Moment of Inertia of a cylinder

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SUMMARY

The moment of inertia of a cylinder with a rotation axis passing through its curved surface and center of mass is defined by the formula MR²/4 + Ml²/12. This formula applies to a cylinder with constant density, radius R, and height l. The discussion emphasizes the integration process required to derive this formula, particularly through slicing the cylinder into discs and applying the moment of inertia formula for a disc about its diameter, along with the parallel axis theorem. Participants express challenges in performing the necessary integration and understanding the variables involved.

PREREQUISITES
  • Understanding of moment of inertia concepts
  • Familiarity with calculus, specifically integration techniques
  • Knowledge of the parallel axis theorem
  • Basic principles of solid geometry
NEXT STEPS
  • Study the derivation of moment of inertia for a disc using integration
  • Learn about the parallel axis theorem in detail
  • Explore advanced integration techniques in calculus
  • Investigate the moment of inertia for different geometric shapes
USEFUL FOR

Students of physics, mechanical engineers, and anyone interested in the mathematical derivation of moment of inertia for cylindrical objects.

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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