Conservation of Mechanical Energy
The mechanical energy
of a system is the sum of its potential energy 
and the kinetic energy 
of the objects within it:
In this section, we examine what happens to this mechanical energy when only conservative forces cause energy transfers within the system—that is, when factional and drag forces do not act on the objects in the system. Also, we shall assume that the system is isolated from its environment; that is, no external force from an object outside the system causes energy changes inside the system.
When a conservative force does work W on an object within the system, it transfers energy between kinetic energy К of the object and potential energy U of the system. From Eq. 7-10, the change 
in kinetic energy is
(8-13)
W and from Eq. 8-1, the change 
in potential energy is
Combining Eqs. 8-13 and 8-14, we find that
(8-15)
In words, one of these energies increases exactly as much as the other decreases. We can rewrite Eq. 8-15 as
8-16
where the subscripts refer to two different instants and thus to two different arrangements of the objects in the system. Rearranging Eq. 8-16 yields
when the system is isolated and only conservative forces act on the objects in the system. In other words:
!► In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system, cannot change.
This result is called the principle of conservation of mechanical energy.(Now you can see where conservative forces got their name.) With the aid of Eq. 8-15, we can write this principle in one more form, as
The principle of conservation of mechanical energy allows us to solve problems that would be quite difficult to solve using only Newton's laws:
► When the mechanical energy of a system is conserved, we can relate the sum of kinetic energy and potential energy at one instant to that at another instant without considering the intermediate motion and without finding the work done by the forces involved.
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