Environmental Engineering Reference
In-Depth Information
d
U
x
d
x
is negative, the force positive
and the particle is driven to higher values of
x
. Thus for system (a), any small
departure from
x
=
x
0
. If we move the particle to
x>x
0
then
x
0
results in a force that works in the same direction as the
displacement. The equilibrium is therefore unstable to small perturbations away
from
x
=
x
0
. In Figure 3.7(b) a minimum in the potential energy is illustrated.
Now the behaviour is reversed and small displacements from equilibrium result in
forces that act in opposition to the displacement. These restoring forces ensure that
small perturbations do not produce large effects; the equilibrium in (b) is stable.
In practice, positions of unstable equilibrium can never be achieved. While we
can imagine setting up a system in unstable static equilibrium by ensuring that the
condition
x
=
x
0
is perfectly observed, this is impossible to achieve in practice.
However, in a situation of stable equilibrium, small perturbations produce small
effects that tend to return the system back to its starting point.
The relationship between potential energy and stability is of course not limited
to systems involving one degree of freedom. A good example of a potential energy
surface in two dimensions is obtained upon considering a marble on the surface
of a hemispherical bowl. Since the gravitational potential energy is proportional to
the height of the marble it can be written as a function of the marble's position
in the horizontal plane using the equation for the surface of the bowl. The contact
forces between the marble and the bowl act as forces of constraint and do not
contribute to the potential energy. Consider first the bowl placed with its rim on
a flat horizontal surface and the opening downwards. If the marble is placed on
the highest point of the outer surface of the upturned bowl then this is a position
of unstable equilibrium, similar to Figure 3.7(a) and is impossible to achieve. If,
however, the bowl sits so that the opening is upwards and the marble is positioned
at the lowest point of the inner surface, this represents stable equilibrium. In this
example, the potential energy can be expressed as a function of the two position
co-ordinates in the horizontal plane,
x
and
y
. To achieve static equilibrium, the
potential energy must be a stationary point with respect to variations in both
x
and
y
. In other words, both the
x
and
y
components of the force in the horizontal plane
must be zero in order to have static equilibrium. Notice that since the potential
energy close to the surface of the Earth is proportional to height, it follows that the
surface of the bowl just happens to provide a visual map of the potential energy
surface appropriate to the marble's motion.
=
3.2.2
The harmonic oscillator
Using the idea of potential energy, we can explore what is perhaps the single most
important physical system in the whole of physics, both classical and quantum: the
simple harmonic oscillator. In so doing we will learn, in this section and the next,
just why an object so ordinary as a stretched spring should provide the prototype
for the behaviour of a vast range of physical systems close to equilibrium. We start
by considering a one-dimensional system for which the only force
F
actingona
particle of mass
m
satisfies Hooke's Law,
F
kx
(think of a stretched spring if
you like). Such a system is known as a simple harmonic oscillator.
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