Sect. 4.10: Coriolis & Centrifugal Forces (Motion Relative to Earth, mainly from Marion) Summary: For motion in an accelerating frame (r), both translating.

Sect. 4.10: Coriolis & Centrifugal Forces (Motion Relative to Earth, mainly from Marion)

• Summary: For motion in an accelerating frame (r), both

translating & rotating with respect to a fixed (f, inertial) frame: Velocities: vf = V + vr + ω r

Accelerations:

ar = Af + ar + ω r + ω (ω r) + 2(ω vr)

Newton’s 2nd Law (inertial frame):

F = maf = mAf + mar + m(ω r)

+ m[ω (ω r)] + 2m(ω vr)

“2nd Law” equation in the moving frame:

mar Feff F - mAf - m(ω r)

- m[ω (ω r)] - 2m(ω vr)

Motion Relative to Earth “2nd Law” in accelerating frame:

Feff mar F - mAf - m(ω r)

- m[ω (ω r)] - 2m(ω vr)

Transformation gave:

Feff F - (non-inertial terms)

• Interpretations:

- mAf : From translational acceleration of moving frame.

- m(ω r): From angular acceleration of moving frame.

- m[ω (ω r)]: “Centrifugal Force”. If ω r: Has magnitude mω2r. Outwardly directed from center of rotation.

- 2m(ω vr): “Coriolis Force”. From motion of particle in moving system (= 0 if vr = 0)

More discussion of last two now!

≈ 0 for motion near Earth

• Motion of Earth relative to inertial frame: Rotation on axis causes small effects! However, this dominates over other (much smaller!) effects:

ω = 7.292 10-5 s-1 ;

ω2Re = 3.38 cm/s2 = Centripetal acceleration at equator

2ωvr 1.5 10-4 v = max Coriolis acceleration

( 15 cm/s2 = 0.015g for v = 105 cm/s)

Even Smaller effects!– Revolution about Sun– Motion of Solar System in Galaxy– Motion of Galaxy in Universe

Also, ω = (dω/dt) ≈ 0

• Coordinate systems (figure): z direction = local vertical

– Fixed: (x,y,z) At Earth center– Moving: (x,y,z) On Earth surface

• Mass m at r in moving system.

• Physical forces in inertial system: F S + mg0

S Sum of non-gravitational forces

mg0 Gravitational force on m

g0 Gravitational field vector, vertical

(towards Earth center; along R in fig).

• From Newton’s Gravitation Law:

g0 = -[(GME)eR]/(R2)

G Gravitational constant, R Earth radius

ME Earth mass, eR Unit vector in R direction

– Assumes isotropic, spherical Earth– Neglects gravitational variations due to oblateness; non-

uniformity; ...

• Effective force on m, measured in moving system is thus: Feff S + mg0 - mAf - m(ω r)

- m[ω (ω r)] - 2m(ω vr)

• Earth’s angular velocity ω is in z direction in inertial system (North): ω ωez

ez unit vector along z

Earth rotation period T = 1 day

ω = (2π)/T = 7.3 10-5 rad/s

(Note: ω 365 ωes)

• ω constant ω 0 Neglect m(ω r)

• Consider mAf term in Feff & use again formalism of last time (rotation instead of translation):

Af = (ω Vf ) = [ω (ω R)]

• Effective force on m is:

Feff S+mg0 - (mω) [ω (r + R)] - 2m(ω vr)

• Rewrite as: Feff S + mg - 2m(ω vr)

Where, mg Effective Weight

g Effective gravitational field (= measured gravitational acceleration, g on Earth surface!)

g g0 - ω [ω (r + R)]

• Considering motion of mass m, at point r near Earth surface. R = |R| = Earth radius. |r| << |R|

ω [ω (r + R)] ω (ω R)

Effective g near Earth surface:

g g0 - ω (ω R)

• If m is at point r far from Earth surface, must consider both R & r terms. Effective g for any r:

g = g0 - ω [ω (r + R)]• Second term = Centrifugal force per unit mass

(Centrifugal acceleration).

• Centrifugal force: – Causes Earth oblateness (g0 neglects). Goldstein

discussion, p 176

– Earth Solid sphere. Earth Viscous fluid with solid crust.

– Rotation “fluid” deforms,

Requator - Rpole 21.4 km

gpole - gequator 0.052 m/s2

– Surface of calm ocean water is g instead of g0.

Deviation of g fromlocal vertical direction!

• Summary: Effective force:

Feff = S + mg - 2m(ω vr) (1)

Where, g = g0 - ω [ω (r + R)] (2)

Often,g g0 - ω (ω R) (3)

These are all we need for motion near the Earth!

Direction of g• Consider: g = g0 - (ω) [ω (r + R)] (2)

• Effective g = Eqtn (2). Consider experiments.

Magnitude of g: Determined by measuring the period of a

pendulum (small θ). DIRECTION of g: Determined by the direction of a “plumb bob” in equilibrium.

• Magnitude of 2nd term in (2):

ω2R 0.034 m/s2 (ω2R)/(g0) 0.35%

• Direction of 2nd term in (2): Outward from the axis of the rotating Earth. Direction of g = Direction of plumb bob = Direction of the vector sum in (2). Slightly different from the “true” vertical line to the Earth’s center. (Figure next page!)

• Direction of plumb bob = Direction of

g = g0 - (ω) [ω (r + R)] (2)

• Figure: (r in figure = r in previous figures!) Deviation of g from g0 direction is exaggerated!

r = R + z where z = altitude

Coriolis Effects• Effective force on m near Earth:

Feff = S + mg - 2m(ω vr)

- 2m(ω vr) = Coriolis force. Obviously, = 0 unless m moves in the rotating frame (moving with respect to Earth’s surface) with velocity vr.

• Figure again:

- 2m(ω vr) = Coriolis force.

• Northern Hemisphere: Earth’s

angular velocity ω is in z direction

in inertial system (North) ω ωez

ez unit vector along z (Figures):

In general, ω has components

along x, y, z axes of the rotating

system. All can have effects,

depending on the direction of vr.

• Most dominant is ω component which is locally vertical in rotating

system, that is ωz Component along local vertical.

- 2m(ω vr) = Coriolis force, Northern hemisphere.

– Consider ωz only for now.

• Particle moving in locally horizontal plane (at Earth surface): vr has no vertical component.

Coriolis force has horizontal component only, magnitude = 2mωzvr & direction to right of particle motion (figure).

Particle is deflected to right of the original direction:

• Magnitude of (locally) horizontal component ofCoriolis force ωz = (locally)

vertical component of ω (Local)

vertical component of ω depends on

latitude! Easily shown:

ωz = ω sin(λ), λ = latitude angle

(figure). ωz = 0, λ =0 (equator);

ωz = ω, λ = 90 (N. pole)

Horizontal component of Coriolis force, magnitude = 2m ωzvr

depends on latitude! 2mωzvr = 2mωvrsin(λ)• All of this the in N. hemisphere! S. Hemisphere: Vertical

component ωz is directed inward along the local vertical. Coriolis force & direction of deflections are opposite of N. hemisphere (left of the direction of velocity vr )

• Coriolis Deflections: Noticeable effects on: • Flowing water (whirlpools)

• Air masses Weather.

Air flows from high pressure

(HP) to low pressure (LP)

regions. Coriolis force deflects it. Produces

cyclonic motion. N. Hemisphere: Right

deflection: Air rotates with HP on right, LP

on left. HP prevents (weak) Coriolis force

from deflecting air further to right.

Counterclockwise air flow!

S. Hemisphere: Left deflection.

(Falkland Islands story)

Bathtub drains!

• More Coriolis Effects on the Weather:• Temperate regions: Airflow is not along pressure isobars due to the

Coriolis force (+ the centrifugal force due to rotating air mass).• Equatorial regions: Sun heating the Earth causes hot surface air to rise

(vr has a vertical component).

In Coriolis force need to account ALSO for (local) horizontal components of ω

Northern hemisphere: Results in cooler air moving

South towards equator, giving vr a horizontal

component . Then, horizontal component of Coriolis

force deflects South moving air to right (West) Trade

winds in N. hemisphere are Southwesterly.

Southern hemisphere: The opposite!

No trade winds at equator because Coriolis force = 0 there

All is idealization, of course, but qualitatively correct!