Does the Triple Integral Formula Apply to a Point-Mass Inside a Spherical Shell?

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SUMMARY

The discussion centers on the application of the Triple Integral Formula to calculate the gravitational potential of a point mass "m" located inside a spherical shell with inner radius R1 and outer radius R2, both having a uniform density ρ. The expression -Gmρ2π(R2² - R1²) is scrutinized for its validity in representing the potential within the shell. Participants clarify that the potential inside a homogeneous sphere can be derived from established formulas for both the interior and exterior of the shell, emphasizing the need to correctly apply the gravitational potential formula GMm/R. The consensus is that the initial expression lacks completeness and requires further refinement.

PREREQUISITES
  • Understanding of gravitational potential and the formula GMm/R
  • Familiarity with spherical coordinates and integration techniques
  • Knowledge of the properties of homogeneous spheres and shells
  • Basic principles of gravitational theory and point mass interactions
NEXT STEPS
  • Study the derivation of gravitational potential inside a homogeneous sphere
  • Learn about the application of the Triple Integral Formula in physics
  • Explore the concept of gravitational potential energy in spherical shells
  • Investigate the implications of density variations in gravitational calculations
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Physics students, gravitational theorists, and anyone involved in advanced mechanics or astrophysics who seeks to understand gravitational potentials in spherical geometries.

cscott
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Does [itex]-Gm\rho2\pi\left(R_2^2-R_1^2\right)[/itex] make sense for the potential of a point-mass "m" inside a spherical shell of radii [itex]R_1< R_2[/itex] and density [itex]\rho[/itex]?

Now I've already found the potential outside of a homogeneous sphere of same density. I'm now asked to use these two results to find the potential inside a homogeneous sphere of again indeity density sphere, using the previous two results.

Can I do this by considering the point on the inside of the shell with a sphere in the space in the middle? Does it make sense to add the two potential formulas using the correct radii?
 
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Expression for potential is GMm/R. So something is missing in the given expression
 

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