$A solid sphere rolls down on an inclined plane of inclination \theta .$
$What is the acceleration as the sphere reaches bottom?$
$\mathbf{[}\mathit{O}\mathit{d}\mathit{i}\mathit{s}\mathit{h}\mathit{a}\mathbf{ }\mathit{J}\mathit{E}\mathit{E}\mathbf{ }\mathbf{2002}\mathbf{,}\mathbf{ }\mathbf{03}\mathbf{;}\mathbf{ }\mathit{C}\mathit{B}\mathit{S}\mathit{E}\mathbf{ }\mathit{P}\mathit{M}\mathit{T}\mathbf{ }\mathbf{2014}\mathbf{]}$
$\left(a\right) \frac{5}{7}gSin\theta \left(b\right) \frac{3}{5}gSin\theta$
$\left(c\right) \frac{2}{7}gSin\theta \left(d\right) \frac{2}{5}gSin\theta$

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

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has a 2 kg mass, 0.2 m radius, and an initial angular velocity of 50 rad s${}^{-1}$. Disc D${}_2$ has 4 kg mass, 0.1 m radius, and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s${}^{-1}$) of the system will be:

1. 60

2. 100

3. 120

4. 40

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.  $\frac{3}{4}$

An inclined plane makes an angle of $30°$ with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to

1.  $\frac{g}{3}$

2.  $\frac{5g}{7}$

3.  $\frac{2g}{3}$

4.  $\frac{5g}{14}$

A disk and a sphere of the same radius but different masses roll off on two inclined planes Of the same altitude and length Which one of the two objects gets to the bottom of the plane first

1.  Disk

2.   Sphere

3.  Both reach at the same time

4.  Depends on their masses

If the equation for the displacement of a particle moving on a circular path is given by $\theta =2t^3+0.5$, where $\theta$ is in radian and t is in second, then the angular velocity of the particle after 2s is 

(1) 8 rad/s

(2) 12 rad/s

(3) 24 rad /s

(4) 36 rad/s

A man weighing \(80\) kg is standing in a trolley weighing \(320\) kg. The trolley is resting on frictionless horizontal rails. If the man starts walking on the trolley with a speed of \(1\) m/s, then after \(4\) s his displacement relative to the ground will be:
1. \(5\) m
2. \(4.8\) m
3. \(3.2\) m
4. \(3.0\) m

A particle moves along a circle of radius $\frac{20}{\mathrm{\pi }}m$ with constant tangential acceleration. If the velocity of the particle is 80 m/s at the end of the second revolution after motion has begin, the tangential acceleration is

(1) 640 $\mathrm{\pi } \mathrm{m}/{\mathrm{s}}^2$

(2) 160 $\mathrm{\pi } \mathrm{m}/{\mathrm{s}}^2$

(3) 40 $\mathrm{\pi } \mathrm{m}/{\mathrm{s}}^2$

(4) 40 $\mathrm{m}/{\mathrm{s}}^2$