3.2 Newton’s Laws Applied to a Biosystem
35
Fig. 3.1 Quantities used to
characterize a mass rotating
on the dotted circle, with
coordinate origin at “O”
v i =
dr i
dt
= ω × r i ,
(3.3)
where ω is the angular velocity of the rotation with direction along the axis of
rotation in the ‘right-hand’ sense. This relation follows from the definition of the
angle and the cross product, and is depicted in Fig. 3.1. In turn, the acceleration of a
coordinate vector under pure rotation becomes
d 2 r i
dt 2 = ω × (ω × r i ) = −
ω
2 r i − (ω · r i ) ω
= − ω
2 ρ i .
(3.4)
The second equality in Eq. (3.4) follows from the “double cross” vector identity,
while the third equality follows by re-expressing r i as z i ω +ρ i , i.e., the radial vector
can always be expressed as a vector, z i ω, along the axis of rotation and another one,
ρ i , perpendicular to the axis. The vector geometry can be seen in Fig. 3.1. The result
in Eq. (3.4) is the infamous centripetal acceleration, a vector pointing inward toward
the axis of rotation of size ω 2 ρ, angular speed square times the radius of the circle
of motion ρ. (The expression may be more familiar as speed squared over radius,
which here is v 2 /ρ.)
The angular momentum of a ‘rigid’ body is usefully written in terms of the
angular velocity of the masses, because under rotation of a rigid body, the angular
velocities for all the masses within are the same, and can be factored out of the
summation:
L =
m i
r
2
i δ − r i r i
· ω ,
(3.5)