Rotational inertia of a THICK spherical shell

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Discussion Overview

The discussion revolves around the calculation of the rotational inertia of a thick spherical shell with inner radius r, outer radius R, and mass M. Participants explore various approaches to derive the formula for the moment of inertia, including comparisons to the moment of inertia of solid spheres and the implications of using different mass values.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a formula for the rotational inertia of a thick spherical shell as [(2/5)M/(R^3-r^3)](R^5-r^5) and questions its validity compared to subtracting the inertia of a larger sphere from a smaller one.
  • Another participant suggests that the original poster should provide their work and asks if they know how to use LaTeX for better readability.
  • A later reply confirms the formula presented by the original poster is correct but emphasizes that the mass used in the inertia formula for the outer sphere differs from that of the inner core, suggesting unique labels for each mass.
  • One participant proposes checking the result by letting the shell thickness approach zero, leading to the moment of inertia of a thin shell, resulting in (2/3)MR^2.
  • Another participant seeks clarification on whether R refers to the inner or outer radius and questions the role of the other radius in the equation.
  • A participant clarifies that \Delta represents the difference between the outer and inner radius, specifically indicating that \Delta R^5 means R^5 - r^5.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to calculating the moment of inertia, with some agreeing on the validity of certain formulas while others raise questions about the assumptions and definitions used. The discussion remains unresolved regarding the best method to derive the moment of inertia for the thick spherical shell.

Contextual Notes

Participants have not reached a consensus on the correct formula for the moment of inertia, and there are unresolved questions about the definitions of the radii and the masses involved in the calculations.

Will
Someone please tell me is I am doing this problem correctly.If I have a thick spherical shell with inner radius r, outer radius R, and mass M, I am getting [(2/5)M/(R^3-r^3)](R^5-r^5). It is not the same thing as subtracting I of large sphere from I of smaller one, different than (2M(R^2-r^2)?

[tex]\frac{\2<br /> <br /> (M(R^5-r^5))}{5(R^3-r^3)}[/tex]
 
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I think we need to see your work. Do you know how to LaTeX your posts?
 
Will said:
Someone please tell me is I am doing this problem correctly.If I have a thick spherical shell with inner radius r, outer radius R, and mass M, I am getting [(2/5)M/(R^3-r^3)](R^5-r^5).
That's correct.
It is not the same thing as subtracting I of large sphere from I of smaller one, different than (2/5)M(R^2-r^2)?
Not exactly. If you treat the hole as a sphere of negative mass, then you can subtract the rotational inertia of each sphere: [itex]I_{shell} = I_{R-sphere} - I_{r-sphere}[/itex]. But realize that the mass of each sphere is different. If you express this answer in terms of the mass of the shell instead of the mass of either sphere, then you will find that you get the same answer as above.
 
Good catch, Doc Al. He is saying that the M in the I = 2/5 MR^2 is not the same for the sphere as it is for the inner core. So you would have to provide unique labels for each.
 
JohnDubYa said:
I think we need to see your work. Do you know how to LaTeX your posts?

Do you mean making my equations in "pretty print"? Please show me where I can learn to do this, its so much easier to read!
 
learning Latex

Will said:
Do you mean making my equations in "pretty print"? Please show me where I can learn to do this, its so much easier to read!
Poke around in this thread for many, many examples: https://www.physicsforums.com/showthread.php?t=8997
 
As a check, let the shell thickness approach zero to get the MI of a thin shell.
[tex]\Delta R^5/\Delta R^3=5R^4/3R^2=(5/3)R^2[/tex]
This is multiplied by [itex](2/5)M[/itex]. The result is the correct answer of
[tex](2/3)MR^2[/tex].
 
krab said:
As a check, let the shell thickness approach zero to get the MI of a thin shell.
[tex]\Delta R^5/\Delta R^3=5R^4/3R^2=(5/3)R^2[/tex]
This is multiplied by [itex](2/5)M[/itex]. The result is the correct answer of
[tex](2/3)MR^2[/tex].


? Does R man radius in or out?
Doesn't the other radius come into the equation?
 
Sorry for the shorthand. [itex]\Delta[/itex] means the difference between the case with R and the case with r. So for example by [itex]\Delta R^5[/itex] means [itex]R^5-r^5[/itex].
 

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