A solid sphere rolls down on an inclined plane of inclination \(\theta\). What is the acceleration as the sphere reaches bottom?
1. \(\frac{5}{7}g\sin\theta\)
2. \(\frac{3}{5}g\sin\theta\)
3. \(\frac{2}{7}g\sin\theta\)
4. \(\frac{2}{5}g\sin\theta\)

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 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

A bob of mass m attached to an inextensible string of length l is suspended from vertical support.  The bob rotates in a horizontal circle with an angular speed $\mathrm{\omega }$ $\mathrm{rad}/\mathrm{s}$ about the vertical.  About the point of suspension.

A wheel is at rest. Its angular velocity increases uniformly and becomes 80 rad/s after 5 s. The total angular displacement is 

1. 800 rad

2. 400 rad

3. 200 rad

4. 100 rad

A constant torque acting on a uniform circular wheel changes its angular momentum from ${\mathrm{A}}_0$ to $4 {\mathrm{A}}_0$ in 4s. The magnitude of this torque is 

1. $\frac{3{\mathrm{A}}_0}{4}$

2. $4 {\mathrm{A}}_0$

3. ${\mathrm{A}}_0$

4. $12 {\mathrm{A}}_0$

A body rolls down an inclined plane without slipping. The fraction of total energy associated with its rotation will be

$1. {\mathrm{K}}^2+ {\mathrm{R}}^2$
$2. \frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$
$3. \frac{{\mathrm{K}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$
$4. \frac{{\mathrm{R}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$

Where k is radius of gyration of the body about an axis passing through centre of mass and R is the radius of the body. 

A solid cylinder rolls down an inclined plane that has friction sufficient to prevent sliding. The ratio of rotational energy to total kinetic energy is

1.  $\frac{1}{2}$

2.  $\frac{1}{3}$

3.  $\frac{2}{3}$

4.  $\frac{3}{4}$

A swimmer while jumping into water from a height easily forms a loop in the air, if

1. He pulls his arms and legs in

2. He spreads his arms and legs

3. He keeps himself straight

4. None of the above