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18.03 Class 11, March 3, 2006
Second order equations: Physical model, characteristic polynomial,
real roots, structure of solutions, initial conditions
[1] F = ma is the basic example.
Take a spring attached to a wall,
spring
mass
dashpot





>
F_ext

______







VVVVVVV
_____


_______
_____________


O  O




>

x
Set up the coordinate system so that at
x = 0
the spring is relaxed.
The cart is influenced by three forces:
the spring, the "dashpot"
(which is a way to make friction explicit), and an external force:
mx"
=
F_spr + F_dash + F_ext
The spring force is characterized by depending only on position:
write
F_spr(x).
If
x > 0
,
F_spr(x) < 0
If
x = 0
,
F_spr(x) = 0
If
x < 0
,
F_spr(x) > 0
I sketched a graph of
F_spr(x)
as a function of
x .
The simplest way to model this behavior (and one which is valid in
general
for small
x ,
by the tangent line approximation) is
F_spr(x)
=
kx
k > 0
the "spring constant."
"Hooke's
Law"
This is another example of a linearizing approxmimation.
The dashpot force is frictional. This means that it depends only on the
velocity.
Write
F_dash(x').
It acts against the velocity:
If
x' > 0 ,
F_dash(x') < 0
If
x' = 0 ,
F_dash(x') = 0
If
x' < 0 ,
F_dash(x') > 0
The simplest way to model this behavior (and one which is valid in
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 Winter '08
 Staff
 Equations, Elementary algebra, Constant of integration

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