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no force at this point, and so it is the equilibrium point. Another way to see
that it is the equilibrium poïnt is that it takes work to move away from đin
either direction. When the two oxygen atoms have settled down, so that no more
--- Trang 275 ---
U(r) œ 1/rŠ
(IF r> đ)
Fig. 14-3. The potential energy between two atoms as a function of
the distance between them.
energy can be liberated from the force between them, they are in the lowest
energy state, and they will be at this separation d. 'This is the way an oxygen
mmolecule looks when it is cold. When we heat it up, the atoms shake and move
farther apart, and we can in fact break them apart, but to do so takes a certain
amount of work or energy, which is the potential energy difference bebween r = đ
and r = œ. When we try to push the atoms very close together the energy goes
up very rapidly, because they repel each other.
The reason we bring this out is that the idea of force is not particularly
suitable for quantum mechanics; there the idea of energw is most natural. We fnd
that although forces and velocities “dissolve” and disappear when we consider
the more advanced forces between nuclear matter and between molecules and so
on, the energy concept remains. 'Pherefore we fnd curves of potential energy In
quantum mechanies books, but very rarely do we ever see a curve for the Íorce
between two molecules, because by that time people who are doing analyses are
thinking in terms of energy rather than of force.
Next we note that if several conservative Íorces are acting on an object at the
same time, then the potential energy of the object is the sum of the potential
energies from each of the separate forces. This is the same proposition that we
mentioned before, because i1f the force can be represented as a vector sum of
forces, then the work done by the total force is the sum of the works done by
the partial forces, and it can therefore be analyzed as changes in the potential
energies of each of them separately. Thus the total potential energy 1s the sum
of all the little pieces.
W© could generalize this to the case oŸ a system of many objects interacting
with one another, like Jupiter, Saturn, Ủranus, etc., or oxygen, nitrogen, carbon,
--- Trang 276 ---
etc., which are acting with respect to one another in pairs due to forces all of
which are conservative. In these circumstances the kinetic energy in the entire
system is simply the sum of the kinetic energies of all of the particular atoms or
planets or whatever, and the potential energy of the system is the sum, over the
pairs of particles, of the potential energy of mutual interaction of a single pair,
as though the others were not there. (This is really not true for molecular forces,
and the formula is somewhat more complicated; it certainly is true for NÑewtonian
gravitation, and ït is true as an approximation for molecular forces. For molecular
forces there is a potential energy, but it is sometimes a more complicated function
of the positions of the atoms than simply a sum oŸ terms from pairs.) In the
special case of gravity, therefore, the potential energy is the sum, over all the
pairs ? and 7, of —ŒGm¿m;/r¡;, as was indicated in Eq. (135.14). Equation (13.14)
expressed mathematically the following proposition: that the total kinetic energy
plus the total potential energy does not change with time. As the various planets
wheel about, and turn and twist and so on, IŸ we calculate the total kinetic energy
and the total potential energy we fñnd that the total remains constant.
14-4 Nonconservative Íorces
We have spent a considerable time discussing conservative forces; what about
nonconservative forces? We shall take a deeper view of this than is usual, and state
that there are no nonconservative forcesl As a matter of fact, all the fundamental
forces in nature appear to be conservative. This is not a consequence of Ñewton”s
laws. In fact, so far as Newton himself knew, the forces could be nonconservative,
as Íriction apparently is. When we say friction øpparenfly 1s, we are taking a
modern view, in which it has been discovered that all the deep forces, the forces
between the particles at the most fundamental level, are conservative.
Tí, for example, we analyze a system like that great globular star cluster that
we saw a picture of, with the thousands of stars all interacting, then the formula
for the total potential energy is simply one term plus another term, etc., summed
over all pairs oŸ stars, and the kinetic energy is the sum of the kinetic energies of
all the individual stars. But the globular cluster as a whole is drifting in space
too, and, if we were far enough away from it and did not see the details, could
be thought of as a single object. Then if forces were applied to i%, some of those
forces might end up driving it forward as a whole, and we would see the center
of the whole thing moving. Ôn the other hand, some of the forces can be, so to
speak, “wasted” in increasing the kinetic or potential energy of the “particles”
--- Trang 277 ---
inside. Let us suppose, for instance, that the action of these forces expands the
whole cluster and makes the particles move faster. 'Phe total energy of the whole
thing is really conserved, but seen from the outside with our crude eyes which
cannot see the confusion of motions inside, and just thinking of the kinetic energy
of the motion of the whole object as though it were a single particle, it would
appear that energy is not conserved, but this is due to a lack of appreciation of
what it is that we see. And that, it turns out, is the case: the total energy of the
world, kinetic plus potential, is a constant when we look closely enough.
'When we study matter in the ñnest detail at the atomic level, it is no always
cøs+ to separate the total energy of a thing into two parts, kinetic energy and
potential energy, and such separation is not always necessary. It is aửnos‡ always
possible to do it, so let us say that it 7s always possible, and that the potential-
plus-kinetic energy of the world is constant. 'Phus the total potential-plus-kinetic
energy inside the whole world is constant, and if the “world” is a piece of isolated
material, the energy is constant if there are no external forces. But as we have
seen, some of the kinetic and potential energy of a thing may be internal, for
instance the internal molecular motions, in the sense that we do not notice ït.
W©e know that in a glass of water everything is jiggling around, all the parts are
moving all the time, so there is a certain kinetic energy inside, which we ordinarily
may not pay any attention to. We do not notice the motion of the atoms, which
produces heat, and so we do not call it kinetic energy, but heat is primarily
kinetic energy. Internal potential energy may also be in the form, for instance, of
chemical energy: when we burn gasoline energy is liberated because the potential
energies of the atoms in the new atomic arrangement are lower than in the old
arrangement. Tt is not strictly possible to treat heat as being pure kinetic energy,
for a little of the potential gets in, and vice versa for chemical energy, so we
put the ©wo together and say that the total kinetic and potential energy inside
an object is partly heat, partly chemical energy, and so on. Anyway, all these
diferent forms of internal energy are sometimes considered as “lost” energy in the
sense described above; this will be made clearer when we study thermodynamics.