Moment of Inertia of a cylinder

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Discussion Overview

The discussion revolves around the moment of inertia of a cylinder, specifically focusing on the integration process to derive the formula for a cylinder with a rotation axis passing through its curved surface and center of mass. Participants express interest in understanding the integration steps rather than simply using the provided formula.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant notes the moment of inertia of a cylinder is given as MR²/4 + Ml²/12 and seeks help with the integration process to derive this result.
  • Another participant suggests slicing the cylinder into discs and using the moment of inertia formula for a disc along with the parallel axis theorem.
  • Some participants express uncertainty about how to integrate a disc and mention that they expect understanding the cylinder's integration would help with the disc.
  • Concerns are raised about the complexity of integrating a disc, particularly regarding the need for cosine functions and multiple variables.
  • One participant states they cannot use the parallel axis theorem, finding it too tricky.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus, with participants expressing various levels of understanding and comfort with the integration process. Some participants propose methods while others express confusion and uncertainty about the steps involved.

Contextual Notes

Participants mention challenges related to the integration process, including the introduction of multiple variables and the application of the parallel axis theorem, which remains unresolved.

loup
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When I am reading a book, I find it is listed that the moment of Inertia of a cylinder is
MR^2/4 + Ml^2/12

It is a cylinder with rotation axis passing through the curve surface and its centre of mass. And its density is constant. With the circile surface raius = R and height = l

Can anybody show me the procedure of the integration? I have tried several times but fail. I just cannot get that answer. It is not homework but I am interested in the process. You know, usually, moment of Inertia is provided in the book, integration is not required. But I am really curious about it. Could anybody please help me?
 
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Hi loup! :smile:

Slice the cylinder into discs.

Use the moment of inertia formula for a disc about its diameter, combined with the parallel axis theorem, and integrate. :wink:
 
The problem is I don't know about how to integrate a disc.
 
And I think the integration of disc actually comes from cylinder. I expected once I finished this cylinder I could do the disc.
 
loup said:
The problem is I don't know about how to integrate a disc.

Slice it into strips parallel to a diameter, and integrate …

what do you get? :smile:
 
The r requires a cosine and there are more than one variable, what I should do?
 
I cannot use parallel axis theorem. I think it is too tricky.
 
loup said:
The r requires a cosine and there are more than one variable, what I should do?

uhh? what equation are you using? :confused:
 

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