Moment of Inertia of a cylinder

In summary, the moment of inertia of a cylinder with a rotation axis passing through its curved surface and center of mass, and a constant density can be calculated using the formula MR^2/4 + Ml^2/12. To find the moment of inertia, the cylinder can be sliced into discs or strips and integrated using the moment of inertia formula for a disc and the parallel axis theorem. However, there may be some difficulties when using the cosine function and multiple variables.
  • #1
loup
36
0
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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  • #2
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:
 
  • #3
The problem is I don't know about how to integrate a disc.
 
  • #4
And I think the integration of disc actually comes from cylinder. I expected once I finished this cylinder I could do the disc.
 
  • #5
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:
 
  • #6
The r requires a cosine and there are more than one variable, what I should do?
 
  • #7
I cannot use parallel axis theorem. I think it is too tricky.
 
  • #8
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:
 

1. What is the formula for calculating the moment of inertia of a cylinder?

The formula for calculating the moment of inertia of a cylinder is I = (1/2)mr², where I is the moment of inertia, m is the mass of the cylinder, and r is the radius of the cylinder.

2. How does the moment of inertia of a cylinder differ from other shapes?

The moment of inertia of a cylinder differs from other shapes because it is dependent on both the mass and the radius of the cylinder, whereas for other shapes it may only be dependent on one or the other.

3. How does the moment of inertia of a hollow cylinder compare to a solid cylinder?

The moment of inertia of a hollow cylinder is less than that of a solid cylinder with the same mass and radius. This is because the mass is distributed further from the axis of rotation in a hollow cylinder, resulting in a larger radius of gyration and a smaller moment of inertia.

4. Can the moment of inertia of a cylinder be negative?

No, the moment of inertia of a cylinder cannot be negative. It is a measure of an object's resistance to changes in its rotational motion and is always a positive value.

5. How does the moment of inertia of a cylinder affect its rotational motion?

The moment of inertia of a cylinder affects its rotational motion by determining how easily it can be rotated. A larger moment of inertia means that more force is required to accelerate the cylinder's rotation, while a smaller moment of inertia means it will rotate more easily.

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