Calculating Gravity Inside Complex Spherical Bodies

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Discussion Overview

The discussion revolves around the calculation of gravitational forces inside complex spherical bodies, particularly focusing on the implications of the shell theorem, the effects of varying densities, and the interactions between different mass distributions. Participants explore theoretical scenarios involving spherical shells, ellipsoids, and plates, as well as the gravitational effects of central masses.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question whether the shell theorem applies to ellipsoids and disks, suggesting that the uniformity of thickness in a sphere is unique.
  • Others argue that for a spherically symmetric mass distribution, the gravitational field at a distance is determined only by the mass within that radius, regardless of density variations.
  • One participant proposes that the shell theorem may not adequately account for gravitational interactions when a mass element is between a central mass and an outer shell, raising concerns about the cancellation of forces.
  • Another participant emphasizes that the shell theorem states that a spherical shell exerts no gravitational force on any mass element within it, regardless of the presence of a central mass.
  • Participants introduce alternative scenarios, such as the gravitational effects of a spherical mass and a round plate, questioning how to calculate the forces in such configurations.
  • There is a suggestion to simply add the gravitational forces from both the spherical mass and the plate acting on a mass element positioned between them.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the shell theorem to various geometries and the nature of gravitational interactions in complex configurations. The discussion remains unresolved, with multiple competing perspectives presented.

Contextual Notes

Some assumptions regarding the uniformity of mass distributions and the conditions under which the shell theorem applies are not fully explored. The implications of varying densities and the specific geometrical configurations are also not definitively established.

dougettinger
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I know about the shell theorem and that the force of gravity due to the mass of the shell is zero inside a spherical shell. I have some questions that may easily be answered.

1. Does this same conclusion hold for an ellipsoid and/or a disk with the shell on the exterior rim ?

2. How is the gravity force inside a spherical solid affected if it has two layers and a central core all with different but homogeneous densities ? Especially inside the middle layer that has the least density and least mass ?
 
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dougettinger said:
1. Does this same conclusion hold for an ellipsoid and/or a disk with the shell on the exterior rim ?
No.
2. How is the gravity force inside a spherical solid affected if it has two layers and a central core all with different but homogeneous densities ? Especially inside the middle layer that has the least density and least mass ?
For a spherically symmetric mass distribution (even with varying densities), the gravitational field at any distance r from the center is only due to the mass within that radius--mass elements at distances greater than r do not contribute to the field at r.
 
dougettinger said:
1. Does this same conclusion hold for an ellipsoid and/or a disk with the shell on the exterior rim ?
You can get that effect with any closed shape, but only in a sphere can the thickness of the walls be the same everywhere. A hollow ellipsoid for example needs to have thicker walls at the ends with the small radius of curvature. The part where the radius of curvature is large needs thinner walls. The exact thickness the walls need to have can be calculated in the same way as the electric charge distribution on a statically charged metal object.
 
Thank you for your unexpected prompt replies. Let me clarify my problem. Let the mass element in question be between a massive sphere in the center and an outer massive shell which you could almost consider being flat with respect to the mass element, although it surrounds the massive sphere. I believe the shell theorem in this case has difficulties canceling the gravity forces of the shell due to the central mass.
Are Doc Al and DrZoidberg tags ?
 
dougettinger said:
Let me clarify my problem. Let the mass element in question be between a massive sphere in the center and an outer massive shell which you could almost consider being flat with respect to the mass element, although it surrounds the massive sphere. I believe the shell theorem in this case has difficulties canceling the gravity forces of the shell due to the central mass.
You have a uniform spherical shell, right? So why would the shell theorem not apply, just like in any other case? How is the presence of the central mass relevant?
 
Hello Doc Al,
The mass element is pulled in two directions; by the central mass and by an element of mass on the shell that cannot be canceled because it is interacting with the central mass.
The formalized shell theorem solution for a mass element inside a hollow sphere is always for the case of an empty sphere except for the mass element.

I am trying to be sure we both understand each other.
 
dougettinger said:
Hello Doc Al,
The mass element is pulled in two directions; by the central mass and by an element of mass on the shell that cannot be canceled because it is interacting with the central mass.
The formalized shell theorem solution for a mass element inside a hollow sphere is always for the case of an empty sphere except for the mass element.

I am trying to be sure we both understand each other.
The shell theorem says: The spherical shell exerts no gravitational force on any mass element within the shell. Doesn't matter if it's empty or not.

In your example there will be a gravitational field within the shell, but that field is due to the central mass and not the shell.
 
Doctor Al,
Thank you for being patient with me. Allow me to propose another example that does not utilize the shell theorem. Assume that the mass element in question is between a spherical mass and a round plate with a specified radius, density, and thickness. The mass element is on a line that is perpendicular to the plate and goes through the center of the spherical mass. The radius of the round plate compared to the radius of the sphere is much larger by a factor of 100. How are the gravity forces exerted on the mass element determined for such a case ?

Thoughtfully, Doug Ettinger
 
Last edited:
dougettinger said:
Doctor Al,
Thank you for being patient with me. Allow me to propose another example that does not utilize the shell theorem. Assume that the mass element in question is between a spherical mass and an infinite plate with a specified density and thickness. The mass element is on a line that is perpendicular to the plate and goes through the center of the spherical mass. How are the gravity forces exerted on the mass element determined for such a case ?
Just add them up. Both the spherical mass and the infinite plate will exert a gravitational attraction on a mass element placed between them.

(Not sure how this relates to your original question.)
 

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