Coriolis Force and the Earth's Rotation,

[BHL 20]

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Last year, in first year I had problems with understanding the Coriolis Force. I asked the lecturer about it and he found a simpler way of explaining it. I thought I had understood. However, I've spent many hours this weekend trying to understand it and it keeps eluding me. That explanation from last year just fails to answer all my questions.

Atm I feel very annoyed with the way lecturers and textbooks approach this topic. They derive the formula for the Coriolis force in a very general way using vectors, which gives absolutely 0 physical understanding. Then they try to bring the physics in by applying the Coriolis force to an object moving through a spinning disc. I feel this is an attempt to appeal to intuition that a lot of people don't have. Plus the example is usually presented in an unclear way where the student has to infer many details by themselves.
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My current understanding of the Coriolis force is this: every point on the surface of the Earth rotates with the same angular velocity ω but the radius (to the axis of rotation) varies with the latitude. As an object moves to a region of different radius its angular momentum must be conserved, i.e. ω₁r₁ = ω₂r₂, so its rotational velocity becomes different to the rotational velocity of the Earth's surface at that point. This change is perceived to be caused by the Coriolis force.

This may seem all good and well, but it seems to me that this force relies a lot on friction to function properly, or else how would the object acquire the Earth's rotational velocity in the first place. It's used to explain wind patterns, and I see no reason why air should be rotating with ω. Can some one tell me this: if a rocket flew in from space and reached a small distance above the Earth's equator, from there it proceeded to fly in a straight line to the North Pole (neglect the Earth's rotation about the sun for the moment), which way would it be deflected? It seems to me it would be deflected in the opposite direction to that which the Coriolis force predicts, as the Earth would be rotating under its straight path.

Even if something just landed on the Earth (no previous rotational velocity) in a region where the coefficient of friction = 0, shouldn't the Earth just rotate "under" it since the object's mass provided inertia but there is no force being applied to it? If so shouldn't the "opposite Coriolis force" be present again? And if the Coriolis force really is dependant on friction, why is it a general feature of rotating frames?

 

[A.T.]

My current understanding of the Coriolis force is this: every point on the surface of the Earth rotates with the same angular velocity ω but the radius (to the axis of rotation) varies with the latitude. As an object moves to a region of different radius its angular momentum must be conserved, i.e. ω₁r₁ = ω₂r₂, so its rotational velocity becomes different to the rotational velocity of the Earth's surface at that point. This change is perceived to be caused by the Coriolis force.

That is just the tangential component of the Coriolis force. There is also a radial component, because moving tangentially in the rotating frame, changes the angular velocity (as seen from the inertial frame) and thus the centripetal force required to stay at constant radius. This change is also attributed to the Coriolis force.

This may seem all good and well, but it seems to me that this force relies a lot on friction to function properly,

Definitely not. Friction is a real (interaction) force that either exist in every frame, or doesn't exist in any frame. Coriolis force is just an coordinate effect, which never exist in the inertial frame, but is always there in the rotating frame, even if there is no interaction. For example: In the rotating frame of the Earth the distant stars are moving in circles, because the Coriolis force (reduced by centrifugal force) acts as the centripetal force.

This video might be helpful too:

 

[mfb]

This may seem all good and well, but it seems to me that this force relies a lot on friction to function properly, or else how would the object acquire the Earth's rotational velocity in the first place.

Do you move relative to the ground right now (like... airplane speed)? If not, you follow the rotational velocity of Earth like everything else, including the atmosphere.

Coriolis force is defined in the rotating system, and this has the rotation by definition.

As an object moves to a region of different radius its angular momentum must be conserved, i.e. ω₁r₁ = ω₂r₂

It does not have to be conserved, but every change has to come from some force acting on the object.

Can some one tell me this: if a rocket flew in from space and reached a small distance above the Earth's equator, from there it proceeded to fly in a straight line to the North Pole (neglect the Earth's rotation about the sun for the moment), which way would it be deflected?

If it does not follow the rotation of earth: as seen from the surface of earth, it first goes west faster than the speed of sound (because Earth rotates under it), the more north it gets the slower this motion gets (reaching to zero at the pole). Therefore, as seen from the surface, some force accelerated it eastwards.
As seen from the spaceship, there is no rotation, no deflection and no Coriolis force.

Even if something just landed on the Earth (no previous rotational velocity) in a region where the coefficient of friction = 0, shouldn't the Earth just rotate "under" it since the object's mass provided inertia but there is no force being applied to it?

Yes, but unless it lands at the equator its ground path will be curved. Why? Because we have the Coriolis force if we view its motion from Earth.To summarize, Coriolis force has nothing to do with friction.

 

[Erland]

As far as I know, the Coriolis force has nothing to do with angular momentum. At least I never thought of it in that way, and I think it is misleading.
In your rocket example, the rocket has all the time no north/south component of angular momentum at all w.r.t the center of the Earth (only a constant east/west component which wouldn't be there if the Earth were a flat disc with the North Pole at its centre, it is there only because the rocket flies along a curved path along the round Earth).

You are right that the Earth is simply rotating under the rocket. But according to an observer at the Earth (moving with the Earth), the rocket is not flying straight to the north, its apparent velocity has also a component to the west. Say, for example, that the speed of the rocket is the same as the rotational speed of the Earth at the equator. Then, when the rocket starts from the equator, the observer at the Earth sees it flying towards the northwest.
This component to the west is seen to decrease, by the Earth observer, as the rocket approaches the North Pole, since the absolute rotational velocity decreases when the latitudes becomes shorter, which means that the Earth observer sees the rocket deflect to the right, so that it finally reaches the North Pole, despite that it initially was seen to fly towards the northwest. This apparent deflection to the right is attributed to the Coriolis force by the Earth observer.

Also, the Coriolis force has nothing to do with friction. The reason that objects at the Earth rotate with the Earth is that they always have done that. When you were born from your mother's womb, your mother was rotating with the Earth, and her velocity was transferred to you. Then, the gravity of the Earth prevents you from flying away in the tangential direction, so that you continue to rotate with the Earth, except for your own relatively small movements at the Earth. The same is true for all earthly objects. This has nothing to do with friction, and the Coriolis force has nothing to do with it.

 

[Erland]

This might be an appropriate place to address an issue which I never saw mentioned anywhere, so I had to figure it out myself, and I think it is pretty cool:

We observers on the Earth see the celestial bodies, Sun, Moon, planets, stars etc. rotate around the Earth once every 23 h 56 m (disregarding their own motions). It should then be possible for us to attribute this apparent circular motion to inertial forces. The resultant inertial force, for each celestial body, should be an apparent centriptal force causing it to rotate around the Earth.
Which known inertial forces have this apparent effect?

Answer: It is a combination of the Coriolis force and the centrifugal force of the body.
Wee see the body move to the west (when we see it in the south, or north if we are located at the southern hemisphere), and then the Coriolis force must deflect it towards the Earth's axis, giving rise to an apparent centripetal force which turns out to be exactly the double of the total apparent centripetal force. This Coriolis force is then partially counteracted by the centrifugal force, directed away from the Earth's axis, and it has the same size as the total apparent centripetal force. The sum of these two inertial forces is then the apparent centripetal force.

This does not seem to be so well known, but it is a useful exercise when learning about inertial forces.

 

[A.T.]

This might be an appropriate place to address an issue which I never saw mentioned anywhere

It's mentioned in the first reply on this thread.

“Everything has been said already, but not yet by everyone.” – Karl Valentin

 

[Erland]

It's mentioned in the first reply on this thread.

Right, so I didn't read this carefully enough. Sorry! Still, I seldom see it mentioned.

Btw, what I wrote about friction and earthly objects should be modified. If a person moves on the Earth over several latitudes, his/her rotation velocity is changed with the latitude. It could be viewed as friction countering the small Coriolis force during the motion.

 

[BHL 20]

That is just the tangential component of the Coriolis force. There is also a radial component, because moving tangentially in the rotating frame, changes the angular velocity (as seen from the inertial frame) and thus the centripetal force required to stay at constant radius. This change is also attributed to the Coriolis force.

?? But the formula for coriolis force is -2mω×v so if ω points straight up and the body is moving either north or south, the right hand screw rule will give a vector which is tangential to the Earth's surface. Am I wrong?

 

[BHL 20]

You are right that the Earth is simply rotating under the rocket. But according to an observer at the Earth (moving with the Earth), the rocket is not flying straight to the north, its apparent velocity has also a component to the west. Say, for example, that the speed of the rocket is the same as the rotational speed of the Earth at the equator. Then, when the rocket starts from the equator, the observer at the Earth sees it flying towards the northwest.
This component to the west is seen to decrease, by the Earth observer, as the rocket approaches the North Pole, since the absolute rotational velocity decreases when the latitudes becomes shorter, which means that the Earth observer sees the rocket deflect to the right, so that it finally reaches the North Pole, despite that it initially was seen to fly towards the northwest. This apparent deflection to the right is attributed to the Coriolis force by the Earth observer..

Thank you, this helped me.

Also, the Coriolis force has nothing to do with friction. The reason that objects at the Earth rotate with the Earth is that they always have done that. When you were born from your mother's womb, your mother was rotating with the Earth, and her velocity was transferred to you. Then, the gravity of the Earth prevents you from flying away in the tangential direction, so that you continue to rotate with the Earth, except for your own relatively small movements at the Earth. The same is true for all earthly objects. This has nothing to do with friction, and the Coriolis force has nothing to do with it.

But what about the atmosphere, why does it rotate with the Earth?

 

[Erland]

But what about the atmosphere, why does it rotate with the Earth?

Because it always did that. Take for example an oxygen molecule in the atmosphere. It was probably once produced by a photosyntetic organism. This organism was rotating with the Earth, hence this motion was transferred to the oxygen molecule. After that, by its own inertia, the molecule wants to continue with the given velocity in the tangential direction, but the centripetal force of gravity from the Earth prevents it from flying away, and it continues to rotate with the Earth, in addition to its own motion in the atmosphere. And since all other molecules in the atmosphere it interacts with also rotate with the Earth, by the same reason, this rotation is not noticed by an observer on the Earth, unless the Coriolis force becomes noticable.

 

[A.T.]

?? But the formula for coriolis force is -2mω×v so if ω points straight up and the body is moving either north or south, the right hand screw rule will give a vector which is tangential to the Earth's surface. Am I wrong?

By tangential/radial I meant with respect to the axis of frame rotation, not to the Earth's surface. Coriolis force exist in all rotating frames, regardless if there is a planet or not.

 

[BHL 20]

By tangential/radial I meant with respect to the axis of frame rotation, not to the Earth's surface. Coriolis force exist in all rotating frames, regardless if there is a planet or not.

Could you please explain what you mean by radial component, and tell me what situations the coriolis force would have a radial component, because this has slightly confused me.

 

[A.T.]

Could you please explain what you mean by radial component,

radial : away or towards the rotation axis of the frame.
tangential : perpendicular to radial and to the rotation axis of the frame.

and tell me what situations the coriolis force would have a radial component,

When v has a tangential component

 

[Erland]

When v has a tangential component

which it does not have if the object is moving only in the north/south direction on the Earth, relative to the Earth's rotating frame.

But notice that this is not so in the rocket example, since in flies straight towards the North Pole only in the inertial frame, while it has a component to the west relative to the Earth frame. The Coriolis force on the rocket indeed has a radial component, directed towards the Earth's axis, that is, effectively to the north. This is what drags the originally northwest flying rocket towards the North Pole.

Other examples are the object without friction landing at the Equator, at velocity 0 in the inertial frame, then sitting there while the Earth slides under it, and the stars etc. going in apparent circles around the Earth (mentioned first by A.T. and then by me, embarrassingly enough without seeing that A.T. mentioned it first). In both these cases, the apparent velocity (in the Earth frame) has only a tangential component, so the Coriolis force has only a radial component, giving rise to an apparent centripetal force, partly counteracted by the centrifugal force, which apparently drives the objects around the Earth.

 

[BHL 20]

which it does not have if the object is moving only in the north/south direction on the Earth, relative to the Earth's rotating frame.

Thanks for this clarification and thanks to everyone who replied. I feel I'm ok now with regards to the coriolis force.