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Physics LibreTexts

13.4: Inertia Tensor

( \newcommand{\kernel}{\mathrm{null}\,}\)

The square bracket term in (13.3.9) is called the moment of inertia tensor, I, which is usually referred to as the inertia tensor

IijNαmα[δij(3kx2α,k)xα,ixα,j]

In most cases it is more useful to express the components of the inertia tensor in an integral form over the mass distribution rather than a summation for N discrete bodies. That is,

Iij=ρ(r)(δij(3kx2k)xixj)dV

The inertia tensor is easier to understand when written in cartesian coordinates rα=(xα,yα,zα) rather than in the form rα=(xα,1,xα,2,xα,3). Then, the diagonal moments of inertia of the inertia tensor are

IxxNαmα[x2α+y2α+z2αx2α]=Nαmα[y2α+z2α]IyyNαmα[x2α+y2α+z2αy2α]=Nαmα[x2α+z2α]IzzNαmα[x2α+y2α+z2αz2α]=Nαmα[x2α+y2α]

while the off-diagonal products of inertia are

Iyx=IxyNαmα[xαyα]Izx=IxzNαmα[xαzα]Izy=IyzNαmα[yαzα]

Note that the products of inertia are symmetric in that

Iij=Iji

The above notation for the inertia tensor allows the angular momentum ??? to be written as

Li=3jIijωj

Expanded in cartesian coordinates

Lx=Ixxωx+Ixyωy+IxzωzLy=Iyxωx+Iyyωy+IyzωzLz=Izxωx+Izyωy+Izzωz

Note that every fixed point in a body has a specific inertia tensor. The components of the inertia tensor at a specified point depend on the orientation of the coordinate frame whose origin is located at the specified fixed point. For example, the inertia tensor for a cube is very different when the fixed point is at the center of mass compared with when the fixed point is at a corner of the cube.


This page titled 13.4: Inertia Tensor is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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