(5.5) |
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(5.6) |
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(5.7) |
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(5.8) |
The normal mode represents the center of mass. The center of
mass behaves as if it were a particle of mass
attached to two
springs (Figure 5.2) and its oscillation frequency is the same
as that of the individual decoupled oscillators
.
The normal mode represents the relative motion of he two masses
which leaves the center of mass unchanged. This can be thought of as
the motion of two particles of mass
connected to a spring of
spring constant
as shown in Figure
5.3. The oscillation frequency of this normal mode
is always higher than that of
the individual uncoupled oscillators (or the center of mass).
The modes
and
are often referred to as the slow mode
and the fast mode respectively.
We may interpret as a mode of oscillation where
the two masses oscillate with exactly the same phase, and
as a
mode where they have a phase difference of
(Figure
5.4). Recollect that the
phases of the two masses are independent when the two masses are
not coupled. Introducing a coupling causes the phases to
be interdependent.
The normal modes have solutions
(5.9) |
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(5.10) |
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(5.11) |
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(5.12) |