A uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two-point
masses m and 2m moving in the same horizontal plane with speeds 2v and v
respectively strike the bar as shown below in the figure and stick to the bar after the collision. Denoting angular velocity about the center of mass, total energy, and center of mass velocity of $\omega$, E and ${\mathrm{v}}_{\mathrm{c}}$ respectively, we have after collision

                      

1.  ${\mathrm{v}}_{\mathrm{c}} = 2 \mathrm{m}/\mathrm{s}$

2.  $\mathrm{\omega } = \frac{3}{5}\frac{\mathrm{v}}{\mathrm{a}}$

3.  $\mathrm{\omega } = \frac{\mathrm{v}}{5\mathrm{a}}$

4.  $\mathrm{E} = \frac{2}{3} {\mathrm{mv}}^2$

A particle of mass m is moving at speed v perpendicular to a rod of length d and mass M = 6m which pivots around a frictionless axle running through its centre. It strikes and sticks to the end of the rod. The moment of inertia of the rod about its centre is $M{d}^2/12$. Then the angular speed  of the system just after the collision is:

1. 2v/3d

2. 2v/d

3. v/d

4. v

In the figure shown, a small solid spherical ball of mass 'm' can move without sliding in a fixed semicircular track of radius R in vertical plane. It is released from the top. The resultant force on the ball at the lowest point of the track is :

1. $\frac{10 \mathrm{mg} }{7}$

2. $\frac{17 \mathrm{mg} }{7}$

3. $\frac{3 \mathrm{mg} }{7}$

4. zero

A uniform thin rod is bent in the form of closed loop ABCDEFA as shown in the figure. The ratio of inertia of the loop about x-axis to that about y-axis is:

1.  >1

2.  <1

3.  =1

4.  =1/2

A hollow sphere is rolling without slipping on the inclined plane. The minimum coefficient of friction is :

1. $\frac{2}{5}\tan \theta$

2. $\frac{2}{3}\tan \theta$

3. $\frac{3}{4}\tan \theta$

4. $\frac{5}{7}\tan \theta$

A body of mass (m) is projected with velocity v₀ at an angle of projection $\mathrm{\theta }$. The magnitude of angular momentum about origin after time t from projection is :

1. Increase with time

2. decrease with time

3. independent of time

4. none of these

A body of mass m slides down an incline and reaches the bottom with a velocity v. If the same mass were in the form of a ring which rolls down this incline, the velocity of the ring at bottom would have been

(1) v

(2) $\sqrt{2} v$

(3) $\left(1/\sqrt{2}\right)v$

(4) $\left(\sqrt{2/5}\right)v$

A rope is wound around a hollow cylinder of mass 3 kg and radius 40cm. What is the angular acceleration of the cylinder,if the rope is pulled with a force of 30 N?

(1) 25$m/s^2$

(2) 0.25 $rad/s^2$

(3) 25 $rad/s^2$

(4) 5 $m/s^2$

Two rotating bodies A and B of masses m and 2m with moments of inertia $I_A$ and $I_B(I_B>I_A)$ have equal kinetic energy of rotation. If $L_{A }$and $L_B$ be their angular momenta respectively, then 

(1) $L_A=\frac{L_B}{2}$

(2) $L_A=2L_B$

(3) $L_B>L_A$

(4) $L_A>L_B$

A solid sphere of mass m and radius R is rotating about its diameter. A soild cyclinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $\left(E_{sphere}/E_{cylinder }\right)$ will be 

(a) 2:3

(b) 1:5

(c) 1:4

(d) 3:1