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

The moment of inertia of a loop of radius R and mass M, about any tangent line in its plane will be

(1) $\frac{3M{R}^2}{2}$

(2) $\frac{M{R}^2}{2}$

(3) $M{R}^2$

(4) $\frac{M{R}^2}{4}$

A uniform rod AB of length l and mass m is free to rotate about point A. The rod is released from rest in horizontal position. Given that the moment of inerita of the rod about A is $\frac{{\mathrm{ml}}^2}{3}$ the initial angular acceleration of the rod will be

$1. \frac{2\mathrm{g}}{3\mathrm{l}}$
$2. \mathrm{mg}\frac{\mathrm{l}}{2}$
$3. \frac{3}{2}\mathrm{gl}$
$4. \frac{3\mathrm{g}}{2\mathrm{l}}$

Three indentical spheres, each of mass 1 kg are kept as shown in figure, touching each other, with their centres on a straight line. If their centres are marked P, Q, R respectively, the distance of centre of mass of the system from P is

$1. \frac{\mathrm{PQ} + \mathrm{PR} + \mathrm{QR}}{3}$
$2. \frac{\mathrm{PQ} +\hspace{0.17em}\mathrm{PR}}{3}$
$3. \frac{\mathrm{PQ} +\hspace{0.17em}\mathrm{QR}}{3}$
$4. \frac{\mathrm{PR} + \mathrm{QR}}{3}$

Two particles of equal mass have coordinates (2m,4m,6m) and (6m,2m,8m). Of these one particle has velocity ${\mathrm{v}}_1=\left(2\hat{\mathrm{i}}\right)$ m/s and another particle has velocity ${\mathrm{v}}_2=\left(2\hat{j}\right)$ m/s at time t = 0. The coordinate of their centre of mass at time t = 1 s will be

1. (4m,4m,7m)

2. (5m,4m,7m)

3. (2m,4m,6m)

4. (4m,5m,4m)