Mathematical Tools for Physics

7—Operators and Matrices 177

Pick the common rectangular, orthogonal basis and evaluate the components of this function. Equation ( 6 ) says
~r=x~e 1 +y~e 2 +z~e 3 so

I(~ei) =

∑

k

Iki~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



 (14)

These are the components of the tensor of inertia. The diagonal elements of the matrix may be familiar;

they are the moments of inertia. x^2 +y^2 is the distance-squared to thez-axis, so the elementI 33 orIzzis 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, and 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