A uniform square plate ABCD has a mass of 10 kg. If two point masses of 5 kg each are placed at the corners C and D as shown in the adjoining figure, then the centre of mass shifts to the mid-point of:
          
1. OH

2. DH

3. OG

4. OF 

At t = 0, the positions of the two blocks are shown. There is no external force acting on the system. Find the coordinates of the center of mass of the system at t = 3 seconds:
   

The mass per unit length of a non-uniform rod of length L is given by $\mathrm{\mu }={\mathrm{\lambda x}}^2$ where $\mathrm{\lambda }$ is a constant and x is the distance from one end of the rod. The distance between the centre of mass of the rod and this end is:

1. $\frac{\mathrm{L}}{2}$

2. $\frac{\mathrm{L}}{4}$

3. $\frac{3\mathrm{L}}{4}$

4. $\frac{\mathrm{L}}{3}$

The center of mass of a system of particles does not depend upon:

A particle is moving with a constant velocity along a line parallel to the positive x-axis. The magnitude of its angular momentum with respect to the origin is:

A rigid body rotates with an angular momentum of \(L.\) If its kinetic energy is halved, the angular momentum becomes:
1. \(L\)
2. \(L/2\)
3. \(2L\)
4. \(L/\)$\sqrt{2}$

A particle rotating on a circular path of the radius $\frac{4}{\mathrm{\pi }}\mathrm{m}$ at 300 rpm reaches 600 rpm in 6 revolutions. If the angular velocity increases at a constant rate, find the tangential acceleration of the particle.

1. 10 m/s²

2. 12.5 m/s²

3. 25 m/s²

4. 50 m/s²

The coordinates of the centre of mass of a uniform plate of shape as shown in the figure are:

A rod is falling down with constant velocity \(V_0\) as shown. It makes contact with hinge A and rotates around it. The angular velocity of the rod just after the moment when it comes in contact with hinge A is:

Two loads ${\mathrm{P}}_1$ $\mathrm{and}$ ${\mathrm{P}}_2$ (${\mathrm{P}}_1$ $>{\mathrm{P}}_2$) are connected by a string passing over a fixed pulley. The center of gravity of loads are initially at the same height. Find the acceleration of the center of gravity of the system:
1. \(\left(\frac{(P_1-P_2)^{\frac{1}{2}}}{P_1+P_2}\right)\text{g}\)
2. \(\left(\frac{P_1-P_2}{P_1+P_2}\right)\text{g}\)
3. \(\left( \frac{P_1-P_2}{P_1+P_2}\right)^2 \text{g}\)
4. \(\left( \frac{P_1+P_2}{P_1-P_2}\right)\text{g}\)