Mathematical Tools for Physics

7—Operators and Matrices 180

As an example, place a disk of massM and radiusRand uniform mass density so that its center is at
(x,y,z) = (R, 0 ,0)and it is lying in thex-yplane. Compute the components of the inertia tensor. First get the
components about the center of mass, using Eq. ( 14 ).

x

z

y

The integrals such as

−

∫

dmxy, −

∫

dmyz

are zero. For fixedyeach positive value ofxhas a corresponding negative value to make the integral add to
zero. It’s odd inx(ory); remember that this is about thecenterof the disk. Next do theI 33 integral.
∫
dm(x^2 +y^2 ) =

∫

dmr^2 =

∫

M

πR^2

dAr^2

For the element of area, usedA= 2πr drand you have

I 33 =

M

πR^2

∫R

0

dr 2 πr^3 =

M

πR^2

2 π

R^4

4

=

1

2

MR^2

For the next two diagonal elements,

I 11 =

∫

dm(y^2 +z^2 ) =

∫

dmy^2 and I 22 =

∫

dm(x^2 +z^2 ) =

∫

dmx^2

Because of the symmetry of the disk, these two are equal, also you see that the sum is

I 11 +I 22 =

∫

dmy^2 +

∫

dmx^2 =I 33 =

1

2

MR^2 (19)