Physics 409/Classical Mechanics
Test #5
19 November 2010
Name
ANSWERS
Answer all parts of each of the two questions.
50
points total. The points value of each part is
indicated.
Begin the answer to each question on a new sheet of paper.
Clearly define all
coordinates and variables. Be sure to include sufficient detail in each answer so that your logic is
clear to me.
(1)(25 pts.)
(a)
(8 pts.)
The inertia tensor
I
)
for a uniform sphere of mass
M
and radius
a
is diagonal. Show
that its principal moments
I
)
are equal to 2
Ma
2
/5.
ANS:
Use spherical coordinates to calculate
I
zz
.
52
2
42
33
00
0
4
2
sin (1 cos
)
2
44
5
3
5
a
zz
M
Ma
M
a
Id
r
d
r
d
aa
ππ
φθ
θ
π
=−
=
=
∫∫
∫
(b) (7 pts.)
A rigid body consists of two identical spheres whose surfaces touch at a point.
Each
sphere is identical to that of part
(a)
. Find the inertia tensor
I
for this object in a body fixed frame
whose origin lies at the center of one sphere.
You may assume that the center of the second
sphere lies on the
z
axis of this body frame.
ANS:
Let sphere 1 have its center at the origin.
Let
sphere 2 have its center a distance 2
a
from the origin. Then
I
=
I
(1)
+
I
(2)
, where
I
(1)
=
I
N
,
I
(2)
=
I
N
+
I
R
, and
I
R
is a diagonal tensor with elements, I
Rxx
=I
Ryy
=
4
Ma
2
, and I
Rzz
= 0. Putting the first 3
equations together, we have
I
= 2
I
N
+
I
R
, where
I
is diagonal with elements
I
xx
=I
yy
= 24
Ma
2
/5,
and I
zz
= 4
Ma
2
/5.
(c)
(5 pts.)
Draw a qualitative sketch of the inertia ellipsoid for the ellipsoid of revolution
defined by the principal moments of part
(b)
.
The inertia ellipsoid is defined by the equation
1 =
I
1
(
ρ
1
)
2
+
I
2
(
ρ
2
)
2
+
I
3
(
ρ
3
)
2
, where the
ρ
i
are the coordinates along the different principal axes and
the elements
I
1
,
I
2
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 Fall '10
 Wilemski
 mechanics, Angular Momentum, Moment Of Inertia, Rigid Body, Rotation, body frame

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