Quick question on coupled pendulum kinetic energy

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SUMMARY

The discussion centers on the kinetic energy of a coupled pendulum system, specifically addressing the absence of \dot{y} terms in the kinetic energy equations for the masses involved. The participants clarify that in the first normal mode, both masses oscillate horizontally, leading to kinetic energy expressions that only include \dot{x} terms. This is due to the coupling by a spring, which restricts vertical movement. The confusion arises from the expectation that coupled pendulums should exhibit behavior similar to regular pendulums, which include vertical motion in their kinetic energy calculations.

PREREQUISITES
  • Understanding of classical mechanics principles, particularly oscillatory motion.
  • Familiarity with coupled oscillators and their dynamics.
  • Knowledge of kinetic energy formulations in mechanical systems.
  • Basic grasp of normal modes in oscillatory systems.
NEXT STEPS
  • Study the derivation of kinetic energy in coupled oscillators.
  • Learn about normal modes and their significance in mechanical systems.
  • Explore the mathematical modeling of coupled pendulum systems.
  • Investigate the effects of spring constants on the motion of coupled pendulums.
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Physics students, mechanical engineers, and anyone interested in the dynamics of coupled oscillatory systems will benefit from this discussion.

cmackeen
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I have found that the kinetic energy term for each mass on a coupled (by a spring) pendulum has no \dot{y} terms, as in the the masses do not move in the y direction at all. What am I missing here? I know they are connected, but in the first normal mode, both masses should move like regular pendulums. Regular pendulums have a \dot{y} component in their kinetic energy, so why are there only \dot{x} terms for kinetic energy? Provided link below

http://csirnetphysics.blogspot.com/2012/11/classical-mechanics-question-1.html

In this diagram, x_1 and x_2 correspond to each mass (m_1 and m_2)
 
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I can't see any correspondence between the diagram and your variables. What are x1 and x2? I don't see any mention of y throughout the derivation.
 

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