Page 216 - Intro to Tensor Calculus

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I 29. In Cartesian coordinates show the Frenet formulas can be written

~
~
~
dT dN dB
~
~
~
~
~
~
= δ × T, = δ × N, = δ × B
ds ds ds
~
~
~
~
where δ is the Darboux vector and is defined δ = τT + κB.
I 30. Consider the following two cases for rigid body rotation.
Case 1: Rigid body rotation about a fixed line which is called the fixed axis of rotation. Select a point 0
on this fixed axis and denote by b e a unit vector from 0 in the direction of the fixed line and denote by ˆ e R
a unit vector which is perpendicular to the fixed axis of rotation. The position vector of a general point
r
in the rigid body can then be represented by a position vector from the point 0 given by ~ = h b e+ r 0 ˆ e R
where h, r 0 and b e are all constants and the vector ˆ e R is fixed in and rotating with the rigid body.
dθ
Denote by ω = the scalar angular change with respect to time of the vector ˆ e R as it rotates about
dt
d dθ
the fixed line and define the vector angular velocity as ~ω = (θ b e)= b e where θ b e is defined as the
dt dt
vector angle of rotation.
d ˆ e R
(a) Show that = b e × ˆ e R .
dθ
d~ r d ˆ e R d ˆ e R dθ
~
(b) Show that V = = r 0 = r 0 = ~ω × (r 0 ˆ e R )= ~ω × (h b e + r 0 ˆ e R )= ~ω × ~r.
dt dt dθ dt
Case 2: Rigid body rotation about a fixed point 0. Construct at point 0 the unit vector ˆ e 1 which is
d ˆ e 1
fixed in and rotating with the rigid body. From pages 80,87 we know that must be perpendicular
dt
d ˆ e 1
to ˆ e 1 and so we can define the vector ˆ e 2 as a unit vector which is in the direction of such that
dt
d ˆ e 1
= α ˆ e 2 for some constant α. We can then define the unit vector ˆ e 3 from ˆ e 3 = ˆ e 1 × ˆ e 2 .
dt
d ˆ e 3
(a) Show that , which must be perpendicular to ˆ e 3 , is also perpendicular to ˆ e 1 .
dt
d ˆ e 3 d ˆ e 3
(b) Show that can be written as = β ˆ e 2 for some constant β.
dt dt
d ˆ e 2
(c) From ˆ e 2 = ˆ e 3 × ˆ e 1 show that =(α ˆ e 3 − β ˆ e 1 ) × ˆ e 2
dt
d ˆ e 1 d ˆ e 2 d ˆ e 3
(d) Define ~ω = α ˆ e 3 − β ˆ e 1 and show that = ~ω × ˆ e 1 , = ~ω × ˆ e 2 , = ~ω × ˆ e 3
dt dt dt
(e) Let ~ = x ˆ e 1 + y ˆ e 2 + z ˆ e 3 denote an arbitrary point within the rigid body with respect to the point 0.
r
r
d~
Show that = ~ω × ~r.
dt
Note that in Case 2 the direction of ~ω is not fixed as the unit vectors ˆ e 3 and ˆ e 1 are constantly changing.
In this case the direction ~ω is called an instantaneous axis of rotation and ~ω, which also can change in
magnitude and direction, is called the instantaneous angular velocity.