Using Great Circles to Understand Motion on a Rotating Sphere David - - PowerPoint PPT Presentation

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Using Great Circles to Understand Motion on a Rotating Sphere David H. McIntyre Oregon State University Department of Physics NSF - Paradigms Tevian Dray, Janet Tate, Rubin Landau 1 Rotating reference frames d r = d r + r

SLIDE 1

Using Great Circles to Understand Motion on a Rotating Sphere

David H. McIntyre Oregon State University Department of Physics

NSF - Paradigms Tevian Dray, Janet Tate, Rubin Landau

SLIDE 2

Rotating reference frames

dr r dt     inertial = dr r dt     rotating + r ω ω ω ω × r r m d2r r dt2      

rotating

= r F − m r ω ω ω ω × r ω ω ω ω × r r

( ) − 2m r

ω ω ω ω × dr r dt     rotating r acent = − r ω ω ω ω × r ω ω ω ω × r r

( )

r acor = −2 r ω ω ω ω × r vr

SLIDE 3

Puck launched from North Pole

SLIDE 4

Great circle coordinate systems

max

z' y' x' z y x

λ φ 0 ≤ φ0 ≤ 2π 0 ≤ λmax ≤ π 2 λ' = 0 tan λ = tan λmax cos φ − φ0

( )

SLIDE 5

Inertial and earthbound great circles

δEarth δinertial vEarth vN vE

A A*

ω

Ω = v R = vE + ωRcosλstart

( )

2 + vN 2

R

SLIDE 6

Puck released from rest

SLIDE 7

Puck initially at rest

5 10 15

Longitude (degrees)

43.6 43.8 44.0 44.2 44.4 44.6 44.8 45.0

A C ω A * Latitude (degrees) lowest order centrifugal deflection

acent

SLIDE 8

Puck launched toward east

SLIDE 9

Puck launched eastward

1 2 3 4 5

Longitude (degrees)

45 44.95

A B C

Coriolis Centrifugal Curvilinear

44.9

A C

ω

B

A * B*

A * B* Latitude (degrees)

acor

F F*

C0

SLIDE 10

Puck launched toward west from London (earth at rest)

SLIDE 11

Puck launched toward north

SLIDE 12

Puck launched northward

A C ω B A * B*

0.4176 0.4174 0.4172 0.4168

Longitude (degrees)

15.4014 15.0

A B C

Coriolis Centrifugal

0.4170 15.4016 15.4018 15.4020 15.4022 15.4024

A * B*

Latitude (degrees)

≈ ≈

B moves away
vEarth ↓
L conserved

SLIDE 13

Puck launched from Vancouver to London v = 1350 mi/hr t = 3.5 hr a ≈ ωv ≈ 0.5% g a ≈ 350 mi/hr/hr D ≈ at2/2 ≈ 2000 mi

SLIDE 14

Puck launched generally

Longitude (degrees)

A

Coriolis Centrifugal

C A* A C

ω

B A* B*

Latitude (degrees)

0.6250 0.6265 0.6255 0.6260 30.3105 30.3110 30.3115 30.3120 30.3125

Curvilinear

A* A* B * F *

SLIDE 15

Interesting earthbound paths

Launch East
vinertial=vEquator
Launch NW
vinertial=vEquator
Launch West
vinertial≈vEarth/6

SLIDE 16

Puck launched to West

SLIDE 17

Summary

Great circles aid understanding of inertial

forces on sphere

Animations on web at

www.physics.orst.edu/~mcintyre/coriolis

Am. J. Phys. 68, 1097 (2000). (Dec. 2000)