Two masses ${\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2, {\mathrm{m}}_1 > {\mathrm{m}}_2$ are connected to the ends of massless rope and allowed to move as shown in the figure. The acceleration of the centre of mass assuming pulley is massless and frictionless, is

1.  \(\frac{m_{1}   -   m_{2}}{m_{1}   +   m_{2}} g\)

2.  0

3.  \({m_1 -m_2^2 \over m_1 + m_2 } g \)

4.  \({m_1 +m_2^2 \over m_1 - m_2 } g \)

$\mathrm{}$

The speed of a uniform spherical shell after rolling down an inclined plane of vertical height h from rest is:

1.  \(\sqrt{\frac{10   g h}{7}}\)

2.    \(\sqrt{\frac{6   g h}{5}}\)

3.  \(\sqrt{\frac{4   g h}{5}}\)

4.  \(\sqrt{2   g h}\)

A uniform rod of mass m is bent into the form of a semicircle of radius R. The moment of inertia of the rod about an axis passing thorugh A and Perpendicular to the plane of paper is

1. $\frac{2}{3}M{R}^2$

2. $M{R}^2$

3. 2$M{R}^2$

4. $\frac{3}{2}M{R}^2$

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. $\frac{M\omega }{4m}$

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

A loaded truck has to take a sharp turn to the left. The center of gravity of the truck can be altered by shifting the concentrated load. To avoid toppling, the load must be shifted 

1.  up and left 

2.  up and right 

3.  down and left 

4.  down and right

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. $2(F_1+F_2)$