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

A rigid body rotates about a fixed axis with a variable angular velocity equal to \(\alpha-\beta t,\) at the time \(t,\) where \(\alpha, ~\beta\) are constants. The angle through which it rotates before its stops is:
1. \(\frac{\alpha^{2}}{2\beta}\)
2. \(\frac{\alpha^{2}-\beta^{2}}{2\alpha}\)
3. \(\frac{\alpha^{2}-\beta^{2}}{2\beta}\)
4. \(\frac{(\alpha-\beta) \alpha}{2}\)

An automobile moves on a road with a speed of 54 kmh${}^{-1}$. The radius of its wheel is 0.45 m and moment of inertia of the wheel about its axis of rotation is 3 kg m${}^2$. If the vehicle is brought to rest in 15s, the magnitude of average torque transmitted by its brakes to the wheel is

1. 8.58 ${\mathrm{m}}^2 {\mathrm{s}}^{-2}$

2. 10.86 ${\mathrm{m}}^2 {\mathrm{s}}^{-2}$

3. 2.86 ${\mathrm{m}}^2 {\mathrm{s}}^{-2}$

4. 6.66 ${\mathrm{m}}^2 {\mathrm{s}}^{-2}$

A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be

(1) $\frac{M\omega }{M+4m}$

(2) $\frac{(M+4m)\omega }{M}$

(3) $\frac{(M-4m)\omega }{M+4m}$

(4) $\frac{M\omega }{4m}$

One hollow and one solid cylinder of the same radius roll down on a smooth inclined plane. Then the foot of the inclined plane is reached by 

1.  solid cylinder earlier 

2.  hollow cylinder earlier 

3.  simultaneously both 

4.  the heavier earlier irrespective of being solid or hollow