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 \(O\)?

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

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. \(\frac{(D-d)M}{M+m}~\text{from the bullet}\)