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In my Physics textbook, it says that if two pendulums of the same natural frequency are placed next to each other and if one is set into vibration, the other starts resonating and when the first one gets damped due to air friction, the vibration of the second one again sets the first into vibration (resonance) and this goes on until all the energy is dissipated as sound. My Question is: If placed in vacuum, will these two complement each others' vibrations and continue vibrating in perpetual motion (because there isn't any propagation of sound in vacuum due to absence of a medium)?

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No, because in a vacuum, there is no way for the two tuning forks (I think you meant this, rather than pendulums) to communicate. The reason a second tuning fork with the same resonance frequency will begin resonating is because, physically, sound waves are hitting it at its natural frequency. Sound waves travel in a medium, so in a vacuum, there's nothing to carry the sound waves and, hence, no communication between the tuning forks.

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    $\begingroup$ You can do the trick with pendulums, too, but it is motion of a common supporting member rather than vibrations in the air that connects them. $\endgroup$ – dmckee May 26 '12 at 15:54
  • $\begingroup$ Is there a way to attach the pendulums/tuning forks to an air column and simultaneously ensure that no sound energy is wasted? $\endgroup$ – Graviton May 26 '12 at 16:17
  • $\begingroup$ @Graviton Sound heats up the medium in which it operates. Always. $\endgroup$ – yo' Aug 21 '15 at 7:55

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