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.  I

3.  $\frac{2}{3}l$

4.  $\frac{3}{2}l$

A wheel is rotating about an axis through its centre at \(720\) r.p.m. 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 N-m is: (given \(I\) = $\frac{24}{\mathrm{\pi }}$ kg ${\mathrm{m}}^2$)
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.  ${\mathrm{I}}_{\mathrm{A}} > {\mathrm{I}}_{\mathrm{B}}$

2.  ${\mathrm{I}}_{\mathrm{A}} = {\mathrm{I}}_{\mathrm{B}}$

3.  ${\mathrm{I}}_{\mathrm{A}} < {\mathrm{I}}_{\mathrm{B}}$

4.  Depends on the actual values of t and r

A particle of mass 2 kg located at the position $(\hat{\mathrm{i}}+ \hat{\mathrm{j}})$ m has a velocity of 2$(+\hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}})$ 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 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?