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236 CHAPTER 8 Conservation of Energy

? How much gravitational potential energy is stored in the upper reservoir, and
how much available electric energy does this represent? (Example 5, page 249)
? When generating power at its maximum capacity, at what rate does the power
plant remove water from the upper reservoir? How many hours can it run?
(Example 10, page 257)
n the preceding chapter we found how to formulate a law of conservation of mechan-
Iical energy for a particle moving under the influence of the Earth’s gravity. Now we
will seek to formulate the law of conservation of mechanical energy when other forces
act on the particle—such as the force exerted by a spring—and we will state the general
law of conservation of energy.As in the case of motion under the influence of gravity, the
conservation law permits us to deduce some features of the motion without having to
deal with Newton’s Second Law.

Online 8.1 POTENTIAL ENERGY OF
10
Concept
Tutorial A CONSERVATIVE FORCE

To formulate the law of conservation of energy for a particle moving under the influ-
ence of gravity, we began with the work–energy theorem [see Eq. (7.24)],
K K W (8.1)
2 1
We then expressed the work W as a difference of two potential energies [see Eq. (7.30)],
W U U (8.2)
2 1
This gave us

K K U U
2 1 2 1
from which we immediately found the conservation law for the sum of the kinetic and
potential energies, K U K U ,or
2 2 1 1
E K U [constant] (8.3)

As an illustration of this general procedure for the construction of the conserva-
tion law for mechanical energy, let us deal with the case of a particle moving under
the influence of the elastic force exerted by a spring attached to the particle. If the par-
JOSEPH LOUIS, COMTE LAGRANGE ticle moves along the x axis and the spring lies along this axis, the force has only an
(1736–1813) French mathematician and x component F , which is a function of position:
x
theoretical astronomer. In his elegant mathe-
F (x) kx (8.4)
matical treatise Analytical Mechanics, x
Lagrange formulated Newtonian mechanics in
Here, as in Section 6.2, the displacement x is measured from the relaxed position of
the language of advanced mathematics and
the spring.The crucial step in the construction of the conservation law is to express the
introduced the general definition of the
work W as a difference of two potential energies. For this purpose, we take advantage
potential-energy function. Lagrange is also
known for his calculations of the motion of of the result established in Section 7.2 [see Eq. (7.17)], according to which the work
planets and for his influential role in securing done by the spring force during a displacement from x to x is
1 2
the adoption of the metric system of units.
2
1
1
W kx kx 2 2 (8.5)
2
2
1
This shows that if we identify the elastic potential energy of the spring as
1
potential energy of spring U kx 2 (8.6)
2