Physics For Scientists And Engineers 6E - part 60
SECTION 8.6 • Energy Diagrams and Equilibrium of a System
237
spring. As the block starts to move, the system acquires kinetic energy and loses an
equal amount of potential energy. Because the total energy of the system must remain
constant, the block oscillates (moves back and forth) between the two points
x " #x
max
and x " % x
max
, called the turning points. In fact, because no energy is lost
(no friction), the block will oscillate between # x
max
and % x
max
forever. (We discuss
these oscillations further in Chapter 15.) From an energy viewpoint, the energy of the
system cannot exceed
therefore, the block must stop at these points and, be-
cause of the spring force, must accelerate toward x " 0.
Another simple mechanical system that has a configuration of stable equilibrium is
a ball rolling about in the bottom of a bowl. Anytime the ball is displaced from its low-
est position, it tends to return to that position when released.
Now consider a particle moving along the x axis under the influence of a conserva-
tive force F
x
, where the U-versus-x curve is as shown in Figure 8.17. Once again, F
x
"
0 at
x " 0, and so the particle is in equilibrium at this point. However, this is a position of
un-
stable equilibrium for the following reason: Suppose that the particle is displaced to
the right (x , 0). Because the slope is negative for x , 0, F
x
" #
dU/dx is positive, and
the particle accelerates away from x " 0. If instead the particle is at x " 0 and is dis-
placed to the left (x - 0), the force is negative because the slope is positive for x - 0,
and the particle again accelerates away from the equilibrium position. The position
x " 0 in this situation is one of unstable equilibrium because for any displacement from
this point, the force pushes the particle farther away from equilibrium. The force pushes
the particle toward a position of lower potential energy. A pencil balanced on its point is
in a position of unstable equilibrium. If the pencil is displaced slightly from its absolutely
vertical position and is then released, it will surely fall over. In general,
configurations
of unstable equilibrium correspond to those for which U(x) is a maximum.
Finally, a situation may arise where U is constant over some region. This is called a
configuration of
neutral equilibrium. Small displacements from a position in this re-
gion produce neither restoring nor disrupting forces. A ball lying on a flat horizontal
surface is an example of an object in neutral equilibrium.
1
2
kx
2
max
;
0
x
U
Negative slope
x > 0
Positive slope
x < 0
Figure 8.17 A plot of U versus x for
a particle that has a position of un-
stable equilibrium located at x " 0.
For any finite displacement of the
particle, the force on the particle is
directed away from x " 0.
Example 8.11 Force and Energy on an Atomic Scale
The potential energy associated with the force between two
neutral atoms in a molecule can be modeled by the
Lennard–Jones potential energy function:
where x is the separation of the atoms. The function U(x) con-
tains two parameters . and / that are determined from experi-
ments. Sample values for the interaction between two atoms in
a molecule are . " 0.263 nm and / " 1.51 0 10
#
22
J.
(A)
Using a spreadsheet or similar tool, graph this function
and find the most likely distance between the two atoms.
Solution We expect to find stable equilibrium when the
two atoms are separated by some equilibrium distance and
the potential energy of the system of two atoms (the mole-
cule) is a minimum. One can minimize the function U(x) by
taking its derivative and setting it equal to zero:
"
4/
&
#
12.
12
x
13
#
#
6.
6
x
7
'
"
0
dU(x)
dx
"
4/
d
dx
&
(
.
x
)
12
#
(
.
x
)
6
'
"
0
U(x) " 4/
&
(
.
x
)
12
#
(
.
x
)
6
'
Solving for x—the equilibrium separation of the two atoms
in the molecule—and inserting the given information yields
x "
We graph the Lennard–Jones function on both sides of
this critical value to create our energy diagram, as shown in
Figure 8.18a. Notice that U(x) is extremely large when the
atoms are very close together, is a minimum when the atoms
are at their critical separation, and then increases again as
the atoms move apart. When U(x) is a minimum, the atoms
are in stable equilibrium; this indicates that this is the most
likely separation between them.
(B)
Determine F
x
(x)—the force that one atom exerts on the
other in the molecule as a function of separation—and ar-
gue that the way this force behaves is physically plausible
when the atoms are close together and far apart.
Solution Because the atoms combine to form a molecule, the
force must be attractive when the atoms are far apart. On the
other hand, the force must be repulsive when the two atoms
are very close together. Otherwise, the molecule would collapse
in on itself. Thus, the force must change sign at the critical sep-
aration, similar to the way spring forces switch sign in the
change from extension to compression. Applying Equation
8.18 to the Lennard–Jones potential energy function gives
2.95 0 10
#
10
m.
Neutral equilibrium
Unstable equilibrium