7—Operators and Matrices 149I(~ei) =
∑
kIki~ek
I(~e 1 ) =
∫
dm
[
~e 1 (x^2 +y^2 +z^2 )−(x~e 1 +y~e 2 +z~e 3 )(x)
]
=I 11 ~e 1 +I 21 ~e 2 +I 31 ~e 3
from which I 11 =
∫
dm(y^2 +z^2 ), I 21 =−
∫
dmyx, I 31 =−
∫
dmzx
This provides the first column of the components, and you get the rest of the components the same
way. The whole matrix is
∫
dm
y^2 +z^2 −xy −xz
−xy x^2 +z^2 −yz
−xz −yz x^2 +y^2
(7.17)
These are the components of the tensor of inertia. The diagonal elements of the matrix may befamiliar; they are the moments of inertia.x^2 +y^2 is the perpendicular distance-squared to thez-axis,
so the elementI 33 (≡Izz) is the moment of inertia about that axis,
∫
dmr^2 ⊥. The other components
are less familiar and are called the products of inertia. This particular matrix is symmetric:Iij=Iji.
That’s a special property of the inertia tensor.
Components of Dumbbell
Look again at the specific case of two masses rotating about an axis. Do it quantitatively.
~v 2 (out)
~r 2 ×m 2 ~v 2
m 2
~r 2
~ω
~r 1
~r 1 ×m 1 ~v 1
~v 1 (in)
m 1
~e 2
~e 1
The integrals in Eq. (7.17) are simply sums this time, and the sums have just two terms. I’m
making the approximation that these are point masses. Make the coordinate system match the indicated
basis, withxright andyup, thenzis zero for all terms in the sum, and the rest are
∫
dm(y^2 +z^2 ) =m 1 r^21 cos^2 α+m 2 r 22 cos^2 α
−
∫
dmxy=−m 1 r 12 cosαsinα−m 2 r^22 cosαsinα
∫
dm(x^2 +z^2 ) =m 1 r^21 sin^2 α+m 2 r^22 sin^2 α
∫
dm(x^2 +y^2 ) =m 1 r^21 +m 2 r^22
The matrix is then