Collecting terms and neglecting terms in
2
ω
′
:
cos
sin
0
g
g
x
x
t
y
y
t
l
l
ω
ω
′
′
+
+
+
±±
±±
=
sin
cos
0
g
g
x
x
t
y
y
t
l
l
ω
ω
′
′
+
−
+
±±
±±
=
5.21
24
sin
T
λ
=
hours
24
73.7
sin19
T
=
=
D
hours
5.22
Choose a coordinate system with the origin at the center of the wheel, the
x
′
and
axes pointing toward fixed points on the rim of the wheel, and the
axis
pointing toward the center of curvature of the track.
Take the initial position of
the
y
′
z
′
x
′
axis to be horizontal in the
V
−
D
G
direction, so the initial position of the
y
′
axis is vertical.
The bicycle wheel is rotating with angular velocity
V
about its axis, so …
b
D
ˆ
l
V
k
b
ω
′
=
D
G
A unit vector in the vertical direction is:
ˆ
ˆ
ˆ
sin
cos
V t
V t
n
i
j
b
b
′
′
=
+
D
D
At the instant a point on the rim of the wheel reaches its highest point:
ˆ
ˆ
ˆ
sin
cos
V t
V t
r
bn
b i
j
b
b
′
′
′
=
=
+
D
D
G
Since the coordinate system is moving with the wheel, every point on the rim is
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 Fall '11
 JohnAnderson
 Geometry, Work, Cos, Polar coordinate system

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