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

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 ω × r r r ω ω ω r   inertial   rotating dt dt m d 2 r ω × d r   r F − m r ω × r ) − 2 m r   ω × r r r ( ω ω ω = ω ω ω ω ω ω   r   rotating   dt 2 dt rotating a cent = − r ω × r a cor = − 2 r r ω × r r ω × r ( ) ω ω ω ω ω ω ω ω ω r v r 2

Puck launched from North Pole

Great circle coordinate systems z z' 0 ≤ φ 0 ≤ 2 π 0 ≤ λ max ≤ π 2 x' y' λ max φ y 0 λ ' = 0 x ( ) tan λ = tan λ max cos φ − φ 0 4

Inertial and earthbound great circles ω 2 + v N ( ) v E + ω R cos λ start 2 Ω = v R = R A* A v N δ Earth δ inertial v E v Earth 5

Puck released from rest

Puck initially at rest 45.0 44.8 Latitude (degrees) lowest order 44.6 centrifugal a cent ω deflection A * A 44.4 44.2 C 44.0 43.8 43.6 5 15 0 10 Longitude (degrees) 7

Puck launched toward east

Puck launched eastward F* * A A F 45 Curvilinear B* B Latitude (degrees) Centrifugal ω a cor C 0 * A A 44.95 B* B C Coriolis C 44.9 2 3 4 0 1 5 Longitude (degrees) 9

Puck launched toward west from London (earth at rest)

Puck launched toward north

Puck launched northward B* B* B B 15.4024 Centrifugal C A * A 15.4022 C Coriolis Latitude (degrees) 15.4020 ω 15.4018 • B moves away • v Earth ↓ 15.4016 • L conserved 15.4014 ≈ * A A 15.0 ≈ 0.4168 0.4170 0.4172 0.4174 0.4176 0 Longitude (degrees) 12

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

Puck launched generally B* B * F 30.3125 Curvilinear * B A A* C A* 30.3120 Latitude (degrees) ω Centrifugal 30.3115 Coriolis C A* 30.3110 A* A 30.3105 0.6250 0.6255 0.6260 0.6265 Longitude (degrees) 14

Interesting earthbound paths • Launch East • Launch NW • Launch West • v inertial ≈ v Earth /6 • v inertial =v Equator • v inertial =v Equator 15

Puck launched to West

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) 17