Can gravity cause an expansion of a sphere?

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Discussion Overview

The discussion revolves around the effects of gravity on the size of a sphere made of an ideal linear material, as presented in the context of elasticity theory. Participants explore whether the sphere will expand or shrink when gravity is applied, referencing a specific problem from a textbook.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a scenario where a sphere's size is analyzed under the influence of gravity, questioning whether it will expand or shrink.
  • Another participant seeks clarification on the specific problem referenced, asking whether the sphere in question is solid or an empty shell.
  • A participant points out that the original source may indicate radial compression up to a certain radius and radial stretching beyond that, questioning the conclusion of overall expansion.
  • Further discussion reveals that a formula implies the sphere undergoes compression as a whole, suggesting that if the solution is correct, it does not contradict the idea of expansion.
  • A participant reflects on a previous calculation that suggested the sphere expanded, acknowledging a potential misunderstanding of the original text's claims.

Areas of Agreement / Disagreement

Participants express differing interpretations of the problem and its implications, with no consensus reached on whether the sphere expands or shrinks under gravity.

Contextual Notes

Participants reference specific pages and problems from a textbook, indicating potential discrepancies in editions and interpretations of the material. The discussion involves integrating radial strain to understand the effects of gravity on the sphere's size.

Jano L.
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Hi everybody,

some time ago our teacher has shown us the following example from the theory of elasticity:

Calculate how the gravity of the sphere changes its size. The sphere is made of ideal linear material (in practice, perhaps some metal) with Young modulus E and Poisson ration \nu. The amount of the material is such that if the gravity did not act, the radius of the sphere would be R_0. Now imagine the gravity is "turned on". Do you think the sphere will shrink or expand?

Teacher said (and the same can be found in Landau Lifgarbagez, Theory of elasticity, p. 21) that the sphere as a whole will actually expand due to gravity.

Do you think such a strange conclusion can be correct?
 
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Jano L. said:
Hi everybody,

some time ago our teacher has shown us the following example from the theory of elasticity:

Calculate how the gravity of the sphere changes its size. The sphere is made of ideal linear material (in practice, perhaps some metal) with Young modulus E and Poisson ration \nu. The amount of the material is such that if the gravity did not act, the radius of the sphere would be R_0. Now imagine the gravity is "turned on". Do you think the sphere will shrink or expand?

Teacher said (and the same can be found in Landau Lifgarbagez, Theory of elasticity, p. 21) that the sphere as a whole will actually expand due to gravity.
Here is that page:

http://books.google.de/books?id=tpY-VkwCkAIC&lpg=PP1&hl=de&pg=PA21#v=onepage&q&f=false

Do you mean problem 12? It is about a spherical cavity in an infinite medium. What sphere do you have in mind? A solid uniform sphere or an empty shell?
 
It is the problem 3, p. 21. The problem 12 with the cavity is at the page 24.
 
Last edited:
Aha, I have a second edition. My apologies. I see it is better not to use the page number but rather the paragraph/problem number. Anyway, I can't wait to read what you think of this...
 
Jano L. said:
Anyway, I can't wait to read what you think of this...
Well, as I said: I don't see yet that the radius increases under gravity. I guess one would have to integrate the radial strain they give.
 
Now I see it, their formula for u implies the sphere as a whole always undergoes compression, so if their solution is correct, there there is no paradox with expansion.

I recall we calculated this in detail and I think we got the result that the sphere expanded. I knew Landau has the same problem and I thought he claims the same thing, but now I see he does not. Most probably we made some mistake.

Thank you for your help,

Jano
 

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