Centre of mass of a solid hemisphere (Feynman way)

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SUMMARY

The discussion focuses on calculating the center of mass of a solid hemisphere using the Law of Pappus, as referenced in Feynman's Lectures, Volume 1, Topic 19.2. The user attempts to apply this method by decomposing the solid hemisphere into an infinite number of semi-circular discs, but encounters a discrepancy with the known result of 3R/8 obtained through calculus. The user seeks clarification on their approach and numerical calculations to reconcile the differences.

PREREQUISITES
  • Understanding of the Law of Pappus
  • Familiarity with the concept of center of mass
  • Basic knowledge of calculus
  • Experience with solid geometry
NEXT STEPS
  • Review the derivation of the center of mass for a solid hemisphere using calculus
  • Study the application of the Law of Pappus in different geometric contexts
  • Examine numerical methods for calculating center of mass
  • Explore Feynman's Lectures, Volume 1, Topic 19.2 for deeper insights
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Students of physics, educators teaching mechanics, and anyone interested in advanced geometric calculations related to center of mass.

saturn_1995
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reference to Feyman lectures vol.1 topic 19.2 locating centre of mass

Feyman gives us the law of pappus to find the centre of mass ,which he proves for semicircular disc and ring.

But when i am trying to extend it to finding the centre of mass of a solid semi-circular solid hemisphere ,i seem to get a different answer from what i get from calculus which is 3R/8.

My approach to problem is using law of pappus extensively:-

1. I assume a solid semi-circular hemisphere into infinite number of semi-circular disc, find each disc's centre of mass using law of pappus.
2. I get a ring with all centre of mass of respective discs.
3. using that ring and law of pappus again i find the centre of mass of that ring.

this value which i find doesn't correspond to 3R/8. is there something wrong in my approach
 
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It would help if you showed the numerical details of your calculation.
 

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