Moment of inertia of combined cylinders

[IBAStudent]

Hi, I've attached a word document with my problem since I've used the Mathtype program (sorry, I didn't quite know how to use the tools on this forum), hope you don't mind :)

 

[SteamKing]

There are ways of calculating the inertia of composite bodies which do not involve evaluating the integrals contained in your attachment.

See: http://www.eng.auburn.edu/users/marghitu/MECH2110/C_4.pdf

By calculating the moment of inertia of each cylinder, one then uses the Parallel Axis Theorem to calculate the moment of inertia of the composite body composed of three cylinders. The attachment also contains a formula to calculate the moment of inertia of a single cylinder given the radius and length, so no integrals are required at all for calculating the result.

 

[IBAStudent]

I didn't quite understand that, sorry. I thought the parallel axis theorem only was for changing the axis of objects? All my three objects are attached at the same axis, so I didn't quite get how I use the theorem. Also, the link you sent didn't contain anything about combined objects?
Thanks for your answer, anyways, I'm just a grasping this concept completely.

 

[SteamKing]

Your description of the three cylinders is ambiguous. You say all three cylinders are attached to the same axis and that you have two smaller cylinders attached to the sides of a big cylinder.

A couple of questions:
1. Are all three cylinders running parallel to one another like this:

a.)
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Or are they like this:

b.)
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If configuration a.), you will need the Parallel Axis Theorem.

Section 4.3 in the attachment shows how to apply the PAT for composite bodies. The procedure composite bodies is analogous to that for combining the inertias of composite areas.

2. Does the density of your cylinders vary as a function of the radius of the cylinder, or do you have three constant density cylinders, with each cylinder having a different value for its density?

 

[IBAStudent]

Hi, sorry for the ambiguous description, the cylinders are glued together like in your b.) illustration, in other words the flat sides are glued together.
The three cylinders are actually just pipes short pipes about 20 and 8 cm in diameter and 2 mm thick, so the density of each cylinder is not constant since the mass is distributed near the rolling surface of the cylinders. So yes, in a way, the mass density is a function of the cylinder's radius, where the density is mostly 0, while it's constant at a distance equal to the radius of the cylinder, since it's like a hoop.

 

[SteamKing]

If these cylinders are pipes, do they contain anything (water, air, etc.?)

For the most part, in calculating the inertia of hollow objects, the inertia of the contents of the cavity is not computed unless there is significant mass present.

 

[IBAStudent]

No, there's just air inside, so the mass of the cavity can be neglected.

 

[SteamKing]

On the last page of my attachment, you will find a formula for calculating the moment of inertia of a hollow cylinder about its centroid. Since the cylinders are of different sizes, you can use the parallel axis theorem to calculate the moment of inertia of the composite body after you determine the location of the centroid of the composite configuration.