7—Operators and Matrices 144Another example of the type of function that I’ll examine is from physics instead of mathematics.
A rotating rigid body has some angular momentum. The greater the rotation rate, the greater the
angular momentum will be. Now how do I compute the angular momentum assuming that I know the
shape and the distribution of masses in the body and that I know the body’s angular velocity? The
body is made of a lot of point masses (atoms), but you don’t need to go down to that level to make
sense of the subject. As with any other integral, you start by dividing the object in to a lot of small
pieces.
What is the angular momentum of a single point mass? It starts from basic Newtonian mechanics,
and the equationF~=d~p/dt. (It’s better in this context to work with this form than with the more
common expressionsF~=m~a.) Take the cross product with~r, the displacement vector from the origin.
~r×F~=~r×d~p/dt
Add and subtract the same thing on the right side of the equation (add zero) to get
~r×F~=~r×
d~p
dt
+
d~r
dt
×~p−
d~r
dt
×~p
=
d
dt
(
~r×~p
)
−
d~r
dt
×~p
Now recall that~pism~v, and~v=d~r/dt, so the last term in the preceding equation is zero because
you are taking the cross product of a vector with itself. This means that when adding and subtracting
a term from the right side above, I was really adding and subtracting zero.
~r×F~ is the torque applied to the point massmand~r×~pis the mass’s angular momentum
about the origin. Now if there are many masses and many forces, simply put an index on this torque
equation and add the resulting equations over all the masses in the rigid body. The sums on the left
and the right provide the definitions of torque and of angular momentum.
~τtotal=
∑
k~rk×F~k=
d
dt
∑
k(
~rk×~pk
)
=
dL~
dt
For a specific example, attach two masses to the ends of a light rod and attach that rod to
a second, vertical one as sketched — at an angle. Now spin the vertical rod and figure out what
the angular velocity and angular momentum vectors are. Since the spin is along the vertical rod, that
specifies the direction of the angular velocity vector~ωto be upwards in the picture. (Viewed from above
everything is rotating counter-clockwise.) The angular momentum of one point mass is~r×~p=~r×m~v.
The mass on the right has a velocity pointingintothe page and the mass on the left has it pointing
out.Take the origin to be where the supporting rod is attached to the axis, then~r×~pfor the mass
on the right is pointing up and to the left. For the other mass both~rand~pare reversed, so the cross
product is in exactly the same direction as for the first mass. The total angular momentum the sum of
these two parallel vectors, and it isnotin the direction of the angular velocity.