PowerPoint slides (pdf) - Physics at Oregon State University
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Using Gre<strong>at</strong> Circles to<br />
Understand Motion on a Rot<strong>at</strong>ing Sphere<br />
David H. McIntyre<br />
<strong>Oregon</strong> St<strong>at</strong>e <strong>University</strong><br />
Department of <strong>Physics</strong><br />
NSF - Paradigms<br />
Tevian Dray, Janet T<strong>at</strong>e, Rubin Landau<br />
1
m d2rr dt 2<br />
⎛ ⎞<br />
⎜<br />
⎝ ⎠<br />
Rot<strong>at</strong>ing reference frames<br />
⎛<br />
⎝<br />
d r r<br />
dt<br />
⎟ rot<strong>at</strong>ing<br />
⎞<br />
⎠ inertial<br />
r<br />
acent =− r ω × r ω × r ( r)<br />
= dr ⎛ r⎞<br />
⎝ dt<br />
⎠ rot<strong>at</strong>ing<br />
+ r ω × r r<br />
= r F − m r ω × r ω × r ( r)−2m<br />
r ω × dr ⎛ r⎞<br />
⎝ dt<br />
r<br />
a cor =−2 r ω × r v r<br />
⎠ rot<strong>at</strong>ing<br />
2
Puck launched<br />
from North Pole
Gre<strong>at</strong> circle coordin<strong>at</strong>e systems<br />
x<br />
z'<br />
z<br />
φ<br />
0<br />
λ<br />
max<br />
x'<br />
y'<br />
y<br />
0 ≤φ 0 ≤ 2π<br />
0 ≤λ max ≤π2<br />
λ' = 0<br />
tan λ=tan λmax cos( φ−φ0) 4
Inertial and earthbound gre<strong>at</strong> circles<br />
ω<br />
A A*<br />
δ Earth<br />
Ω= v<br />
R =<br />
v E<br />
δ inertial<br />
v E +ωRcosλ start<br />
( ) 2 + v N 2<br />
R<br />
v Earth<br />
v N<br />
5
Puck released<br />
from rest
L<strong>at</strong>itude (degrees)<br />
45.0<br />
44.8<br />
44.6<br />
44.4<br />
44.2<br />
44.0<br />
43.8<br />
A<br />
Puck initially <strong>at</strong> rest<br />
ω<br />
C<br />
A*<br />
a cent<br />
43.6<br />
0 5 10 15<br />
Longitude (degrees)<br />
lowest order<br />
centrifugal<br />
deflection<br />
7
Puck launched<br />
toward east
L<strong>at</strong>itude (degrees)<br />
45<br />
44.95<br />
44.9<br />
A<br />
Puck launched eastward<br />
A<br />
ω<br />
B<br />
B<br />
A*<br />
F A*<br />
F*<br />
C<br />
B *<br />
a cor<br />
C 0<br />
B*<br />
Coriolis<br />
0 1 2 3 4 5<br />
Longitude (degrees)<br />
Curvilinear<br />
Centrifugal<br />
C<br />
9
Puck launched<br />
toward west from<br />
London (earth <strong>at</strong><br />
rest)
Puck launched<br />
toward north
L<strong>at</strong>itude (degrees)<br />
15.4024<br />
15.4022<br />
15.4020<br />
15.4018<br />
15.4016<br />
15.4014<br />
≈<br />
0<br />
B<br />
15.0 A<br />
Puck launched northward<br />
≈<br />
B*<br />
Centrifugal<br />
A*<br />
0.4168<br />
0.4170<br />
Coriolis<br />
0.4172<br />
Longitude (degrees)<br />
0.4174<br />
C<br />
0.4176<br />
A<br />
B<br />
ω<br />
B *<br />
C<br />
A *<br />
• B moves away<br />
• vEarth ↓<br />
• L conserved<br />
12
Puck launched<br />
from Vancouver<br />
to London<br />
v = 1350 mi/hr<br />
t = 3.5 hr<br />
a ≈ ωv ≈ 0.5% g<br />
a ≈ 350 mi/hr/hr<br />
D ≈ <strong>at</strong> 2 /2 ≈ 2000 mi
L<strong>at</strong>itude (degrees)<br />
Puck launched generally<br />
30.3125 Curvilinear<br />
30.3120<br />
30.3115<br />
30.3110<br />
30.3105<br />
A*<br />
A*<br />
F*<br />
B*<br />
Centrifugal<br />
Coriolis<br />
A<br />
A<br />
Longitude (degrees)<br />
A*<br />
0.6250 0.6255 0.6260 0.6265<br />
B<br />
ω<br />
A*<br />
B*<br />
C<br />
C<br />
14
• Launch East<br />
• v inertial =v Equ<strong>at</strong>or<br />
Interesting earthbound p<strong>at</strong>hs<br />
• Launch NW<br />
• v inertial =v Equ<strong>at</strong>or<br />
• Launch West<br />
• v inertial ≈v Earth /6<br />
15
Puck launched to<br />
West
Summary<br />
• Gre<strong>at</strong> circles aid understanding of inertial<br />
forces on sphere<br />
• Anim<strong>at</strong>ions on web <strong>at</strong><br />
www.physics.orst.edu/~mcintyre/coriolis<br />
• Am. J. Phys. 68, 1097 (2000). (Dec. 2000)<br />
17
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