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)

The M.I. of a body about the given axis is 1.2 kg x ${\mathrm{m}}^2$ initially the body at rest. In order to the rotational kinetic energy of 1500 J, the angular acceleration of 25 rad/${\sec }^2$ must be about that axis for the duration of

1.  4 sec

2.  2 sec

3.  8 sec

4.  10 sec

A loop rolls down on the inclined plane. The fraction of its total kinetic energy that is associated with rotational motion is

1.  1: 2

2.  1: 3

3.  1: 4

4.  2: 3

A ball kept in a closed box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass

1. of the box remains constant 

2. of the box plus the ball system remains constant

3. of the ball remains constant

4. of the ball relative to the box remains constant

A solid sphere is rolling on a frictionless surface, as shown in the figure with a translational velocity of v m/s. If a sphere climbs up to a height h, then the value of v would be:
                 

1. $\ge \sqrt{\frac{10}{7}\mathrm{gh}}$                                                   

2. $\ge \sqrt{2\mathrm{gh}}$

3. $2\mathrm{gh}$                                                             

4. $\frac{10}{7}\mathrm{gh}$