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Conservation of Mechanical Energy

From the previous section we know that the work done by a conservative force in terms of the change in potential energy is given by
DU = -WC (i)

where U is the potential energy and WC is the work done by a conservative force.
From the work-energy theorem, we know that
Wnet = DK

Where Wnet represents the sum of work done by all the forces acting on the mass.
If a particle is subject to only conservative forces, then
WC = Wnet = DK

Thus, the equation (i) becomes,
DU = -DK
or DU + DK = 0 (ii)

The equation (ii) tells us that the total change in potential energy plus the total change in kinetic energy is zero if only conservative forces are acting on the system.

That is, there is no change in the sum of K plus U.
D(K + U) = 0 (iii)
or DE = 0 where E = K + U

The quantity E = K + U is called the total mechanical energy.

According to equation (iii), when only conservative forces act, the change in total mechanical energy of a system is zero, in other words, If only conservative forces perform work on and within a system of masses, the total mechanical energy of the system is conserved.

Alternatively, the equation (iii) may be written as
(Kf + Uf) - (Ki + Ui) = 0
or Kf + Uf = Ki + Ui (iv)

SinceDE = 0, integrating both sides,
we get E = constant.

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