(
)
(
)
(
)
1
2
3
1
2
2
1
1
3
2
1
2
1
3
3
2
0
I
I
I
I
I
I
I
I
I
ω
ω ω
−
−
+
−
+
=
±±
ω ω
1
1
1
0
K
ω
ω
+
=
±±
,
(
)(
)
(
)(
)
3
2
2
1
3
2
3
1
2
2
1
2
3
1
3
1
2
I
I
I
I
I
I
I
I
K
I I
I I
ω
ω
−
−
−
−
= −
+
(a) For
3
ω
large and
2
0
ω
=
,
so
1
0
K
>
1
1
1
0
K
ω
ω
+
=
±±
is the harmonic oscillator
equation.
1
ω
oscillates, but remains small.
Motion is stable for initial rotation
about the 3 axis if the 3 axis is the principal axis having the largest or smallest
moment of inertia.
(b) For
3
0
ω
=
and
2
ω
large,
1
0
K
<
so
1
1
1
0
K
ω
ω
+
=
±±
is the differential equation for
exponential growth of
1
ω
with time:
1
1
k t
k t
Ae
Be
ω
−
=
+
1
=
.
Motion is unstable
for the initial rotation mostly about the principal axis having the median moment
of inertia.
9.20
0 since either
xy
i
i
i
i
I
m x y
=
∑
i
x
or
is zero for all six particles.
Similarly, all the
other products of inertia are zero.
Therefore the
coordinate axes are principle axes.
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 Fall '11
 JohnAnderson
 Inertia, Work, Moment Of Inertia, Complex number, 2m, initial rotation

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