In an orbital motion, the angular momentum vector is 

1. Along the radius vector

2. Parallel to the linear momentum

3. In the orbital plane

4. Perpendicular to the orbital plane

A particle of mass \(m\) moves with a constant velocity along \(3\) different paths, \(DE, OA\) and \(BC\). Which of the following statements is not correct about its angular momentum about point \(O\)?

$\mathrm{A}<mspace\; width="1em"></mspace>\mathrm{particle}<mspace\; width="1em"></mspace>\mathrm{of}<mspace\; width="1em"></mspace>\mathrm{mass}<mspace\; width="1em"></mspace>0.2<mspace\; width="1em"></mspace>\mathrm{kg}<mspace\; width="1em"></mspace>\mathrm{is}<mspace\; width="1em"></mspace>\mathrm{moving}<mspace\; width="1em"></mspace>\mathrm{in}<mspace\; width="1em"></mspace>\mathrm{a}<mspace\; width="1em"></mspace>\mathrm{circle}<mspace\; width="1em"></mspace>\mathrm{of}<mspace\; width="1em"></mspace>\mathrm{radius}<mspace\; width="1em"></mspace>1\mathrm{m}<mspace\; width="1em"></mspace>\mathrm{with}<mspace\; width="1em"></mspace>$
$\mathrm{f}=2/\mathrm{\pi }<mspace\; width="1em"></mspace>\mathrm{rps},<mspace\; width="1em"></mspace>\mathrm{then}<mspace\; width="1em"></mspace>\mathrm{its}<mspace\; width="1em"></mspace>\mathrm{angular}<mspace\; width="1em"></mspace>\mathrm{momentum}<mspace\; width="1em"></mspace>\mathrm{is}-$
$\left(\mathrm{A}\right)<mspace\; width="1em"></mspace>0.8<mspace\; width="1em"></mspace>{\mathrm{kgm}}^2/\mathrm{s}<mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace>\left(\mathrm{B}\right)<mspace\; width="1em"></mspace>2<mspace\; width="1em"></mspace>{\mathrm{kgm}}^2/\mathrm{s}<mspace\; width="1em"></mspace>$
$\left(\mathrm{C}\right)<mspace\; width="1em"></mspace>8<mspace\; width="1em"></mspace>{\mathrm{kgm}}^2/\mathrm{s}<mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace>\left(\mathrm{D}\right)<mspace\; width="1em"></mspace>16<mspace\; width="1em"></mspace>{\mathrm{kgm}}^2/\mathrm{s}<mspace\; width="1em"></mspace>$

In free space, a rifle of mass \(M\) shoots a bullet of mass \(m\) at a stationary block of mass \(M\) at a distance \(D\) away from it. When the bullet has moved through a distance \(d\) towards the block, the centre of mass of the bullet-block system is at a distance of:
1. \(\frac{D-d}{M+m}~\text{from the bullet}\)
2. \(\frac{md+ MD}{M+m}~\text{from the block}\)
3. \(\frac{2md+ MD}{M+m}~\text{from the block}\)
4. \(\frac{(D-d)M}{M+m}~\text{from the bullet}\)

Blocks A and B are resting on a smooth horizontal surface given equal speeds of 2 m/s in the opposite sense as shown in the figure.

At t = 0, the position of blocks are shown, then the coordinates of centre of mass at t = 3s will be 

1.  (1, 0)

2.  (3, 0)

3.  (5, 0)

4.  (2.25, 0)

A wheel is at rest. Its angular velocity increases uniformly and becomes 80 rad/s after 5 s. The total angular displacement is 

1. 800 rad

2. 400 rad

3. 200 rad

4. 100 rad

Moment of inertia of an object does not depend upon

A force $\overrightarrow{\mathrm{F}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>4\hat{\mathrm{i}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>5\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>3\hat{\mathrm{k}}$ is acting on a point $\overrightarrow{{\mathrm{r}}_1}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>2\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>3\hat{\mathrm{k}}$. The torque acting about a point $\overrightarrow{{\mathrm{r}}_2}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>3\hat{\mathrm{i}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>2\hat{\mathrm{j}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>3\hat{\mathrm{k}}$ is

1.  0

2.  $42\hat{\mathrm{i}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>30\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>6\hat{\mathrm{k}}$

2.  $42\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>30\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>6\hat{\mathrm{k}}$

4.  $42\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>30\hat{\mathrm{j}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>6\hat{\mathrm{k}}$

A thin uniform circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with an angular velocity $\omega$. Another disc of the same dimensions but of mass \(\frac{1}{4}M\) is placed gently on the first disc co-axially. The angular velocity of the system will be:

When a torque acting upon a system is zero, then which of the following will be constant