Problem Inertia Tensor and Kinetic Energy of Two Cones Joined
at Their Tips
The two cones are joined at their tips. Each cone is a prolate ellipsoid with
I
1
=
I
2
>I
3
. One cone is oriented with its principal symmetry axis along the
x
axis of a body frame, while the other cone has its symmetry axis along the
y
axis of the body frame. Because the inertia tensor is extensive in the mass, the
inertia tensor for this composite body is simply the sum of the inertia tensors
for the two separate bodies. For the cone along the
x
axis, we have
I
(
x
)
=
⎛
⎝
I
3
00
0
I
1
0
00
I
1
⎞
⎠
,
(1)
and for the cone along the
y
axis, we write
I
(
y
)
=
⎛
⎝
I
1
00
0
I
3
0
00
I
1
⎞
⎠
.
(2)
In both Eqs.(1) and (2) we see that the location of
I
3
depends upon the orien
tation of the cone along the body axes. Adding Eqs.(1) and (2), we obtain the
inertia tensor for the composite system
I
=
⎛
⎝
I
1
+
I
3
00
0
I
1
+
I
3
0
00
2
I
1
⎞
⎠
.
(3)
This tells us that the original coordinateaxesst
i
l
lserveaspr
inc
ipa
laxes
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 Fall '10
 Wilemski
 mechanics, Energy, Inertia, Kinetic Energy, Moment Of Inertia, Euclidean geometry, Polar coordinate system, inertia tensor

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