Environmental Engineering Reference
In-Depth Information
Solution 3.2.3 We use Eq. (3.19) to get from the potential energy to an expression
for the force on the particle:
d U
d x
4 αx 3 .
=−
=−
F(x)
kx
4 αx 3 . Notice
that F a is still a restoring force since it depends on an odd-power of x, i.e. it points
towards the origin (x
This deviates from Hooke's Law by the addition of the term F a =−
0 ) whatever the sign of x. As a result, we can still expect
the particle to oscillate about the origin. However, Newton's Second Law now leads
to the equation of motion
=
m d 2 x
d t 2
4 αx 3 ,
=−
kx
where m is the mass of the particle. A simple sine or cosine function with fixed
frequency cannot satisfy this equation and the motion is more complicated. Such a
system is referred to as an anharmonic oscillator.
3.2.3
Motion about a point of stable equilibrium
Finally, we are ready to reveal why springs are so important in physics. Consider
a particle at a point of stable equilibrium x
=
x 0 in a potential U(x) . What motion
do we expect for small departures from equilibrium? We may expand the potential
as a Taylor series about x 0 as
x 0 +
x 0 +
d 2 U
d x 2
d U
d x
1
2 (x
x 0 ) 2
U(x)
U(x 0 )
+
(x
x 0 )
...
(3.29)
d x 2 x 0
d x x 0 =
0and d 2 U
d U
Since the equilibrium is stable we must have
0. We set the
d x 2 x 0 =
d 2 U
constant
k and redefine our scale of potential energy such that U(x 0 )
=
0.
Thus
1
2 k(x
x 0 ) 2
U(x)
(3.30)
and so, for small enough departures from equilibrium, the potential energy of (and
hence the restoring forces acting on) the particle are identical to those of the
harmonic oscillator.
The above argument may be generalised to three dimensions leading to three
terms in the potential energy:
1
2 k x (x
1
2 k y (y
1
2 k z (z
x 0 ) 2
y 0 ) 2
z 0 ) 2 ,
U(x,y,z)
+
+
(3.31)
where the three spring constants, k x k y and k z , correspond to the generally different
restoring forces in the x, y and z directions, respectively. The equilibrium position
is given by the co-ordinates (x 0 ,y 0 ,z 0 ) .
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