Frequency of Small Oscillations

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SUMMARY

The discussion focuses on determining the optimal distance \( d \) from the center of a uniform coin that maximizes the frequency of small oscillations when the coin is pivoted. The potential energy \( V(x) \) is approximated using the second derivative \( V''(x_0) \), and the angular frequency \( \omega \) is defined as \( \sqrt{V''(x)/m} \). Participants emphasize treating the coin as a physical pendulum and suggest calculating the moment of inertia about the pivot point as a critical step in solving the problem.

PREREQUISITES
  • Understanding of physical pendulums
  • Knowledge of potential energy and its derivatives
  • Familiarity with moment of inertia calculations
  • Basic principles of oscillatory motion
NEXT STEPS
  • Calculate the moment of inertia for a uniform coin about a pivot point
  • Explore the relationship between potential energy and angular frequency in oscillatory systems
  • Study the effects of varying pivot points on the frequency of oscillations
  • Investigate the mathematical derivation of the frequency formula for physical pendulums
USEFUL FOR

Students in physics, particularly those studying mechanics and oscillatory motion, as well as educators seeking to clarify concepts related to physical pendulums and their dynamics.

Piglet1024
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1. A uniform coin with radius R is pivoted at a point that is a distance d from its center. The coin is free to swing back and forth in the vertical plane defined by the plane of the coin. For what value of d is the frequency of small oscillations largest?



2. V(x)\equivpotential energy; V(x)\approx\frac{1}{2}V"(x_{o})(x-x_{o})^{2} ; omega is equal to the square root of V"(x)/m



3. I have no idea what to do. My textbook is vague and my notes from lecture are not sufficient. I don't want a solution, just a push in the right direction
 
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Treat this as a physical pendulum. Begin by finding the moment of inertia about the pivot.
 

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