Consider a system of two identical pendulums swinging in parallel planes and connected at the top by a flexible string. (A
recently published Demonstration considered the related version of two pendulums with their bobs connected by a massless spring.) We describe here a more classic form of the problem, going back to the time of Huygens.
Each pendulum consists of a mass
suspended from a fixed support by a massless string of length
, with its coordinate described by the angle
from the vertical. Under the action of gravity, the equation of motion is given by
. Restricting ourselves to small-amplitude oscillations, we can approximate
, leading to the elementary solution
, where
, the natural frequency of the pendulum, independent of
. For the coupled pendulum system, to the same level of approximation, the Lagrangian can be written
where
is a measure of the coupling between the two pendulums mediated by the connecting string. The equations of motion are readily solved to give
.
The system has two normal modes of vibration. In the in-phase mode, starting with
, the two pendulums swing in unison at a frequency of
. In the out-of-phase mode, starting with
, they swing in opposite directions at a frequency of
.
A more dramatic phenomenon is resonance. If one pendulum is started at
and another at
, then energy will periodically flow back and forth between the two pendulums.
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