Environmental Engineering Reference
In-Depth Information
to escape. Rearranging gives
2
GM
r
E
=
v
0
.
3.2.1
The stability of mechanical systems
In Chapter 2 we defined static equilibrium to hold when the net force on a
particle is zero. Suppose we have a system in which all the forces acting on the
particle are conservative, and which the particle is free to move in one dimension
only (we go beyond one dimension in Chapter 9). As we have seen, we can then
define a potential energy
U(x)
where
x
represents the position of the particle.
Using Eq. (3.1), d
W
=−
d
U
=
F(x)
d
x
and so
d
U(x)
d
x
F(x)
=−
.
(3.19)
The condition of static equilibrium can thus be written
d
U
d
x
=
0
,
(3.20)
i.e. the potential energy has a stationary point where the force is zero. This sta-
tionary point can be either a maximum or a minimum, as illustrated in Figure 3.7.
Figure (a) illustrates the potential energy in the case that there is a maximum at
x
x
0
. The particle is in static equilibrium at this point, but if we consider even
tiny departures from
x
=
x
0
we see that the equilibrium is unstable. For example,
moving the particle to
x<x
0
results in a potential energy surface with positive
gradient. Thus
F
=
d
U
d
x
=−
is negative and the force drives the particle away from
Unstable
Stable
U
(
x
)
U
(
x
)
F
< 0
F
> 0
F
> 0
F
< 0
x
0
x
x
0
x
(a)
(b)
Figure 3.7 Potential energy as a function of distance for two one-dimensional systems. In
(a) a maximum in
U
results in unstable equilibrium: for
x
=
x
0
the force acts to move the
particle away from
x
0
.In(b)for
x
=
x
0
the force acts to move the particle towards
x
0
and
the equilibrium is stable.