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where h is the object’s altitude above the earth’s surface. As an object falls to earth, its Section 2.1
1
2
potential energy mgh decreases and its kinetic energy mv increases. Provided the Classical Mechanics
2
effect of air friction is negligible, the total mechanical energy K V remains constant
as the object falls.
We have considered a one-particle system. Similar results hold for a many-particle
system. (See H. Goldstein, Classical Mechanics, 2d ed., Addison-Wesley, 1980,
sec. 1-2, for derivations.) The kinetic energy of an n-particle system is the sum of the
kinetic energies of the individual particles:
1 n
# # #
K K K K a m v 2 (2.23)
1
2
n
i i
2 i 1
Let the particles exert conservative forces on one another. The potential energy V of
the system is not the sum of the potential energies of the individual particles. Instead,
V is a property of the system as a whole. V turns out to be the sum of contributions due
to pairwise interactions between particles. Let V be the contribution to V due to the
ij
forces acting between particles i and j. One finds
V a a V ij (2.24)
i j 7i
The double sum indicates that we sum over all pairs of i and j values except those with i
equal to or greater than j.Terms with i j are omitted because a particle does not exert
a force on itself. Also, only one of the terms V and V is included, to avoid counting
12
21
the interaction between particles 1 and 2 twice. For example, in a system of three parti-
cles, V V V V .Ifexternal forces act on the particles of the system, their con-
13
23
12
tributions to V must also be included. [V is defined by equations similar to (2.17).]
ij
One finds that K V E mech is constant for a many-particle system with only
conservative forces acting.
The mechanical energy K V is a measure of the work the system can do. When
a particle’s kinetic energy decreases, the work–energy theorem w K [Eq. (2.16)]
says that w, the work done on it, is negative; that is, the particle does work on the sur-
roundings equal to its loss of kinetic energy. Since potential energy is convertible to
kinetic energy, potential energy can also be converted ultimately to work done on the
surroundings. Kinetic energy is due to motion. Potential energy is due to the positions
of the particles.

EXAMPLE 2.1 Work

A woman slowly lifts a 30.0-kg object to a height of 2.00 m above its initial
position. Find the work done on the object by the woman, and the work done by
the earth.
The force exerted by the woman equals the weight of the object, which from
2
Eq. (2.6) is F mg (30.0 kg) (9.81 m/s ) 294 N. From (2.10) and (2.11),
the work she does on the object is
x 2
w F1x2 dx F¢x 1294 N212.00 m2 588 J
x 1
The earth exerts an equal and opposite force on the object compared with the
lifter, so the earth does 588 J of work on the object. This work is negative