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. \(120\)

Assertion : The speed of whirlwind in a tornado is alarmingly high.

Reason    : If no external torque acts on a body, its angular velocity remains conserved.

1. If both assertion and reason are true and reason is the correct explanation of assertion.

2. If both assertion and reason are true but reason is not the correct explanation of assertion.

3. If assertion is true but reason is false.

4. If both assertion and reason are false.

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?

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

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