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 $\mathrm{m}/{\mathrm{s}}^2$

A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizotal and moment of inertia about it is I. A weight mg is attached to the end of the cord and falls from rest. After falling through a distance h, the angular velocity of the wheel will be

(1) $\sqrt{\frac{2gh}{I+mr}}$

(2) $\sqrt{\frac{2mgh}{I+m{r}^2}}$

(3) ${\left[\frac{2mgh}{I+2m}\right]}^{\frac{1}{2}}$

(4) $\sqrt{2gh}$

A square lamina is as shown in the figure. The moment of inertia of the frame about the three axes is $I_1,$ $I_{\mathrm{2,}}$ $and$ $I_3$ respectively. Select the correct alternative.

     
1. $I_2=I_3>I_1$
2. $I_1>I_2>I_3$
3. $I_2=I_3<I_1$
4. $I_1<I_2<I_3$

A body falling vertically downward under gravity breaks into two parts of unequal masses. The center of mass of the parts taken together shifts horizontally towards

1.  heavier piece 

2.  lighter piece 

3.  does not shift horizontally 

4.  depends on the vertical velocity at the time of breaking 

A solid sphere, disc, and solid cylinder all Of the same mass and made up Of same material are allowed to roll down (from rest) on an inclined plane, then

1.  Solid sphere reaches the bottom first

2.  Solid sphere reaches the bottom late

3.  The disc will reach the bottom first

4.  All of them reach the bottom at the same time

From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia do the remaining part of the disc about a perpendicular axis, passing through the centre

$1. 15 {\mathrm{MR}}^2/32$
$2. 13 {\mathrm{MR}}^2/32$
$3. 11 {\mathrm{MR}}^2/32$
$4. 9 {\mathrm{MR}}^2/32$

The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is

$1. \sqrt{2} : 1$
$2. \sqrt{2} : \sqrt{3}$
$3. \sqrt{3} :\hspace{0.17em}\sqrt{2}$
$4. 1 : \sqrt{2}$

A wheel comprises of a ring of radius R and mass M and three spokes of mass m each. The moment of inertia of the wheel about its axis is

1.  $\left(\mathrm{M} + \frac{\mathrm{m}}{4}\right){\mathrm{R}}^2$

2.  $\left(\mathrm{M} + \mathrm{m}\right){\mathrm{R}}^2$

3.  $\left(\mathrm{M} + 3\mathrm{m}\right){\mathrm{R}}^2$

4.  $\left(\frac{\mathrm{M} + \mathrm{m}}{2}\right){\mathrm{R}}^2$