THEORY
Vibration
is the motion of an object back and forth over the same ground.
The most important example of vibration is
simple harmonic motion
(SHM).
One system that manifests SHM is a mass, m, attached to a spring where k is
the spring constant. Let the system reside on a horizontal table with no
friction present. If the mass is pulled back a distance x from the equilibrium
it will experience a spring force that obeys Hooke’s law,
F =  k x.
Since this is the net force on m, by Newton’s second law,
k x = m d
2
x / dt
2
OR
d
2
x / dt
2
+
ϖ
2
x
=
0
where
ϖ
2
≡
k/m,
and
ϖ
is called the
(angular) frequency
of vibration = 2
π
/T, T =
period
=1/f,
and f is the
frequency
.
Here is the first case where we are confronted with a nonconstant
acceleration. The above equation is called a
differential equation
because it
contains both a function, x(t), as well as derivatives of the function.
We will solve the differential equation two ways:
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 Spring '08
 Chen
 Physics, Energy, Force, Mass, Simple Harmonic Motion

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