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7.4 Gravitational Potential Energy 221

If we let the particle push or pull on some obstacle (such as the wheel of the grand-
father clock) during its descent from y to y , then the total amount of work that we
1 2
can extract during this descent is equal to the work done by gravity; that is, it is equal
to U U (U U ) U, or the negative of the change of potential energy.
2 1 2 1
Of course, the work extracted in this way really arises from the Earth’s gravity—the par-
ticle can do work on the obstacle because gravity is doing work on the particle. Hence
the gravitational potential energy is really a joint property of the particle and the Earth; it
is a property of the configuration of the particle–Earth system.
If the only force acting on the particle is gravity, then by combining Eqs. (7.24)
and (7.30) we can obtain a relation between potential energy and kinetic energy.
According to Eq. (7.24), the change in kinetic energy equals the work, or K K W;
2 1
and according to Eq. (7.30), the negative of the change in potential energy also equals
the work: W U U . Hence the change in kinetic energy must equal the nega-
2 1
tive of the change in potential energy:
K K U U
2 1 2 1
CHRISTIAAN HUYGENS (1629–1695)
We can rewrite this as follows: Dutch mathematician and physicist. He invented
the pendulum clock, made improvements in the
K U K U (7.32)
2 2 1 1 manufacture of telescope lenses, and discovered
This equality indicates that the quantity K U is a constant of the motion; that the rings of Saturn. Huygens investigated the
is, it has the same value at the endpoint as it had at the starting point. We can express theory of collisions of elastic bodies and the theory
of oscillations of the pendulum, and he stated the
this as
Law of Conservation of Mechanical Energy for
K U [constant] (7.33) motion under the influence of gravity.
The sum of the kinetic and potential energies is called the mechanical energy of the
particle. It is usually designated by the symbol E:

E K U (7.34) mechanical energy
This energy represents the total capacity of the particle to do work by virtue of both
its speed and its position.
Equation (7.33) shows that if the only force acting on the particle is gravity, then
the mechanical energy remains constant:
E K U [constant] (7.35) Law of Conservation of
Mechanical Energy
This is the Law of Conservation of Mechanical Energy.
Since the sum of the potential and kinetic energies must remain constant during
the motion, an increase in one must be compensated by a decrease in the other; this
means that during the motion, kinetic energy is converted into potential energy and vice
versa. For instance, if we throw a baseball straight upward from ground level (y 0),
the initial kinetic energy is large and the initial potential energy is zero. As the base-
ball rises, its potential energy increases and, correspondingly, its kinetic energy decreases,
so as to keep the sum of the kinetic and potential energies constant. When the base-
ball reaches its maximum height, its potential energy has the largest value, and the
kinetic energy is (instantaneously) zero. As the baseball falls, its potential energy
decreases, and its kinetic energy increases (see Fig. 7.22).
Apart from its practical significance in terms of work, the mechanical energy is
very helpful in the study of the motion of a particle. If we make use of the formulas for
K and U, Eq. (7.35) becomes
1
2
E mv mg y [constant] (7.36)
2