Condition for Simple Harmonic Motion

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SUMMARY

The discussion focuses on determining the least ratio of R/r necessary for a small object mounted on a hoop to perform simple harmonic motion (SHM) within a semi-cylinder shaped trough. The key conclusion is that for SHM to occur, the object must be in a stable equilibrium position where its potential energy is minimized at the mean position. The mathematical expression of this condition is critical for solving the problem effectively.

PREREQUISITES
  • Understanding of simple harmonic motion (SHM)
  • Knowledge of potential energy concepts in physics
  • Familiarity with equilibrium positions in mechanics
  • Basic mathematical skills for expressing physical conditions
NEXT STEPS
  • Research the mathematical conditions for stable equilibrium in SHM
  • Study potential energy graphs and their implications in mechanics
  • Explore the relationship between radius ratios in oscillatory systems
  • Investigate previous discussions on similar SHM problems for deeper insights
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Students studying physics, particularly those focusing on mechanics and oscillatory motion, as well as educators seeking to explain the principles of simple harmonic motion in practical scenarios.

utkarshakash
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Homework Statement


a small object is mounted to the perimeter of a hoop of radius r. The mass of the object and the hoop is the same. The hoop is placed into a fixed semi-cylinder shaped rough trough of radius R, such that the small mass is at the top. Find the least R/r ratio such that the object performs simple harmonic motion.

The Attempt at a Solution



For the object to perform SHM it must be in a stable equilibrium position. So, its potential energy must be minimum at the mean position. But how do I express it mathematically?
 
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Pranav-Arora said:
You have the right idea. I discussed the very same problem in past here.

Look: https://www.physicsforums.com/showthread.php?t=663246

Hmm that's a really long post and I think I should better leave this question as it would be too time consuming. btw Thanks for help!
 

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