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 $O$?

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

A sphere is rolling down a plane of inclination $\theta$ to the horizontal.  The acceleration of its centre down the plane is 

1. g sin $\theta$                   

2. less than g sin $\theta$

3. greater than g sin $\theta$

4. zero

A bob of mass m attached to an inextensible string of length l is suspended from vertical support.  The bob rotates in a horizontal circle with an angular speed $\mathrm{\omega }$ $\mathrm{rad}/\mathrm{s}$ about the vertical.  About the point of suspension.

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)