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when the system is isolated and only conservative forces act on the objects in the
system. In other words:

8-2 CONSERVATION OF MECHANICAL ENERGY 185

In an isolated system where only conservative forces cause energy changes, the

kinetic energy and potential energy can change, but their sum, the mechanical

energyEmecof the system, cannot change.

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 consider-

ing the intermediate motionandwithout finding the work done by the forces involved.

This result is called the principle of conservation of mechanical energy.(Now you
can see where conservativeforces got their name.) With the aid of Eq. 8-15, we
can write this principle in one more form, as

EmecKU0. (8-18)
The principle of conservation of mechanical energy allows us to solve
problems that would be quite difficult to solve using only Newton’s laws:

Figure 8-7 shows an example in which the principle of conservation of
mechanical energy can be applied: As a pendulum swings, the energy of the

Figure 8-7A pendulum, with its mass
concentrated in a bob at the lower end,
swings back and forth. One full cycle of
the motion is shown. During the cycle the
values of the potential and kinetic ener-
gies of the pendulum – Earth system vary
as the bob rises and falls, but the mechani-
cal energy Emecof the system remains
constant. The energy Emeccan be
described as continuously shifting between
the kinetic and potential forms. In stages
(a) and (e), all the energy is kinetic energy.
The bob then has its greatest speed and is
at its lowest point. In stages (c) and (g), all
the energy is potential energy. The bob
then has zero speed and is at its highest
point. In stages (b), (d), (f), and (h), half
the energy is kinetic energy and half is
potential energy. If the swinging involved
a frictional force at the point where the
pendulum is attached to the ceiling, or a
drag force due to the air, then Emecwould
not be conserved, and eventually the
pendulum would stop.

(a)

U K

(b)

U K

(c)

U K

(d)

U K

(e)

U K

(h)

U K

(f)

U K

(g)

U K

v = +vmax

v = 0

v = –vmax

v = 0

v

v

v

v

v

v

All potential

energy

All potential

energy

The total energy

does not change

(it is conserved).

All kinetic energy

All kinetic energy