Coriolis force caused by tangential velocity

[Swamp Thing]

From another recent thread I learned that you see a Coriolis force if an object in a rotating reference frame moves along a tangent at some velocity v. (I was already familiar with the case where the velocity is radial).

I still find it a little counter-intuitive that the force has the same magnitude irrespective of whether the velocity is directed radially or tangentially. When you move radially towards the center of rotation, you are dashing headlong from a "fast lane" into a "slow lane", so you will soon see a growing mismatch between your own tangential velocity and the local velocity of the surface that you are running over. Fair enough.

But when you move along the tangent, you are following approximately the same direction that the local surface is following anyway, only faster. So you are merely sidling or drifting, in an incidental manner, from a "fast lane" into a "superfast lane" where the local surface velocity is trying to leave you behind.

So intuitively, the Coriolis force in the second example should be an order smaller than in the first. Now obviously, my intuition is wrong -- but is there a simple, correct intuitive picture that will make it at once clear why the two cases produce exactly the same Coriolis force?

 

[Anachronist]

It isn't a force, it's an effect that has the appearance of a force to an observer in the rotating reference frame. To an observer outside that reference frame, no forces are acting on the object other than the usual (centripetal, gravity).

If you sit on a rotating disk and roll a frictionless ball toward the center, you'll see a deflection in its path because your point of view changes with respect to the straight path of the ball. If you roll the ball in the direction of rotation, you'll also see a deflection in its path, because your path diverges from the straight path of the ball. In either case (and everything in between), the deflection appears as if a force were acting on the ball. It's an effect of the reference frame, not an actual force.

In the case of vortices in our atmosphere, a low pressure region attracts air toward it. As the air moves toward it (assuming the pressure region is relatively stationary with respect to the rotation of the planet), it doesn't head directly for the low pressure region but deflects to the right (when viewed from above in the northern hemisphere) no matter what direction from which the air approaches the low pressure region. Air converging in all directions deflected to the right will create a whorl to the left (counterclockwise) around the low-pressure zone.

 

[Dale]

my intuition is wrong -- but is there a simple, correct intuitive picture that will make it at once clear why the two cases produce exactly the same Coriolis force?

Since your intuition is wrong I don’t think that a correct picture will possibly be intuitive. So I think you should focus on simple and correct, discarding intuitive.

The simplest correct explanation is simply the formula for the Coriolis force: $-2m \mathbf{\Omega} \times \mathbf v$. Since this cross product is the same in both cases the acceleration is the same in both cases.

 

[A.T.]

So you are merely sidling or drifting, in an incidental manner,...

The reason for the radial Coriolis force has nothing to do with sidling or drifting.

One intuitive way to think about the radial Coriolis force, is as a velocity dependent modification of the centrifugal force. In fact you could order the inertial force terms in a rotating frame by direction, and lump the radial Coriolis force with radial centrifugal force. But for mathematical and historical reasons we separate them as position dependent term (centrifugal) and a velocity dependent term (Coriolis).

 

[Swamp Thing]

One intuitive way to think about the radial Coriolis force, is as a velocity dependent modification of the centrifugal force.

Ok, that's a nice way to make sense of it.

 

[Swamp Thing]

And I see how I mis-analyzed the second scenario. The acceleration / force due to "lane changing" is not only an order of magnitude less than Case-1, it's actually zero. But a different phenomenon has now kicked in, namely "modified centrifugal force" which can be interpreted as a new force with the same magnitude, different direction.

What is cool is that the same elegant cross product formula encompasses both of these phenomena (which, after all, belong to the same general class of inertial forces).

 

[FactChecker]

Something going tangentially is moving toward an increasing height and simultaneously falling behind the radius that it would have been at if it was traveling in a circle. It's not clear to me how the two effects can be intuitively summed, but it may not be so surprising that the total magnitude of the deviation is the same as the example that you intuitively accept.

 

[Dr.D]

The Coriolis "force" is not at all. It is an "inertial reaction" term, that is a mass x acceleration term.

Recall Newton's 3rd law about every FORCE having an equal and opposite reaction force? Well, where is the reaction to the "Coriolis Force"? It cannot be found because it does not exist.

 

[jbriggs444]

Try this intuition on for size...

If you have a fixed velocity that you are measuring against rotating directions then that velocity will appear to rotate anti-spinward. That rotation amounts to an acceleration at right angles to the current direction. The higher the velocity, the greater the required acceleration. The higher the rotation rate, the greater the required acceleration.

 

[FactChecker]

Try this intuition on for size...

If you have a fixed velocity that you are measuring against rotating directions then that velocity will appear to rotate anti-spinward. That rotation amounts to an acceleration at right angles to the current direction. The higher the velocity, the greater the required acceleration. The higher the rotation rate, the greater the required acceleration.

This is my favorite way of looking at the difference between the velocity derivative in a rotating frame versus acceleration due to a force. Suppose there is a coordinate system (XYZ) attached to an airplane pilot's head, where X is straight forward from his head and Y is out his left ear. When he looks forward, the velocity can be Mach 1 in the X direction. When he turns his head to look right, the X velocity becomes 0 and the Y velocity becomes Mach 1. That can happen in a second, so the velocity derivatives are huge. But there is no force and no physical acceleration. This example shows how important it is to distinguish between physical acceleration due to a force and the derivative of velocity.

 

[zanick]

The Coriolis "force" is not at all. It is an "inertial reaction" term, that is a mass x acceleration term.

Recall Newton's 3rd law about every FORCE having an equal and opposite reaction force? Well, where is the reaction to the "Coriolis Force"? It cannot be found because it does not exist.

actually, you could have a force due to Coriolis, that would be the force in opposition in slowing tangentially, a north bound object from the equator. there would have to be a force to slow the object down, or it would break a conservation law. without the "force" there would be an apparent deflection , called Coriolis.

 

[Dr.D]

Every force has a reaction. Where is the reaction in your example, @zanick ?

 

[zanick]

Every force has a reaction. Where is the reaction in your example, @zanick ?

assume have a 1000mph bullet train going north bound from the equator.
the force of the "train" acting on the east side of the rail and the rail acting back on the train. suddenly, a fictitious force or deflection is not so "apparent" anymore. ;) similar to centrifugal force of your head hitting the side window of a car in circular path. centripetal force (proper) is the window... centrifugal force ( head hitting window) inertial force/fictitious force. or you can look at it from a conservation of angular momentium... the train going north spins up the earth, requiring a force

 

[FactChecker]

Due to the Coriolis tendency to drift, there may be a force required to prevent an object from drifting, just like a centrifugal force has an associated centripetal force to keep an object "in place" in the rotating coordinate system. The "equal and opposite" would be seen in the infinitesimal change in the rotation of the Earth.

 

[Swamp Thing]

... there would have to be a force to slow the object down, or it would break a conservation law.

... there may be a force required to prevent an object from drifting ... The "equal and opposite" would be seen in the infinitesimal change in the rotation of the Earth.

These remarks dovetail very nicely with something that I've come to understand in the last couple of days, as a result of thinking about Veritasium's video on the Dzhanibekov Effect.

If we have a free rigid body that is rotating with no external torque, then of course angular momentum has to be conserved. However, there can be "internal" off-axis torques due to unbalanced centrifugal forces, akin to the dynamic balancing problem -- e.g. when balancing crankshafts and vehicle wheels. These torques will try to shift the orientation of the object with respect to the angular momentum vector. When that happens, the moment of inertia around the direction of angular momentum could (e.g.) increase, which would require a decrease in angular velocity in order to conserve angular momentum.

Moreover, an individual component of the system may happen to move towards the angular momentum axis, while another may move away from it, requiring a redistribution of energy/momentum from one element to another.

So in a rigid freely evolving system, Coriolis forces are merely the mechanisms that mediate the change in angular velocity and the internal redistribution of energy described above. The reaction forces -- Dr D asked about those -- those are just the structural rigidity of the system trying to prevent elastic distortions within the system. If we imagine the system as a massless wireframe with point masses attached to certain nodes with strong rubber bands, then the Coriolis reaction forces are the rubber band forces that remain after you factor out the "conventional" centrifugal forces.

Through those rubber bands, the point masses are all constantly "negotiating" with each other (sending forces/torques to each other) to collectively determine how the system is going to move, such that each element has to swerve as little as possible from its constant-velocity motion, while still respecting Newton's laws and global conservation of angular momentum. A heavy point mass has more "say" in the negotiation that a light one, for example. This negotiation aspect is reflected in certain simulation algorithms by solving nonlinear simultaneous equations to determine what has to happen during a small time step.

Edit: I am not too happy with the way I have expressed this:

However, there can be "internal" off-axis torques due to unbalanced centrifugal forces...

but it will have to do for the moment.

 

[Dale]

the force of the "train" acting on the east side of the rail and the rail acting back on the train.

Neither of those are the Coriolis force. @Dr.D is correct, inertial forces do not follow Newton’s 3rd law.

 

[zanick]

I agree... and also, I didnt say that both of them were "Coriolis force".. that's how you know you are in a non -inertial reference frame, as things don't follow Newton mechanics . I think we would agree, that Coriolis is an apparent deflection not a force for the reason you mentioned. so, what would that force be called if you prevented the apparent deflection with a force?

 

[Dale]

what would that force be called if you prevented the apparent deflection with a force?

Depends on the nature of the real force. If it were a contact force then I would call it a contact force. If it were a tension then I would call it a tension. If it were an electromagnetic force then I would call it an electromagnetic force. It doesn’t have a special name.

I didnt say that both of them were "Coriolis force"..

You sort of did. You listed them as an answer to the question about what is the reaction to the Coriolis force. Since those two forces are reaction forces to each other then it sure seems like you are describing one of them as the Coriolis force.

 

[zanick]

Due to the Coriolis tendency to drift, there may be a force required to prevent an object from drifting, just like a centrifugal force has an associated centripetal force to keep an object "in place" in the rotating coordinate system. The "equal and opposite" would be seen in the infinitesimal change in the rotation of the Earth.

agreed... for example, an airliner has to decal laterally only 1.5mph per min while traveling north bound.. very small force guiding the plane trimmed to the west, but still a force.. technically, what would you call that force... as you mentioned, centripical force could be analogous to this...… "anti Coriolis force" ? ;)

 

[jbriggs444]

technically, what would you call that force

Why must it have a name?

 

[Swamp Thing]

Why must it have a name?

Indeed! "The Force that can be named is not the True Force"

 

[zanick]

Why must it have a name?

to help explain it best. for example... centrifugal force is not a true force, but its called "centrifugal"... sure we call these forces , inertial , or pseudo, 'not true force" , etc. the term" Coriolis " force is used on occasion, can we feel good about using it here?

 

[zanick]

Depends on the nature of the real force. If it were a contact force then I would call it a contact force. If it were a tension then I would call it a tension. If it were an electromagnetic force then I would call it an electromagnetic force. It doesn’t have a special name.

You sort of did. You listed them as an answer to the question about what is the reaction to the Coriolis force. Since those two forces are reaction forces to each other then it sure seems like you are describing one of them as the Coriolis force.

Thanks Dale... I just want to be as correct as possible , especially when discussing the roots of terms with my son who is in his first year of physics now. I like your comment. …. its more of a "reaction" to Coriolis and could be a" contact force" in my example. but, as I have always understood it, it is an apparent deflection in the non inertial reference frame. the point was, could it be called" corilos", similarly to how centrifugal is used.

 

[jbriggs444]

to help explain it best. for example... centrifugal force is not a true force, but its called "centrifugal"... sure we call these forces , inertial , or pseudo, 'not true force" , etc. the term" Coriolis " force is used on occasion, can we feel good about using it here?

But what is it that you want to explain?

You want to explain the nature of the force that an airplane pilot needs to command in order to keep his plane on a "straight" course? Surely that comes under the heading of aerodynamics and control systems rather than under the heading of physics.

 

[vanhees71]

I'd not call inertial forces "fictitious", but that's another story...

 

[zanick]

But what is it that you want to explain?

You want to explain the nature of the force that an airplane pilot needs to command in order to keep his plane on a "straight" course? Surely that comes under the heading of aerodynamics and control systems rather than under the heading of physics.

not really, the reaction to those forces would be what a pilot controls . the cause could be Coriolis or it could be the jet stream... coincidentally those "forces" would be pretty similar in magnitude and direction on the aircraft flying north bound from the equator. the question of what we are trying to explain, IS the cause and effect. what's wrong with being as accurate and descriptive as possible for a more complete understanding? just the term "pseudo force", or "non force" is not well understood . its not always easy to explain a force that is reference frame dependent.

 

[zanick]

I'd not call inertial forces "fictitious", but that's another story...

do tell... can you summarize... I don't usually , but when I hear it ,I don't disagree. should I? :)

 

[Dr.D]

assume have a 1000mph bullet train going north bound from the equator.
the force of the "train" acting on the east side of the rail and the rail acting back on the train. suddenly, a fictitious force or deflection is not so "apparent" anymore. ;)

If there is a load on the wheel flange, it is a constraint force.

 

[Dr.D]

could it be called" corilos", similarly to how centrifugal is used

When applying Newton's Second Law (Sum F = m*a), neither a "coriolis force" nor a "centrifugal force" belong on the left side of the equation. They belong on the right side.

 

[zanick]

If there is a load on the wheel flange, it is a constraint force.

which is a variant of a "contact" force , correct? and, could or would you call it "due to Coriolis"?

 

[Dr.D]

I would simply call a constraint force by that name and not try to attribute it to any particular source other than the fact that it constrains (guides) the motion.

 

[zanick]

When applying Newton's Second Law (Sum F = m*a), neither a "coriolis force" nor a "centrifugal force" belong on the left side of the equation. They belong on the right side.

yes, I understand, and is the reason why I though using that naming convention logic that calling it" Coriolis force" would be appropriate. Using the example of the train and the "constraining " force caused by Coriolis, there would be no net force as there would be no acceleration or velocity to the east traveling north. Unconstrained... in the non inertial frame of refence, you would see the "m" and an "a" to the east, following Newton, wouldn't there have to be an "F"?

 

[vanhees71]

do tell... can you summarize... I don't usually , but when I hear it ,I don't disagree. should I? :)

Well, in a non-inertial frame you have inertial forces. They are defined by writing $\vec{F}=m \vec{a}$ also in the non-inertial frame. Then in addition to the "true forces" (i.e., some mediated by the electroweak or strong interaction) you have to add the inertial forces on the left-hand side of this equation. They are then not fictitious in any way.

If you work in a general covariant way, of course, no such forces belong to the left-hand side but occur on the right-hand side of the equation, but I don't know any textbook that takes this point of view. All I know lump the inertial forces to the left-hand side and thus they call them forces.

You can argue about gravity. In Newtonian physics you consider it a "true force", in relativistic physics you have to use general relativity, and there gravitational (local!) effects on a test particle are usually reinterpreted as the effect of free motion in curved spacetime. In this sense you cannot distinguish between gravitational interaction of a test particle and inertial forces in a sufficiently small spacetime region. Would you then call gravity a "fictitious forces", because you do so for the inertial forces in Newtonian physics (or special relativity) in accelerated reference frames?

This is of course all no problem, because you always end up with the same equations of motion to solve.

 

[zanick]

I would simply call a constraint force by that name and not try to attribute it to any particular source other than the fact that it constrains (guides) the motion.

Thank you... sorry it took so long to get to this, but if that is the best naming convention, ill use that from now on to avoid error and/or confusion. I've been using "apparent deflection due to Coriolis " and rarely "Coriolis force" ill stop using the later. Now the challenge is to describe how the "apparent" deflection is actually real, if you are looking at it in the non inertial reference frame. ;)

 

[zanick]

Well, in a non-inertial frame you have inertial forces. They are defined by writing $\vec{F}=m \vec{a}$ also in the non-inertial frame. Then in addition to the "true forces" (i.e., some mediated by the electroweak or strong interaction) you have to add the inertial forces on the left-hand side of this equation. They are then not fictitious in any way.

If you work in a general covariant way, of course, no such forces belong to the left-hand side but occur on the right-hand side of the equation, but I don't know any textbook that takes this point of view. All I know lump the inertial forces to the left-hand side and thus they call them forces.

You can argue about gravity. In Newtonian physics you consider it a "true force", in relativistic physics you have to use general relativity, and there gravitational (local!) effects on a test particle are usually reinterpreted as the effect of free motion in curved spacetime. In this sense you cannot distinguish between gravitational interaction of a test particle and inertial forces in a sufficiently small spacetime region. Would you then call gravity a "fictitious forces", because you do so for the inertial forces in Newtonian physics (or special relativity) in accelerated reference frames?

This is of course all no problem, because you always end up with the same equations of motion to solve.

GREAT, thanks for that ^^^^^^ it reinforces the correct way of looking at the relationships.