Question
Thu October 27, 2011 By: Mayank Kumar

# centre of mass of two particle system ..... explain.

Fri October 28, 2011

Consider two particles A and B of masses m1 and m2, respectively. Take the line joining A and B as the X-axis. Let the coordinates of the particles at time 't' be x1 and x2. Suppose no external force acts on the system. The particles A and B, however, exert forces on each other and these particles accelerate along the line joining them. Suppose the particles are initially at rest and the force between them is attractive. The particles will then move along the line AB as shown in the figure.

As time passes, x1 and x2 change and hence, X changes and the centre of mass moves along the X-axis. Velocity of the centre of mass at time t is,

The acceleration of the centre of mass is

Suppose the magnitude of the forces between the particles is F. As the only force acting on A towards B, is F, its acceleration is
The force on B is (-F) and hence,

Substituting this in equation (1)

This means, the velocity of the centre of mass does not change with time. But as we assumed initially, the particles are at rest. Thus, v1 =then Vcm has to be zero. Hence, the centre of mass remains fixed and does not change with time.

Thus, if no external force acts on a two-particle system and its centre of mass is at rest, initially it remains fixed even when the particles individually move and accelerate.

If the external forces do not add up to zero, the centre of mass is accelerated and is given by

If we have a single particle of mass m on which a force acts, its acceleration would be the same as. Thus, the motion of the centre of mass of a system is identical to the motion of a single particle of mass equal to the mass of given system, acted upon by the same external forces that act on the system.

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