Two particles of equal mass have coordinates \((2~\text{m}, 4~\text{m}, 6~\text{m})\) and \((6~\text{m},2~\text{m},8~\text{m})\). Of these one particle has velocity ${\mathrm{}}_{}$\(v_1 = 2\hat i~\text{m/s}\) and another particle has velocity \(v_2 = 2\hat j ~\text{m/s}\) at time \(t=0\). The coordinate of their centre of mass at time \(t=1~\text{s}\) will be:
1. \((4~\text{m}, 4~\text{m}, 7~\text{m})\)
2. \((5~\text{m}, 4~\text{m}, 7~\text{m})\)
3. \((2~\text{m}, 4~\text{m}, 6~\text{m})\)
4. \((4~\text{m}, 5~\text{m}, 4~\text{m})\)

The M.I. of a body about the given axis is 1.2 kg x ${\mathrm{m}}^2$ initially the body at rest. In order to the rotational kinetic energy of 1500 J, the angular acceleration of 25 rad/${\sec }^2$ must be about that axis for the duration of

1.  4 sec

2.  2 sec

3.  8 sec

4.  10 sec

A loop rolls down on the inclined plane. The fraction of its total kinetic energy that is associated with rotational motion is

1.  \(1\over 2\)

2.  \(1\over 3\)
3.  \(3\over 2\)

4.  \(2\over 3\)

A ball kept in a closed box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass

1. of the box remains constant 

2. of the box plus the ball system remains constant

3. of the ball remains constant

4. of the ball relative to the box remains constant

Three indentical spheres, each of mass 1 kg are kept as shown in figure, touching each other, with their centres on a straight line. If their centres are marked P, Q, R respectively, the distance of centre of mass of the system from P is

$1. \frac{\mathrm{PQ} + \mathrm{PR} + \mathrm{QR}}{3}$
$2. \frac{\mathrm{PQ} +\hspace{0.17em}\mathrm{PR}}{3}$
$3. \frac{\mathrm{PQ} +\hspace{0.17em}\mathrm{QR}}{3}$
$4. \frac{\mathrm{PR} + \mathrm{QR}}{3}$

A small disc of radius 2 cm is cut from a disc of radius 6 cm. If the distance between their centres is 3.2 cm, what is the shift in the centre of mass of the disc?

1.  0.4 cm

2.  2.4 cm

3.  1.8 cm

4.  1.2 cm

A circular plate of diameter d is kept in contact with a square plate of an edge as shown in the figure. The density of the material and the thickness are the same everywhere. The center of mass of the composite system will be

1.  inside the circular plate 

2.  inside the square plate

3.  at the point of contact 

4.  outside the system

Four masses are fixed on a massless rod as shown in the figure. The moment of inertia about the axis P is about

1.  2 kg-${\mathrm{meter}}^2$

2.  1 kg-${\mathrm{meter}}^2$

3.  0.5 kg-${\mathrm{meter}}^2$

4.  0.3 kg-${\mathrm{meter}}^2$

A particle moves in a force field given by $\overrightarrow{\mathrm{F}} = \hat{\mathrm{r}} \mathrm{F} \left(\mathrm{r}\right)$, where r is the unit vector along with the position vector $\overrightarrow{\mathrm{r}}$, then which is true? 

1.  The torque acting on the particle is not zero 

2.  The torque acting on the particle produces an angular acceleration in it 

3.  The angular momentum of the particle is conserved 

4.  The angular momentum of the particle increases.

A disc is rotating With an angular speed of m. If a child sits on it, which of the following is conserved? 

1.  kinetic energy 

2.  potential energy 

3.  linear momentum 

4.  angular momentum