Calculating moment of inertia of a filled cylinder

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Discussion Overview

The discussion centers on calculating the moment of inertia of a filled cylinder rolling down an inclined plane. Participants explore the implications of density, viscosity, and the mathematical formulation of the moment of inertia in this context.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the calculation of the moment of inertia, noting the complexity of the mass density as a function of position and volume differential.
  • Another participant suggests that if the density is uniform, the moment of inertia can be treated as that of a solid cylinder, although the mass may affect the inertia value.
  • A different viewpoint is presented regarding the effect of a liquid with zero viscosity, stating that it does not contribute to the moment of inertia despite adding mass to the cylinder.
  • A follow-up question is posed about the scenario where the liquid has a nonzero viscosity, indicating a shift in the discussion's focus.

Areas of Agreement / Disagreement

Participants express differing views on the role of viscosity in the moment of inertia calculation, and there is no consensus on how to approach the problem when considering non-uniform density or viscosity effects.

Contextual Notes

The discussion does not resolve the complexities of integrating the mass density function or the implications of viscosity on the moment of inertia, leaving these aspects open for further exploration.

asinewave
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Hi all,

I was wondering how to calculate the moment of inertia of a cylinder that is filled with a substance and is rolling down an inclined plane.

I know that I = int(r^2 dm), and I know what r is... but I gather that dm may be a complicated function of mass density-position and some volume differential.

I was wondering if someone can explain to me how this works ... I may have to find the density of the substance, right? Is that the only way I can find it?

Also, I understand that I may very well end up with an approximation, but that' sfine.. as long as its reasonably accurate. I want a numerical answer (even if it is a reasonable approximation).. not an exact integral.

Thanks!
 
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If the density is uniform, then the moment of inertia is that of a solid cylinder (what changes is the amount of mass, which may change the value of the inertia, but the expression is the same).
 
A sine wave-
If the liquid has zero viscosity, it will not rotate as the cylinder rolls down the inclined plane. So although it contributes mass to the cylinder, it contributes nothing to the moment of inertia.
Bob S
 
Hi all,

Thanks for the replies. What if the liquid has a nonzero viscosity?
 

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