1. \((0,0,0)\)
2. \((7.5, 7.5, 7.5)\)
3. \((1,2,3)\)
4. \((4,4,4)\)

The position of a particle is given by \(\vec r = \hat i+2\hat j-\hat k\) and momentum \(\vec P = (3 \hat i + 4\hat j - 2\hat k)\). The angular momentum is perpendicular to:

A body falling vertically downward under gravity breaks into two parts of unequal masses. The center of mass of the parts taken together shifts horizontally towards

1.  heavier piece 

2.  lighter piece 

3.  does not shift horizontally 

4.  depends on the vertical velocity at the time of breaking 

A particle of mass m moves along line PC with velocity v as shown. What is the angular momentum of the particle about O

1. \(\text {mvL}\)

2. \(mvl\)

3. \(mvr\)

4. \(0\)

A solid sphere, disc, and solid cylinder all of the same mass and made up of the same material are allowed to roll down (from rest) on an inclined plane, then,

Two rings of the same radius and mass are placed such that their centers are at a common point and their planes are perpendicular to each other. The moment of inertia of the system about an axis passing through the center and perpendicular to the plane of one of the rings is (mass of the ring and radius = r)

1.  $\frac{1}{2}{\mathrm{mr}}^2$

2.  ${\mathrm{mr}}^2$

3.  $\frac{3}{2}{\mathrm{mr}}^2$

4.  $2{\mathrm{mr}}^2$

In a rectangle ABCD (BC = 2AB). The moment of inertia along which axis will be minimum?
               
1. BC
2. BD
3. HF
4. EG

A wheel comprises of a ring of radius R and mass M and three spokes of mass m each. The moment of inertia of the wheel about its axis is

1.  \(\left(M   +   \frac{m}{4}\right) R^{2} \)

2.  \(\left(M   +   m\right) R^{2} \)

3.  \(\left(M   +   3 m\right) R^{2} \)

4.  \(\left(\frac{M   +   m}{2}\right) R^{2}\)

The angular velocity of a body changes from $\frac{1}{25}$ rev./sec to 1 rev./sec. without applying any torque. The ratio of its radius of gyration in two situations is :

1.  5: 1

2.  1: 5

3.  1: 25

4.  25: 1