A cylinder of mass 'M' is suspended by two strings wrapped around it as shown. The acceleration 'a' and the tension T when the cylinder falls and the string unwinds itself are, respectively,
             

1.  $\mathrm{a} = \mathrm{g}, \mathrm{T} = \frac{\mathrm{Mg}}{2}$

2.  $\mathrm{a} = \frac{g}{2}, \mathrm{T} = \frac{\mathrm{Mg}}{2}$

3.  $\mathrm{a} = \frac{g}{3}, \mathrm{T} = \frac{\mathrm{Mg}}{3}$

4.  $\mathrm{a} = \frac{2g}{3}, \mathrm{T} = \frac{\mathrm{Mg}}{6}$

An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver ?

1. 350 N-m

2. 440 N-m

3. 531 N-m

4. 628 N-m

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?

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

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

1.  mass of object

2.  mass distribution

3.  angular velocity

4.  axis of rotation

A force $\overrightarrow{\mathrm{F}} = 4\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is acting on a point $\overrightarrow{{\mathrm{r}}_1} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. The torque acting about a point $\overrightarrow{{\mathrm{r}}_2} = 3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ is

1.  0

2.  $42\hat{\mathrm{i}} - 30\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

2.  $42\hat{\mathrm{i}} + 30\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

4.  $42\hat{\mathrm{i}} + 30\hat{\mathrm{j}} - 6\hat{\mathrm{k}}$