Environmental Engineering Reference
In-Depth Information
Thus
ηη
∗
=
ω
1
+
ω
2
=
A
2
exp
(
2
kt/I )
decays exponentially. In the lab any
initial wobble dies away and the disc then spins only about the symmetry
axis.
10.8 To show this use
−
1
2
ω
·
1
2
ω
·
=
=
ω
T
L
I
and differentiate the product with respect to time in the principal-axis,
body-fixed frame where
I
is constant and diagonal and hence in which
ω
·
I
ω
=
ω
·
I
ω
=
ω
·
τ
.
10.9 Write the angular momentum relative to the origin as
R
i
),
=
+
×
˙
+
L
m
i
(
r
i
R
)
(
r
i
i
expand the brackets and use the definition of
R
i
m
i
r
i
R
=
i
m
i
.
(a) Use cylindrical polar co-ordinates
(r,θ,z)
for the integrals with
z
along
the symmetry axis:
10.10
sfasfd
ρ
h
0
d
z
zR/h
0
d
r
2
π
0
d
θr
3
,
=
I
z
3
m
and the density of the cone
ρ
=
πR
2
h
. For the other two principal
moments of inertia evaluate
ρ
h
0
d
z
zR/h
0
d
r
2
π
0
d
θ(r
3
cos
2
θ
rz
2
).
I
x
=
I
y
=
+
ω
(b) We shall compute the components of
in the direction of the principal
axes and then use
1
2
I
z
ω
z
.
There are two sources of rotation contributing to
1
2
I
x
(ω
x
+
ω
y
)
T
=
+
: the precessional
rotation of the cone about its apex (which points vertically) and the spin
of the cone about its symmetry axis. If
α
ω
tan
−
1
(R/h)
is the half-angle
of the cone, then the precession gives rise to a contribution to
ω
z
of
ω
sin
α
and
ω
x
+
=
ω
y
=
ω
2
cos
2
α
. The spin contributes only to
ω
z
and is
−
equal to
ω/
sin
α
. To see this note that the base of the cone must travel
a distance
l
2
πR/
sin
α
for each complete precession of the cone, i.e.
the cone must spin about its symmetry axis at a rate of
ωl/(
2
πR)
. Hence
ω
z
=
=
ω
sin
α
−
ω/
sin
α
. Express sin
α
and cos
α
in terms of
R
and
h
to
get the result.
