Lecture 21  Wednesday May 20th
[email protected]
Key words: Moments of inertia, center of mass, work
21.1 Moments of inertia and center of mass
Let
W
be a solid with density
δ
(
x, y, z
) at point (
x, y, z
). Then the
moments of inertia
of
W
with respect to the coordinate axes are
I
x
=
ZZZ
W
(
y
2
+
z
2
)
dV
I
y
=
ZZZ
W
(
x
2
+
z
2
)
dV
I
z
=
ZZZ
W
(
x
2
+
y
2
)
dV.
These quantities measure the difficulty with which an object is rotated about the axes. As
an exercise, one can determine whether it is harder to rotate a sphere
x
2
+
y
2
+
z
2
= 1 with
uniform density
δ
(
x
) = 1 around the
z
axis or to rotate the cylinder
x
2
+
y
2
≤
1
, z
≤
4
/
3
with uniform density
δ
(
x
) = 1 around the
z
axis. Note that these objects clearly have
the same mass. The
center of mass
of a solid object
W
in
n
dimensions is at the point
w
= (
w
1
, w
2
, . . . , w
n
) defined by
w
i
=
Z Z
· · ·
Z
W
x
i
δ
(
x
)
m
dV
where
m
is the mass of the object.
Example 1.
Find the center of mass of
H
=
{
(
x, y, z
) :
x
2
+
y
2
+
z
2
≤
1
, z
≥
0
}
with
uniform density.
Solution.
We can assume that
δ
(
x
) = 1 is the density function for the hemisphere,
and the hemisphere is defined by
x
2
+
y
2
+
z
2
≤
1 where
z
≥
0. Then the mass of the
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Enright
 Math, Force, Work, Cos, Coordinate system, iy

Click to edit the document details