Angular Velocity and Momentum - Kurt Nalty

The magnitude of the angular velocity per unit time is
|˜ω| =
|⃗a + ⃗a × ⃗v|
1 + v 2 (30)
Finally, note that angular velocity has the same units as energy and momentum.
2 Angular Momentum
Classically, the angular momentum of a body is defined by
⃗L = ⃗r × (m⃗v) = ⃗r × ⃗p (31)
where L ⃗ is the angular momentum, m is the mass, ⃗v is the velocity, and ⃗r is
the vector from the origin (usually at the axis of rotation) to the particle. The
term ⃗p represents the linear momentum of the particle.
I don’t want to get bogged down by speculating about mass yet, so I’ll leave
m alone for the moment. Likewise, from the previous discussion of intrinsic
angular velocity, I identified r as the radius of curvature r = 1/κ.
The time rate of change of angular momentum is torque.
⃗T = d⃗ L
dt = ⃗r × (m⃗a) = ⃗r × ⃗ f (32)
where ⃗ f is the force on the particle.
More interesting, the curl of classical angular momentum is proportional to
linear momentum.
⃗∇ × ⃗ L = −2m⃗v = −2⃗p (33)
When two equal opposing forces separated by a distance are applied to an
object, we have a torque due to this couple.
⃗T = (⃗r 1 − ⃗r 2 ) × ⃗ f (34)
If we substitute the radius of curvature in the definition of ⃗ L, we obtain the
rather strange looking equation
This can be recognized as
⃗r = 1 κ ⃗n = v2 (⃗v × (⃗a × ⃗v))
|⃗a × ⃗v| 2 (35)
⃗L = mv4 (⃗a × ⃗v)
|⃗a × ⃗v| 2 (36)
⃗L = m 1 ⃗ω
κ2 (37)
= m⃗r × (⃗r × ⃗ω) (38)
= m⃗r × ⃗v (39)
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