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. $\frac{M\omega }{4m}$

Let g be acceleration due to gravity on the surface of earth and T be the rotational kinetic energy of the earth. Suppose the earth's radius decreases by 2%. Keeping all other quantities same (even $\omega$)

1.  g decreases by 2% and T decreases by 4%

2.  g decreases by 4% and T decreases by 2%

3.  g increases by 4% and T decreases by 4%

4.  g decreases by 4% and T increases by 4%

The angular momentum of a system of particles is conserved

1. when no external force acts upon the system

2. when no external torque acts upon the system

3. when no external impulse acts upon the system

4. when axis of rotation remains same

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

ABC is an equilateral triangle with O as its cente. ${\overrightarrow{F}}_1, \overrightarrow{F_2} and \overrightarrow{F_3}$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of $\overrightarrow{F_3}$ is

1. $F_1+F_2$

2. $F_1-F_2$

3. $\frac{F_1+F_2}{2}$

4. $2(F_1+F_2)$

The motor of an engine is rotating about its axis with an angular velocity of 100 rpm.  It comes to rest is 15 s, after being switched off.  Assuming constant angular deceleration.  What are the numbers of revolutions made by it before coming to rest?

1. 12.5         

2. 40             

3. 32.6           

4. 15.6

$If F be a force acting on a particle having the position vector r and \tau be the torque of the force about the origin, then;$
$\left(a\right) r.\tau =0 and F.\tau =0 \left(b\right) r.\tau =0 and F.\tau \ne 0$
$\left(c\right) r.\tau \ne 0 and F.\tau \ne 0 \left(d\right)r.\tau \ne 0 and F.\tau =0$ 

A T joint is formed by two identical rods A and B each of mass m and length L in the XY plane as shown. Its moment of inertia about axis passing through A and perpendicular to the plane of the joint,

1.  $\frac{2{\mathrm{mL}}^2}{3}$

2.  $\frac{{\mathrm{mL}}^2}{12}$

3.  $\frac{{\mathrm{mL}}^2}{6}$

4.  None of these

A hollow sphere of diameter 0.2 m and mass 2 kg is rolling on an inclined plane with velocity v = 0.5 m/ s. The kinetic energy of the sphere is

1.  0.1 J

2.  0.3 J

3.  0.5 J

4.  0.42 J