What is the proof behind the Coriolis force formula?

In summary, the Coriolis force, given by the formula -2m\vec{}\omegax\vec{}v, is a result of using a rotating reference frame and arises when expressing Newton's second law in a rotating frame. It can be derived by changing coordinate systems from xyz to rho, theta, phi or from the corresponding transformation quaternion. The Coriolis force is a pseudo force and is proportional to the mass of an object. It is just one of the various fictitious forces that appear in the second derivative of the position vector from the perspective of a rotating observer.
  • #1
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43
0
When I studied "Earth Science" , I found a formula of Coriolis Force which is

-2m[tex]\vec{}\omega[/tex]x[tex]\vec{}v[/tex]

I am wondering how the formula is proven?


I need a careful proof and diagram

THX!
 
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  • #2
The Coriolis Effect a result of using a rotating reference frame. I recommend getting comfortable with the Coriolis effect in 2d first. The site below is a nice site to get used to the polar coordinate basis and how to take derivatives using r, theta which are unit vectors that are dependent on time.

Notice the equation for acceleration:
img937.png


the
[itex]
2\dot{r} \dot{ \theta }\bf{e_{ \theta}
[/itex]
is the Coriolis effect in 2d
here is a nice video describing this:
 
Last edited by a moderator:
  • #3
LogicalTime said:
The Coriolis Effect a result of using a rotating reference frame. I recommend getting comfortable with the Coriolis effect in 2d first. The site below is a nice site to get used to the polar coordinate basis and how to take derivatives using r, theta which are unit vectors that are dependent on time.

Notice the equation for acceleration:
img937.png


the
[itex]
2\dot{r} \dot{ \theta }\bf{e_{ \theta}
[/itex]
is the Coriolis effect in 2d
here is a nice video describing this:


I have known the effects of Coriolis force

I want to see the "3-D" condition of Coriolis Force

and the "critical" proof(by mathematics and physic)...

Thanks you for displaying the vedio!
 
Last edited by a moderator:
  • #4
hmm not sure what we are looking for by "critical proof", basically when you are using a different coordinate system these pseudo forces just show up, so being able to change the coordinate system seems to do it. Maybe I am misunderstanding what you are asking, dunno.
 
  • #5
I never took a "proof-writing" math class, but I think that's what he's asking for (am I right abc?). Or perhaps a derivation of the equation?
 
  • #6
But this just requires changing coordinate systems from xyz to rho,theta,phi. If you prove that then the Coriolis effect is just a side effect of the proof. Am I missing something in my reasoning Lurch?
 
  • #7
The Coriolis force is a pseudo force (aka inertial force, aka fictional force) that arises from wanting to express Newton's second law in a rotating frame. So, to understand how this term arises it is necessary to delve at least a little into rotating reference frames.

I'm not going to prove this here, but

[tex]
\frac{d}{dt_I}(\boldsymbol{u}}) =
\frac{d}{dt_R}(\boldsymbol{u}}) +
\boldsymbol{\omega} \times \boldsymbol{u}
[/tex]

where [itex]\boldsymbol{u}[/itex] is any vector quantity and [itex]\boldsymbol{\omega}[/itex] is the angular velocity of the rotating frame with respect to some inertial frame. The subscripts I and R on the time derivative of [itex]\boldsymbol{u}[/itex] denote the time derivative of [itex]\boldsymbol{u}[/itex] as observed by someone fixed in the inertial and rotating frames, respectively.

You can find a hand-waving proof of this conjecture in most sophomore/junior level classical physics texts. Good, solid proofs are hard to find. Some upper-level aerospace engineering texts do so. The result is an immediate consequence of the time derivative of the transformation matrix from the rotating to inertial frame (which takes a page or so to derive) or from the corresponding transformation quaternion (which takes a quarter page or so to derive).

The above expression in operator form becomes

[tex]
\frac{d}{dt_I} =
\frac{d}{dt_R} +
\boldsymbol{\omega} \times
[/tex]

Using this expression, the second time derivative of the position vector from the perspective of inertial and rotating observers are related via

[tex]
\aligned
\frac{d^2\boldsymbol{x}}{dt^2_I} &=
\frac{d}{dt_I}\left(\frac{d\boldsymbol{x}}{dt_I}\right) \\
&=
\frac{d}{dt_R}\left(\frac{d\boldsymbol{x}}{dt_I}\right) +
\boldsymbol{\omega} \times \left(\frac{d\boldsymbol{x}}{dt_I}\right) \\
&=
\frac{d}{dt_R}\left(\frac{d\boldsymbol{x}}{dt_R} +
\boldsymbol{\omega} \times \boldsymbol{x} \right) +
\boldsymbol{\omega} \times
\left(\frac{d\boldsymbol{x}}{dt_R} +
\boldsymbol{\omega} \times \boldsymbol{x}\right) \\
&=
\frac{d^2\boldsymbol{x}}{dt^2_R} +
\frac{d\boldsymbol{\omega}}{dt} \times \boldsymbol{x} +
2\,\boldsymbol{\omega} \times \frac{d\boldsymbol{x}}{dt_R} +
\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \boldsymbol{x})
\endaligned
[/tex]

Newton's second law holds in an inertial frame, so the left-hand side is just Fext/m. Solving for the second derivative of the position vector as observed in the rotating frame yields for constant-mass objects

[tex]
m\,\frac{d^2\boldsymbol{x}}{dt^2_R} =
\boldsymbol{F}_{ext} -
\left(
m\,\frac{d\boldsymbol{\omega}}{dt} \times \boldsymbol{x} +
2\,m\,\boldsymbol{\omega} \times \frac{d\boldsymbol{x}}{dt_R} +
m\,\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \boldsymbol{x})
\right)
[/tex]

The three terms proportional to mass are various fictitious forces (all fictitious forces are proportional to mass). As the Earth's rotation rate is nearly constant, so the first fictitious force term can be safely ignored for earth-based observations. The second fictitious force term is the Coriolis force while the final term is centripetal force.
 

1. What is the Coriolis Force formula?

The Coriolis Force formula is a mathematical equation that describes the force that acts on an object in motion due to the Earth's rotation. It takes into account the object's velocity, the Earth's angular velocity, and the latitude of the object's location.

2. How is the Coriolis Force formula derived?

The Coriolis Force formula is derived from the laws of motion and the principles of circular motion. It also takes into account the Earth's rotation and the fact that the Earth is a non-inertial reference frame. The derivation can be complex and involves advanced mathematical concepts such as vector calculus.

3. What is the significance of the Coriolis Force formula?

The Coriolis Force formula is significant because it helps explain many phenomena in the atmosphere and oceans, such as the direction of winds and ocean currents. It also plays a crucial role in weather patterns and global climate patterns.

4. Are there any limitations to the Coriolis Force formula?

Yes, the Coriolis Force formula has some limitations. It is only accurate for objects moving at relatively large velocities, such as those found in the atmosphere and oceans. It also does not take into account other forces that may be acting on the object, such as friction.

5. How is the Coriolis Force formula used in practical applications?

The Coriolis Force formula is used in a variety of practical applications, including meteorology, oceanography, and engineering. It helps scientists and engineers understand and predict the behavior of objects in motion, such as weather patterns, ocean currents, and the flight paths of airplanes and missiles.

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