What is the product of inertia and how does it relate to rotation?

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The product of inertia is a concept that extends the idea of moment of inertia, which quantifies an object's resistance to rotational motion about a specific axis. While moment of inertia provides information about the distribution of mass relative to an axis, it does not account for how mass is distributed in relation to other axes. The product of inertia is essential for understanding rotational stability and balance. For example, in a ceiling fan, a balanced fan has a zero product of inertia, indicating no wobble during operation. However, if a weight is added to one of the blades, the product of inertia becomes non-zero, affecting the fan's stability. This highlights the importance of the product of inertia in analyzing systems where mass distribution impacts rotational dynamics. Further exploration of its mathematical derivation is sought for deeper understanding.
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I'm wondering what the product of inertia is.

What I mean is this, while I know the mathematical formula for it, I don't understand the underlying concept.

For instance, I understand the moment of inertia, 'I', because I can relate it to the kinetic energy of rotation of a body. I also understand it in terms of rotation about an axis.

So that, 1/2mv^2 = 1/2mwr^2 ... where v = wr, v is the linear velocity, w the angular velocity, and r the radius, or radial distance, of a point from the center of rotation of the body. So, I = mr^2 for each mass unit of the body, or in integral form: I = ∫ r^2 dm

However, the product of inertia isn't clear to me. I don't see the relation to kinetic energy or rotation (what's the axis of rotation?). My text gives a rather thin explanation for it. Can anyone give a concrete description of what it is, with examples?

Many thanks for any help.
 
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Hi Thales,

Welcome to the forums.

The product of inertia is related to the moments of inertia.

Imagine you have a ceiling fan which is operating correctly (no wobble). The fan has a specific non-zero moment of inertia about the center of rotation. That value for the moment of inertia does not relate any information about the balance of the object.

For example, if you attached a weight to one of the fan blades, the moment of inertia would go up, but there would be no way to distinguish it from a evenly distributed fan with heavier blades.

The product of inertia is what you need to determine rotational stabilities. In the case of the balanced fan, the product of inertia is zero, and there is no wobble. With the weight on one blade, the product of inertia becomes non-zero and it affects the rotational stability of the fan.

Hopefully that helps,
 
Thanks Enigma. That definitely helped.

Although I'm still trying to make sense out of the math for the product of inertia. I'd like to see how it was derived.
 

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