294
Rotation of a Rigid Object About a Fixed Axis
P10.22
IM
x m
Lx
=+
−
2
2
a
f
dI
dx
Mx
m L x
=−
−
=
22
0
af
(for an extremum)
∴=
+
x
mL
Mm
dI
dx
mM
2
2
; therefore
I
is minimum when the axis of
rotation passes through
x
mL
=
+
which is also the center
of mass of the system. The moment of inertia about an axis
passing through
x
is
mL
m
m
L
Mm
LL
CM
=
+
L
N
M
O
Q
P
+−
+
L
N
M
O
Q
P
=
+
=
2
1
µ
where
=
+
Mm
.
x
M
m
L
L
−
x
x
FIG. P10.22
Section 10.5
Calculation of Moments of Inertia
P10.23
We assume the rods are thin, with radius much less than
L
.
Call the junction of the rods the origin of coordinates, and
the axis of rotation the
z
axis.
For the rod along the
y
axis,
Im
L
=
1
3
2
from the table.
For the rod parallel to the
z
axis, the parallelaxis theorem
gives
r
m
L
mL
F
H
G
I
K
J
≅
1
1
4
2
2
2
axis of rotation
z
x
y
FIG. P10.23
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This note was uploaded on 12/14/2011 for the course PHY 203 taught by Professor Staff during the Fall '11 term at Indiana State University .
 Fall '11
 Staff
 Physics

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