Center of mass of solid hemisphere by using angle as variable.

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SUMMARY

The discussion focuses on calculating the center of mass of a solid hemisphere using angular variables. The user initially attempted to derive the center of mass using the equation y(cm) = ∫y dm / ∫ dm = 1/M (∫y dm) but encountered discrepancies in their results. After reviewing their calculations, the user identified an error in their approach. This highlights the importance of careful variable selection and integration techniques in physics problems.

PREREQUISITES
  • Understanding of center of mass concepts
  • Familiarity with integral calculus
  • Knowledge of solid geometry, specifically hemispheres
  • Experience with variable substitution in integrals
NEXT STEPS
  • Review the derivation of center of mass for different geometric shapes
  • Study integration techniques in calculus, focusing on variable substitution
  • Explore the application of polar coordinates in solving physics problems
  • Practice problems involving the center of mass of composite bodies
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Students studying physics, particularly those focusing on mechanics and calculus, as well as educators seeking to enhance their teaching methods in these subjects.

AlchemistK
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Homework Statement


I've just started with center of mass, and instead of the method in the book, I tried solving the center of mass for a solid hemisphere using angle as a variable, but the answer didn't match.
A small disk of mass "dm" is taken, which subtends an angle "dθ" at the center.The total mass of the hemisphere is "M". Please refer to the attachment for the figure and my full attempt.
Where have i gone wrong?

Homework Equations



y(cm) = ∫y dm / ∫ dm = 1/M (∫y dm)

The Attempt at a Solution


Please refer to the attachment.
 

Attachments

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Mod Note: moved to calculus & beyond. Please post in the correct forum next time!
 
Ah. Never mind, I found my mistake. Very idiotic of me.
 

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