If the net external forces acting on the system of particles is zero, then which of the following may vary ?

1. Momentum of the system

2. Velocity of centre of mass

3. Position of centre of mass

4. None of the above

A constant torque of 1000 N-m turns a wheel of moment of inertial 200 $\mathrm{kg}-{\mathrm{m}}^2$ about an axis through its centre. Its angular velocity after 3 s is

1. 1 rad/s

2. 5 rad/s

3. 10 rad/s

4. 15 rad/s

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 circular disc of radius r has a uniform thickness.  A circular hole of diameter equal to the radius of the disc has been cut out as shown.  The centre of gravity of the remaining disc lies on the diameter of the disc at a distance x to the left of the centre of the original disc.  The value of x is

1. r
2. \({ r \over 2}\)
3. \({ r \over 4}\)
4. \({ r \over 6}\)$\frac{\mathrm{}}{}$

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. \(120\)

Given below are two statements: 

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