Moment of Inertia of a cylinder

AI Thread Summary
The moment of inertia of a cylinder with a rotation axis through its curved surface and center of mass is given by the formula MR^2/4 + Ml^2/12. The discussion revolves around the integration process to derive this formula, with participants suggesting slicing the cylinder into discs and applying the moment of inertia formula for a disc along with the parallel axis theorem. However, there is confusion regarding the integration of a disc, particularly with the variables involved and the need for cosine functions. Participants are seeking clarity on the integration steps and the correct application of the relevant equations. The conversation highlights the complexity of deriving the moment of inertia through integration.
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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