13.4: Inertia Tensor
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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
Iij≡N∑αmα[δij(3∑kx2α,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(3∑kx2k)−xixj)dV
The inertia tensor is easier to understand when written in cartesian coordinates r′α=(xα,yα,zα) rather than in the form r′α=(xα,1,xα,2,xα,3). Then, the diagonal moments of inertia of the inertia tensor are
Ixx≡N∑αmα[x2α+y2α+z2α−x2α]=N∑αmα[y2α+z2α]Iyy≡N∑αmα[x2α+y2α+z2α−y2α]=N∑αmα[x2α+z2α]Izz≡N∑αmα[x2α+y2α+z2α−z2α]=N∑αmα[x2α+y2α]
while the off-diagonal products of inertia are
Iyx=Ixy≡−N∑αmα[xαyα]Izx=Ixz≡−N∑αmα[xαzα]Izy=Iyz≡−N∑αmα[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=3∑jIijωj
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.