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# Plate Motions on a Sphere

(Minster, J-B, T.H. Jordan, Present-day plate motions, J. Geophys. Res., 85, p. 5331-5354, 1978. DeMets et al., Current plate motions, Geophys. J. R. Astr. Soc., 101, p. 425- 478, 1990. DeMets, C., R. G. Gordon, and D. F. Argus, Geologically current plate motions, Geophys. J. Int., 181, 1-80, 2010)

v

r

Δ

ω

# Given:

ω angular velocity vector ⎜⎛ rad⎞⎟ ⎝ s⎠

r position on earth (m)

# Calculate:

v velocity vector at position r

m s ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Of course, the velocity of the plate must be tangent to the surface of the earth so the velocity is the cross product of the position vector and the angular velocity vector.

v=ω×r

(2)

or

v = ˆ i ω y z ω z y ( ) ˆ j ω x z ω z x ( ) + ˆ k ω x y ω y x (

)

(3)

where i, j, and k are unit vectors. The magnitude of the velocity is given by

v = ω r sin(Δ)

(4)

where Δ is the angle between the position vector and the angular velocity vector. It is given by the following formula.

cos(Δ) =

ωr ωr

(5)

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