Quantizing Radii of a Mass in Circular Orbit

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SUMMARY

The discussion focuses on determining the allowed radii of an object of mass m moving in a circular orbit under a central force defined by F=-kr. The application of the Bohr quantization condition (L=mvr=nh) leads to several potential formulas for the allowed radii, including rn=(n4h4/m2k)1/4, rn=(n2h2/mk)1/4, rn=(n2h2/mk)1/3, and rn=(n2h2/mk)1/2. The participants emphasize the necessity of calculating velocity by equating the force to the centripetal acceleration before applying the quantization condition. The discussion concludes that without additional information, it is impossible to definitively compute the quantized levels.

PREREQUISITES
  • Understanding of circular motion and centripetal acceleration
  • Familiarity with the Bohr quantization condition (L=mvr=nh)
  • Knowledge of force laws, specifically F=-kr
  • Basic algebra for solving equations involving variables
NEXT STEPS
  • Study the derivation of the centripetal acceleration in circular motion
  • Learn about the implications of the Bohr quantization condition in quantum mechanics
  • Explore different force laws and their effects on orbital motion
  • Investigate the mathematical techniques for solving equations with multiple variables
USEFUL FOR

This discussion is beneficial for physics students, educators, and anyone interested in the application of quantum mechanics to classical motion scenarios, particularly in understanding orbital dynamics and quantization principles.

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This is yet another sample exam problem which I don't quite know how to approach. Any help would be greatly appreciated.

5) Consider an object of mass m that moves in a circular orbit caused by a central
force given by F=-kr, where k is a constant. Suppose that the Bohr quantization
condition (L=mvr=nh) is applied to this motion. What are the allowed radii?
a. rn=(n4h4/m2k)1/4, n = 1,2,…
b. rn=(n2h2/mk)1/4, n = 1,2,…
c. rn=(n2h2/mk)1/3, n = 1,2,…
d. rn=(n2h2/mk)1/2, n = 1,2,…
e. Not enough information given to compute levels
f. Radii for this force law cannot be quantized
 
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You first find v by requiring the force to produce the centrepital acceleration of circular motion. Then set L=mvr=nh and solve for r.
 

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