Heavy Top Angular Momentum Problem
(a)Long Way
Designate the body frame unit vectors as
i
,
j
,and
k
, and assume they lie
along the principal axes of the body.
The space frame unit vectors will be
designated as
x
,
y
,and
z
. Since it is easiest to write
L
in the body frame as
L
=
I
1
(
ω
1
i
+
ω
2
j
)+
I
3
ω
3
k
,
we need to evaluate the components of the gravitational torque
N
in the body
frame. The torque can be written as
N
=
−
Mgl
k
×
z
,
(1)
and it’s not di
ﬃ
cult to express
z
in terms of the body frame unit vectors, so that
one can easily work out the cross product.
Equivalently, we see from Eq.(1)
and Figure 5.7 that
N
always lies along the line of nodes (the intermediate
x
axis used in de
f
ning the Euler angles). Since the line of nodes always lies in the
body
i

j
plane and makes the (Euler) angle
ψ
with
i
,wecanwr
itebyinspect
ion
N
1
=
N
·
i
=
N
cos
ψ,
(2)
N
2
=
N
·
j
=
−
N
sin
ψ,
(3)
and
N
3
=0
,
(4)
where
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 Fall '10
 Wilemski
 mechanics, Angular Momentum, Momentum, Trigraph, NZ, frame unit vectors

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