What is the relationship between rotational inertia and the inertia tensor?

AI Thread Summary
Moment of inertia is a tensor quantity known as the mass moment of inertia tensor, which is crucial in rigid body dynamics. It is neither a scalar nor a vector, but rather a second-order tensor that describes how mass is distributed relative to an axis of rotation. In three-dimensional problems, the full inertia tensor is essential, while in two-dimensional scenarios, the effective mass moment of inertia can be derived from the inertia tensor. The relationship between rotational inertia and the inertia tensor is expressed mathematically, emphasizing the importance of the mass distribution in calculating rotational dynamics. Understanding this tensorial nature is key for analyzing rotational motion accurately.
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Homework Statement



What is moment of inertia?

Homework Equations



N/A

The Attempt at a Solution



Some people said that it is not a scalar and also not a vector.

But other said that if it is not a vector, it must be a scalar.

However, another said that it is actually a tensor.

Can anyone tell me what it is?
 
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For a mass distribution, there is a tensor quantity called the mass moment of inertia tensor.

In rigid body dynamics problems, in 3D, the full inertia tensor is significant. In 2D, the quantity that is significant for the rotational inertia is related to the inertia tensor as
Is = (n) {n}
where
Is = scalar effective mass MOI
(n) = row form for unit vector along the axis of rotation
{n} = column fomr for unit vector along the axis of rotation
= mass MOI matrix (2nd order tensor)
 

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