Harmonic oscillator and simple pendulum time period

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Discussion Overview

The discussion revolves around the differences in the time period of oscillation between a harmonic oscillator (HO) and a simple pendulum. Participants explore the implications of amplitude and length on the period of oscillation, considering both theoretical and practical aspects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that the time period T of a harmonic oscillator remains constant regardless of amplitude A or length l, provided stiffness k and mass m are unchanged.
  • Others argue that for a simple pendulum, the time period T changes with length l, as expressed in the formula T = 2π√(l/g), and that the mass does not affect the period in this context.
  • A participant suggests that the pendulum should take the same time T for one complete oscillation regardless of length, but acknowledges confusion when confronted with the formula indicating otherwise.
  • Some participants clarify that the period of the pendulum is independent of the angle θ, as long as the small angle approximation is valid, and that changing length l effectively alters the period.
  • One participant introduces the concept that the simple pendulum has a different equation of motion than the harmonic oscillator, leading to a dependence on amplitude when the small angle approximation is not valid.
  • Another participant references Huygens' work on cycloidal motion, noting that a cycloidal pendulum can maintain a constant period independent of amplitude.
  • Several participants discuss the practical implications of pendulum clocks and the adjustments made to maintain accurate timekeeping.
  • Some participants provide mathematical expansions for the period when the small angle approximation is not applicable, indicating that amplitude does influence the period in such cases.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of length and amplitude on the period of oscillation. While some agree on the formulas presented, others challenge the initial intuitions about pendulum behavior, leading to a nuanced discussion without a clear consensus.

Contextual Notes

Limitations include the dependence on the small angle approximation for the pendulum and the differing equations of motion for the harmonic oscillator and simple pendulum. The discussion also highlights the potential for confusion when applying intuitive reasoning to these concepts.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of physics, particularly those interested in oscillatory motion, pendulum mechanics, and the mathematical modeling of physical systems.

  • #31
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  • #32
malawi_glenn said:
We need a "pendulum clock" thread and insight article :)
The timing is poor because people lost interest in mechanical clocks in the 1970s because electronics became the future. However, the Internet didn't soar until the 1990s. That leaves little left online about the subject. However, some traditional books still survive. Here's one I just found.

Clockmaking - Past And Present:,' By The Late Lord Grimthorpe
This vintage book contains a complete guide to clocking making. This text is a veritable must-have for anyone with a keen interest in clocks and watches, and includes detailed, interesting information on the history of clock making, descriptions of the inner machinations and composition of clocks, and much more besides. Although old, the information contained herein is timeless, and will be of as much utility to modern readers as it was to those contemporary with its original publication. The chapters of this book include: A history of clocks and watches, Materials, Tools, Wheels and pinions, Escapements, Pendulums, Motive power, Striking mechanisms, Lantern clocks, Long case clocks, Bracket clocks, The age of a movement, Clock hands, British clocks for export, etcetera. We are republishing this antiquarian volume now in a modern, affordable edition complete with a new introduction on the history of clocks and watches.

By the way, I hope that Google's parent survives long enough for the 40 million books scanned by Google Books, to all become public domain so that their contents can become as searchable as post-book information. Otherwise, the knowledge of millennia may become as lost to mankind as the former contents of the Library at Alexandria.
 
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  • #33
Lord Grimthorpe? You couldn't make it up!
 
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  • #34
anorlunda said:
Atmospheric clocks use harvested energy to rewind the spring, not to drive the clock directly.
I wasn't aware these are still made. Thanks.
 
  • #35
Harvesting energy? Is that what is taught in field theory courses?
1660425619158.png
 
  • #37
The force applied to the mass is the same no matter how long it is (at least for small angles), so the tangential velocity is the same too. The angular velocity is not the same because of:

w = v/L

The angular velocity for a longer L is less, and for a shorter L more. You need more time to pass the same angle with longer L because arc length is longer.
 

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