Conservation of Energy in a System of Particles:



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             The conservation of energy states that the total amount of energy in any isolated system remains constant but can't be recreated, although it may change forms, e.g. friction turns kinetic energy into thermal energy. Thus, the energy conservation is given by,

This is the conservation of mechanical energy for a system of particles and the subscripts 1 and 2 imply the initial and final total energy.

The total mechanical energy for an isolated system undergoing rotational motion is constant. It can be written as follows:

E1 = E2

[ KE+PE]1 ==[ KE+PE]2

[(1/2)mv2 + (1/2)Icmω2 + PE]1 =[ (1/2)mv2 + (1/2)Icmω2 + PE]2

Here, m is the mass of the rigid body, v is its linear velocity, ω is its angular velocity and Icm is the center of mass of the rigid body consisting of a lot of tiny particles and PE is the potential energy of the system. The subscripts 1 and 2 imply the initial and final stage mechanical energy.

For a system of particles, there is a center of mass where all of the mass of the system is concentrated. Then the velocity of center of mass of the system is given by,

Now, the kinetic energy of the system in any collision during an inelastic collision is given by

Thus, the final kinetic energy of the system is equal to the kinetic energy of its center of mass.