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

   

$1.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\mathrm{L}$
$2.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}\mathrm{L}$
$3.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{{\mathrm{m}}_2\mathrm{L}}{{\mathrm{m}}_1+<mspace\; width="1em"></mspace>{\mathrm{m}}_2}$
$4.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{{\mathrm{m}}_1\mathrm{L}}{{\mathrm{m}}_1+<mspace\; width="1em"></mspace>{\mathrm{m}}_2}$

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 wheel is rotating about an axis through its centre at \(720~\text{rpm}.\) It is acted upon by a constant torque opposing its motion for \(8\) seconds to bring it to rest finally. The value of torque in \((\text{N-m })\) is:
(given \(I=\frac{24}{\pi}~\text{kg.m}^2)\) 
1. \(48\)
2. \(72\)
3. \(96\)
4. \(120\)

A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 2r and thickness $\frac{\mathrm{t}}{4}$. The relation between moments of inertia ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ is

1.  ${\mathrm{I}}_{\mathrm{A}}<mspace\; width="1em"></mspace>><mspace\; width="1em"></mspace>{\mathrm{I}}_{\mathrm{B}}$

2.  ${\mathrm{I}}_{\mathrm{A}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>{\mathrm{I}}_{\mathrm{B}}$

3.  ${\mathrm{I}}_{\mathrm{A}}<mspace\; width="1em"></mspace><<mspace\; width="1em"></mspace>{\mathrm{I}}_{\mathrm{B}}$

4.  Depends on the actual values of t and r

A particle of mass 2 kg located at the position $(\hat{\mathrm{i}}+<mspace\; width="1em"></mspace>\hat{\mathrm{j}})$ m has a velocity of 2$(+\hat{\mathrm{i}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+\hspace{0.17em}\hat{\mathrm{k}})$ m/s. Its angular momentum about z-axis in kg-${\mathrm{m}}^2/\mathrm{s}$ is :

1. +4

2. +8

3. -4

4. -8

A solid 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. \(a=g,~T=\dfrac{Mg}{2}\)
2. \(a=\dfrac{g}{2},~T=\dfrac{Mg}{2}\)
3. \(a=\dfrac{g}{3},~T=\dfrac{Mg}{3}\)
4. \(a=\dfrac{2g}{3},~T=\dfrac{Mg}{6}\)

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 \(O\)?