Kinetic Energy of the system of particles

KINETIC ENERGY OF THE SYSTEM OF PARTICLES

Let there be n number of particles in a system of particles and these particles possess some motion. The motion of the ith particle of this system depends on the external force acting on it.

Let at any time the velocity of ith particle be then its kinetic energy would be

Let be the position vector of the ith particle w.r.t. O and be the position vector of the centre of mass w.r.t. , as shown in the figure 1.16, then

where is the position vector of centre of mass of the system w.r.t. O.

Differentiating the equation 2 we get

where vi is the velocity of ith particle and vCM is the velocity of centre of mass of system of particle.

Putting equation 3 in 1 we get, 

The sum of kinetic energy of all the particles can be obtained from equation 4

Now last term in equation (5) is equal to zero

Therefore, kinetic energy of the system of particles is,

is the kinetic energy obtained as if all the mass were concentrated at the centre of mass

is the kinetic energy of the system of particle w.r.t. the centre of mass.

Hence it is clear from equation (6) that kinetic energy of the system of particles consists of two parts: the kinetic energy obtained as if all the mass were concentrated at the centre of mass plus the kinetic energy of motion of all particles about the centre of mass.

If there were no external force acting on the particle system then the velocity of the centre of mass of the system will remain constant and kinetic energy of the system would also remain constant.