Moment of Inertia of Hollow Cylinder Derivation

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Discussion Overview

The discussion revolves around the derivation of the moment of inertia for a uniform hollow cylinder. Participants explore the mathematical formulation and the implications of using different mass definitions in the context of the derivation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the correctness of the derivation, presenting a formula that involves the mass of both the whole solid cylinder and the missing piece.
  • Another participant suggests that the derivation is correct but highlights that it uses the mass of the whole cylinder instead of the mass of the hollow cylinder, proposing a substitution to arrive at the standard expression.
  • A third participant agrees with the initial derivation but notes that to express the moment of inertia using only the mass of the outer shell, a different mass definition must be applied.
  • A later reply indicates that the confusion arises from the specific application of the formula rather than a fundamental error in the derivation itself.

Areas of Agreement / Disagreement

Participants express differing views on the correctness of the derivation, with some asserting it is fine as written while others suggest modifications for clarity. The discussion remains unresolved regarding the best approach to express the moment of inertia.

Contextual Notes

Participants rely on specific definitions of mass and the geometry of the hollow cylinder, which may affect the interpretation of the moment of inertia. The discussion does not resolve the implications of these definitions on the derivation.

BrainSalad
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For a uniform, hollow cylinder, why is this derivation wrong?

M = mass of whole solid cylinder
m = mass of missing cylindrical piece
R = radius of whole cylinder
r = radius of missing cylindrical piece

moment of inertia = moment of inertia of whole cylinder - moment of inertia of missing cylindrical piece

I = MR2/2 - mr2/2

m/M = pi*r2*h/pi*R2*h = r2/R2

m = M*r2/R2

I = MR2/2 - M*r4/2R2

I = MR4/2R2 - M*r4/2R2

I = M/2R2*(R4 - r4)
 
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It seems correct, it is only that the result contains in its expression the mass M of the whole cylinder and not the mass of the corresponding hollow cylinder. If you substitute M=\frac{M_{h}R^2}{R^2-r^2} you ll get the usual expression for the inertia of a hollow cylinder I=\frac{1}{2}M_h(R^2+r^2).
 
Last edited:
It's fine as written. If you want to express ##I## using the mass of the outer shell only, call it ##M^\prime,## then you have to use ##M^\prime = M - m = M (1 - r^2/R^2)## to get the usual form for ##I##.
 
Thanks guys. Just a matter of confusion due to the specific application of the formula.
 

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