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Law of Conservation of Energy

For a body or an isolated system by work–energy theorem we have
But according to the definition of potential energy in a conservative field,
So, from (5) and (6), we have
For an isolated system or body in presence of conservative forces, the sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend upon time. This is known as the law of conservation of mechanical energy.
Δ(K + U) = ΔE = 0 [as E is constant in a conservative field]
∴ ΔK + ΔU = 0
i.e., if the kinetic energy of the body increases, its potential energy will decrease by an equal amount and vice versa.

Law of conservation of total energy

If some non-conservative force such as friction is also acting on the particle, the mechanical energy is no more constant. It changes by the amount of work done by the frictional force.
Δ(K + U) = ΔE = Wf
where Wf is the work done against friction
The lost energy is transformed into heat and the heat energy developed is exactly equal to loss in mechanical energy.
We can, therefore, write
ΔE Q = 0 ΔE + Q = 0
where Q is the heat produced. This shows that if the forces are both conservative and non-conservative, it is not the mechanical energy alone which is conserved, but it is the total energy, may be heat, light, sound, or mechanical, etc., which is conserved.


In other words: Energy may be transformed from one kind to another but it cannot be created or destroyed. The total energy in an isolated system is constant. This is the law of conservation of energy.

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