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. \(\frac{2}{5}g\sin\theta\)

Given below are two statements: 

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

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 \(A_0\) to \(4A_0\) in \(4~\text{s}\). The magnitude of this torque is:
1. \(\dfrac{3A_0}{4}\)
2. \(4A_0\)
3. \(A_0\)
4. \(12A_0\)

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

$1.<mspace\; width="1em"></mspace>{\mathrm{K}}^2+<mspace\; width="1em"></mspace>{\mathrm{R}}^2$
$2.<mspace\; width="1em"></mspace>\frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$
$3.<mspace\; width="1em"></mspace>\frac{{\mathrm{K}}^2}{{\mathrm{K}}^2+<mspace\; width="1em"></mspace>{\mathrm{R}}^2}$
$4.<mspace\; width="1em"></mspace>\frac{{\mathrm{R}}^2}{{\mathrm{K}}^2+<mspace\; width="1em"></mspace>{\mathrm{R}}^2}$

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