Point masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are placed at the opposite ends of a rigid of length L and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\mathrm{\omega }}_0$ is minimum is given by

   

$1. \mathrm{x} =\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\mathrm{L}$
$2. \mathrm{x} = \frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}\mathrm{L}$
$3. \mathrm{x} = \frac{{\mathrm{m}}_2\mathrm{L}}{{\mathrm{m}}_1+ {\mathrm{m}}_2}$
$4. \mathrm{x} = \frac{{\mathrm{m}}_1\mathrm{L}}{{\mathrm{m}}_1+ {\mathrm{m}}_2}$

A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force '$\overrightarrow{\mathrm{F}}$' is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C

1.  \(\frac{4}{3} l\)

2.  \(l\)

3.  \(\frac{2}{3} l\)

4.  \(\frac{3}{2} l\)

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 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)

A flywheel is in the form of a uniform circular disc of radius 1 m and mass 2 kg. The work which must be done on it to increase its frequency of rotation from 5 rev ${\mathrm{s}}^{-1}$ to 10 rev ${\mathrm{s}}^{-1}$ is approximately

1.  1.5 x ${10}^2$ J

2.  3.0 x ${10}^2$ J

3.  1.5 x ${10}^3$ J

4.  3.0 x ${10}^3$ J