Calculating Gravity Inside Complex Spherical Bodies

[dougettinger]

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 ?

 

[Doc Al]

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.

 

[DrZoidberg]

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.

 

[dougettinger]

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 ?

 

[Doc Al]

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?

 

[dougettinger]

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.

 

[Doc Al]

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.

 

[dougettinger]

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

 

[Doc Al]

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.)