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238 CHAPTER 8 Conservation of Energy

PROBLEM-SOLVING TECHNIQUES ENERGY CONSERVATION

To obtain an expression for the total mechanical energy, you
CONTRIBUTIONS TO THE MECHANICAL ENERGY
must include terms for the different kinds of energy that are
present:
KIND OF CONTRIBUTION TO TOTAL
ENERGY APPLICABLE IF MECHANICAL ENERGY
1 Begin with an expression for the energy at one point
[Eq. (8.8)]. Kinetic energy Particle is in K mv 2
1
2
motion
2 And an expression for the energy at another point
[Eq. (8.9)]. Gravitational Particle is moving
potential up or down near U mgy
3 Then use energy conservation to equate these expres- energy the Earth’s surface
sions [Eq. (8.10)].
1
Elastic Particle is U kx 2
2
potential subject to a
With the appropriate expression for the mechanical
energy spring force
energy, you can apply energy conservation to solve some prob-
lems of motion. As illustrated in the preceding example, this
involves the three steps outlined in Section 7.4 and 8.1.

To formulate the law of conservation of mechanical energy for a particle moving
under the influence of some other force, we want to imitate the above construction. We
will be able to do this if, and only if, the work performed by this force can be expressed
as a difference between two potential energies, that is,

W U U (8.12)
2 1
If the force meets this requirement (and therefore permits the construction of a con-
servation law), the force is called conservative. Thus, the force of gravity and the force
of a spring are conservative forces. Note that for any such force, the work done when
the particle starts at the point x and returns to the same point is necessarily zero, since,
1
with x x , Eq. (8.12) implies
2 1
W U U 0 (8.13)
1 1
This simply means that for a round trip that starts and ends at x , the work the force
1
does during the outward portion of the trip is exactly the negative of the work the
force does during the return portion of the trip, and therefore the net work for the
y round trip is zero (see Fig. 8.3).Thus, the energy supplied by the force is recoverable:
Work is done during
outward trip. the energy supplied by the force during motion in one direction is restored during the
x 1 return motion in the opposite direction. For instance, when a particle moves down-
O x ward from some starting point, gravity performs positive work; and when the particle
moves upward, returning to its starting point, gravity performs negative work of a
Opposite work is done
during inward trip. magnitude exactly equal to that of the positive work.
The requirement of zero work for a round trip can be used to discriminate between
FIGURE 8.3 A particle starts at a point x
1 conservative and nonconservative forces. Friction is an example of a nonconservative force.
and returns to the point x after completing
1 If we slide a metal block through some distance along a table and then slide the block
some round trip. If the force is conservative,
back to its starting point, the net work is not zero.The work performed by the friction
the work done is zero, because the work for
the outward portion of the trip is opposite force during the outward portion of the motion is negative, and the work performed
to that for the inward portion. by the friction force during the return portion of the trip is also negative—the friction