For what value of d is the frequency of small oscillations largest?

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SUMMARY

The discussion focuses on determining the optimal distance d from the center of a coin pivoted for maximizing the frequency of small oscillations. The relevant equation for frequency is established as Frequency = (1/2π)√(k/m). The user attempts to derive the relationship by applying Newton's second law in the tangential direction and considers the moment of inertia of a disk using the parallel axis theorem. The correct approach involves calculating the moment of inertia and applying torque to find the angular frequency.

PREREQUISITES
  • Understanding of angular motion and oscillations
  • Familiarity with the moment of inertia, particularly for disks
  • Knowledge of the parallel axis theorem
  • Basic calculus for deriving maxima and minima
NEXT STEPS
  • Study the moment of inertia of a disk and its applications
  • Learn about the parallel axis theorem in detail
  • Explore the derivation of angular frequency in oscillatory motion
  • Investigate the relationship between torque and angular acceleration
USEFUL FOR

Physics students, mechanical engineers, and anyone studying dynamics and oscillatory systems will benefit from this discussion.

Tonyt88
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Homework Statement


A coin of radius R is pivoted at a point that is distance d from the 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?



Homework Equations



Frequency = 1/T = (1/2(pi))(k/m)^1/2

x = A cos( (omega)(t) + phi )



The Attempt at a Solution



I assume I have to find some derivative in order to maximize the value, just don't know where to start.
 
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Would this be correct?

F = ma in the tangental direction

-mg sin(theta) = ma

-mg sin(theta) = m((R+d) theta '' ) (note theta '' is 2nd derivative)

thus omega = (g/(R+d))^1/2

Or is this totally the wrong idea?
 
You have to use the moment of inertia of a disk, and then use the parallel axis theorem so that you can apply torque=I alpha to the coin.
 

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