Coriolis effect, Conservation of Angular Momentium, Planes

[zanick]

I have a couple of questions that i thougth this group could help me with.
1. A plane (SR71) takes off from the equator, with a lateral speed, relative to space of 1000mph. (earth rotational speed) say it takes an hour to get there so, its going 10,000mph or something. . Tt flys over the north pole. According to conservation of angular momentum, since the radius is being shortened, doesn't the plane's rotation around the Earth speed up? since the radius is going down, the speed should go up.
intuitively, i can see how a Coriolis effect could turn the plane to the right but it seems that there should be an effect that speeds up the spin of the plan vs what it had when it took off at the equator (say Brazil flying to the north pole)

2. The same plane takes off from Brazil and lands on the north pole. has the KE changed to PE? (because the plane was rotating with the Earth (relative to space) at 1000mph and now its sitting on the north pole at 0mph. It feels like i can compare this to someone at the outer edge of a merry-go-round walking to the center (using some force) and stores up KE as PE, vs the people at the edge of the merry-go-round. ...if he walked to the center, the Coriolis effect would make him press to the left to make it to the center , otherwise he would miss the center and end up at the edge again being pulled right. (for a counterclockwise merry-go-round)

3. Someone asked me why you wouldn't feel going from 1000mph to 0 (lateral speed as you go from the equator to the north pole) over say 1 hour (in some real fast SR71 or something) i said, the speed sounds impressive and so does the change, but over an hour, the rate of change isnt. ...that's less than .01g (0.1m/s/s) , so hardly within the range of human perception.

 

[Sharkey4123]

I'm not entirely sure I understand what the first question is. But in essence there will be some effect on the Earth's rotation by a plane traversing the globe. I'd even presume that as you walk across the surface of the Earth you are altering the angular momentum. But I think the key point is to note that the amount you are likely to alter the angular momentum by is absolutely miniscule to the point it is negligible.

As for point 2, you should think about how your frame of reference changes things. For example, the Earth might be rotating at say, 1000mph, but in the Earth's frame of reference it is stationary. So the only contributions to the plane's KE come from it's motion relative to the Earth.

You could look at it from another frame, for example the centre of the solar system. In this case you would need to calculate the velocity of the plane as it moves in the solar system. This would include contributions from the Earth's orbit, the Earth's rotation and the plane's motion across the Earth. This is analogous to your merry go round example, the people on the edge of the merry go round don't have any kinetic energy (in the frame of the merry go round).

I hope helps, or at least gets you thinking a little about how you're choosing to view this problem.

 

[zanick]

I'm not entirely sure I understand what the first question is. But in essence there will be some effect on the Earth's rotation by a plane traversing the globe. I'd even presume that as you walk across the surface of the Earth you are altering the angular momentum. But I think the key point is to note that the amount you are likely to alter the angular momentum by is absolutely miniscule to the point it is negligible.

As for point 2, you should think about how your frame of reference changes things. For example, the Earth might be rotating at say, 1000mph, but in the Earth's frame of reference it is stationary. So the only contributions to the plane's KE come from it's motion relative to the Earth.

You could look at it from another frame, for example the centre of the solar system. In this case you would need to calculate the velocity of the plane as it moves in the solar system. This would include contributions from the Earth's orbit, the Earth's rotation and the plane's motion across the Earth. This is analogous to your merry go round example, the people on the edge of the merry go round don't have any kinetic energy (in the frame of the merry go round).

I hope helps, or at least gets you thinking a little about how you're choosing to view this problem.

That was the kind of input i was looking for. I kind of thought of that when asking the question.. (no KE change relative to the Earth ). however going to the merry-go round example, if you work your way to the center of the merry-go -round. it takes some force , over a distance, and time, so that work (energy) has to be conserved... doesn't the peson's PE go up? doesn't the merry-go-round go a little faster with some of its mass now going to the axis of rotation?
as far as the first question, it also deals with KE and PE of the plane itself. relative to other same planes on the ground in Brazil, is the SR71 storing up PE going to the north pole? (vs those on the ground still in brazil) i know its relative to the earth, but relative to space doesn't the north pole plane have less KE? as it isn't moving relative to the sun let's say.

 

[russ_watters]

I have a couple of questions that i thougth this group could help me with.
1. A plane (SR71) takes off from the equator, with a lateral speed, relative to space of 1000mph. (earth rotational speed) say it takes an hour to get there so, its going 10,000mph or something.

An SR-71 travels about 3,000 mph, but ok...

It flys over the north pole. According to conservation of angular momentum, since the radius is being shortened, doesn't the plane's rotation around the Earth speed up? since the radius is going down, the speed should go up.

The plane isn't fixed to a rail or rod or cable, it is an airplane with control surfaces and can fly in any direction it wants. Still, have you ever played "Tether-Ball"? That's where you have a volleyball like ball attached to a pole with a rope and you hit it and make the rope wrap around the pole. What happens if you just lightly toss it over top of the pole? (Answer: Nothing interesting)

2. The same plane takes off from Brazil and lands on the north pole. has the KE changed to PE?

If you did this thought experiment with a train instead of a plane, the trip to the pole would cause the Earth's rotation to speed up. But since the plane isn't attached to the Earth in any way, there is no effect and all rotational kinetic energy/momentum are just lost to drag.

It feels like i can compare this to someone at the outer edge of a merry-go-round walking to the center (using some force) and stores up KE as PE, vs the people at the edge of the merry-go-round.

Unlike the plane vs the earth, the person walking on the merry-go-round is attached to the merry-go-round.

3. Someone asked me why you wouldn't feel going from 1000mph to 0 (lateral speed as you go from the equator to the north pole) over say 1 hour (in some real fast SR71 or something) i said, the speed sounds impressive and so does the change, but over an hour, the rate of change isnt. ...that's less than .01g (0.1m/s/s) , so hardly within the range of human perception.

Right.

 

[zanick]

An SR-71 travels about 3,000 mph, but ok...

The plane isn't fixed to a rail or rod or cable, it is an airplane with control surfaces and can fly in any direction it wants. Still, have you ever played "Tether-Ball"? That's where you have a volleyball like ball attached to a pole with a rope and you hit it and make the rope wrap around the pole. What happens if you just lightly toss it over top of the pole? (Answer: Nothing interesting)

If you did this thought experiment with a train instead of a plane, the trip to the pole would cause the Earth's rotation to speed up. But since the plane isn't attached to the Earth in any way, there is no effect and all rotational kinetic energy/momentum are just lost to drag.

Unlike the plane vs the earth, the person walking on the merry-go-round is attached to the merry-go-round.

Right.

Thanks Russ...however, i wonder if the tether ball analogy holds up with the factor that the Earth does have the atmosphere that acts like an attachment of things flying through it, no? on Mars would the effect be more like a tetherball toss over the top? maybe the plane can speed up the Earth , via its effect on the air its flying in as it decelerates laterally. this all rolls into my next question about the Coriolis effect. as the plane does go northward, its lateral speed is coming from a higher speed, thus curving the planes path to the right. along the way, the pilot must bank the wings ever so slightly to counteract this force. true?

 

[A.T.]

along the way, the pilot must bank the wings ever so slightly to counteract this force. true?

As you noted yourself, the lateral acceleration is tiny compared to other disturbances like air currents.

Also note that the vertical component of the Coriolis force (on East-West flights) affects the amount of required lift, and thus fuel consumption. It's a measurable effect, but also small compared to jetstreams etc:
http://naca.central.cranfield.ac.uk/reports/arc/rm/3680.pdf

 

[zanick]

As you noted yourself, the lateral acceleration is tiny compared to other disturbances like air currents.

Also note that the vertical component of the Coriolis force (on East-West flights) affects the amount of required lift, and thus fuel consumption. It's a measurable effect, but also small compared to jetstreams etc:
http://naca.central.cranfield.ac.uk/reports/arc/rm/3680.pdf

I figured it was not even a rounding error, but under closer examination, airlines have to account for it. kind of amazing when you think about it. who would have thought... how does it effect east west flights though? i thought it would only north to south or visa versa, that creates requirements to increase control input to keep on track. as i understand it, the north bound flight, would have to correct to the west to avoid the Coriolis effect taking the plane to the east if uncorrected. is this how it works?

 

[A.T.]

how does it effect east west flights though?

See the link.

 

[zanick]

See the link.

so, its a factor.. not as much as north south travel, but still a component.

 

[A.T.]

...not as much as north south travel,..

No idea what you are comparing here. But the impact on the required lift magnitude (and thus fuel consumption) is far greater for the vertical effect, than for the lateral effect.

 

[zanick]

No idea what you are comparing here. But the impact on the required lift magnitude (and thus fuel consumption) is far greater for the vertical effect, than for the lateral effect.

im comparing the effect of an object traveling north from the equator being more effected by the coriolis effect than it would by east west travel. as you travel north, the east west speed changes , thus altering your trajectory relative to the orignal direction. correct?

 

[A.T.]

im comparing the effect of an object traveling north from the equator being more effected by the coriolis effect than it would by east west travel.

At what latitude? At the equator the always horizontal effect from S-N travel is zero, and grows as you move towards the poles. The effect from E-W travel is always there, and goes from vertical (equator) to horizontal (near the poles).

And since the vertical component has more impact on the effective weight magnitude than the horizontal, I don't see where the effect from S-N travel would be more relevant than the effect from E-W.

 

[zanick]

At what latitude? At the equator the always horizontal effect from S-N travel is zero, and grows as you move towards the poles. The effect from E-W travel is always there, and goes from vertical (equator) to horizontal (near the poles).

And since the vertical component has more impact on the effective weight magnitude than the horizontal, I don't see where the effect from S-N travel would be more relevant than the effect from E-W.

as i understand it, if i traveled 1000miles north vs 1000miles south vs traveling 1000mile east or west, , there would be no coriolis effect for the east west travel, whereas due to conservation of angular momentum, going north or south would have a proportional effect on the change of of the direction of travel.

 

[zanick]

As russ pointed out, if you are not a train on tracks, where going north would have an effect of speeding the Earth up, like the arms of an iceskater, traveling in the air, and not connected, to the Earth other than through the atmosphere. where the steering to stay on a north path would cause the small amounts of drag to slow the horizontal(east west ) speed to zero as it passes over the north pole. that makes me think of an other question. if you are a particle on the Earth at the north pole.. do you have a potential energy higher than a particle at the equator? the logic I am using is if you are in the center of a merry-go-round, and you step toward the outside, you get shot out to the outer circumference. does the same thing happen on the Earth but with incredibly small forces?

 

[jbriggs444]

as i understand it, if i traveled 1000miles north vs 1000miles south vs traveling 1000mile east or west, , there would be no coriolis effect for the east west travel, whereas due to conservation of angular momentum, going north or south would have a proportional effect on the change of of the direction of travel.

The component of the Coriolis force in the horizontal direction is the same for east-west travel as for north-south travel.

 

[jbriggs444]

the logic I am using is if you are in the center of a merry-go-round, and you step toward the outside, you get shot out to the outer circumference. does the same thing happen on the Earth but with incredibly small forces?

Centrifugal force is why the ground at the equator is higher (farther from the center of the earth) than the ground at the poles. By exactly enough so that the net of centrifugal force versus the slope of the land balances out. Everything looks level and the ground forms an equipotential surface.

 

[A.T.]

there would be no coriolis effect for the east west travel

That's not correct. E-W travel causes Coriolis acceleration, which is mostly S-N near the poles, and mostly UP-DOWN near the equator:

https://en.wikipedia.org/wiki/Coriolis_force#Rotating_sphere

The UP-DOWN component is sometimes called Eötvös effect, but it's really just a component of the Coriolis force. For airplane fuel consumption this is the most relevant Coriolis component. But still quite small, compared to other factors like jetstreams.

 

[zanick]

That's not correct. E-W travel causes Coriolis acceleration, which is mostly S-N near the poles, and mostly UP-DOWN near the equator:

https://en.wikipedia.org/wiki/Coriolis_force#Rotating_sphere

The UP-DOWN component is sometimes called Eötvös effect, but it's really just a component of the Coriolis force. For airplane fuel consumption this is the most relevant Coriolis component. But still quite small, compared to other factors like jetstreams.

Sounds like the Coriolis effect going north bound has to do with conservation of angular momentum, and not centrifugal force. when going east to west, you say, Coriolis is higher? how is that possible? there is no change in angular momentum . the plane is now traveling faster around the equator as viewed by the sun going east than it would be going west.. that increase in velocity would make the plane lighter flying eastbound and heavier flying westbound due to going west bound almost negating centrifugal forces if the plane could fly 1000mph. Is this a correct way of looking at the problem?

In thinking again about it, going north to south would create a similar change in weight of the aircraft as it went from the equator to the poles , however seemingly less because the speeds going west to east at the equator would be the Earth rotational speed plus the planes speed. if the plane was traveling 1000mph due to the Earth rotation, it would also be adding to this, via its air speed of say, 1000mph. (2000mph net velocity to create centrifugal force) the north south would the air speed, of 1000mph, only while at the equator and going lower as it traveled to the poles. so it would have a north bound velocity of 1000mph, and a spinning velocity of 1000mph, so is that equal to the east west forces? :)

 

[A.T.]

Coriolis effect going north bound has to do with conservation of angular momentum, and not centrifugal force.

Coriolis is simply the velocity dependent term, while centrifugal is the position dependent term, of the total inertial force in the rotating frame:

https://en.wikipedia.org/wiki/Rotating_reference_frame#Newton.27s_second_law_in_the_two_frames

 

[zanick]

Coriolis is simply the velocity dependent term, while centrifugal is the position dependent term, of the total inertial force in the rotating frame:

https://en.wikipedia.org/wiki/Rotating_reference_frame#Newton.27s_second_law_in_the_two_frames

centrifugal would be position centric and velocity centric.

Coriolis is simply the velocity dependent term, while centrifugal is the position dependent term, of the total inertial force in the rotating frame:

https://en.wikipedia.org/wiki/Rotating_reference_frame#Newton.27s_second_law_in_the_two_frames

So, it seems that the coriolis effect is not a factor when traveling , east to west.. is it not? centrifugal force would go up and it is based on angular velocity, as well as radius.

 

[A.T.]

centrifugal would be position centric and velocity centric.

Look at the wiki link I gave you. The centrifugal term depends on the position vector (r) while the Coriolis term depends on the velocity vector (v). This is simply a convention. Nothing deep behind this.

So, it seems that the coriolis effect is not a factor when traveling , east to west.. is it not?

Wrong, look at the formula. Apply the cross product for this case.

 

[zanick]

Look at the wiki link I gave you. The centrifugal term depends on the position vector (r) while the Coriolis term depends on the velocity vector (v). This is simply a convention. Nothing deep behind this.Wrong, look at the formula. Apply the cross product for this case.

I don't see it.. can you give me a brief example of why? my logic is that the only thing effecting a west to east travel at some velocity, would be a change in centrifugal force.

 

[A.T.]

my logic is that the only thing effecting a west to east travel at some velocity, would be a change in centrifugal force.

It's not about logic, but about naming conventions. What you call "change in centrifugal force" is called the radial component of the Coriolis force.

 

[zanick]

It's not about logic, but about naming conventions. What you call "change in centrifugal force" is called the radial component of the Coriolis force.

Got it.. so the faster the plane goes the lighter it becomes . so, if traveling north from the equator. the faster it goes the more eastward it is deflected and the lighter it becomes. are the forces adititve , just after leaving the equator going north? say, a .03% reduction in gravity due to being at the equator and a .03% (ball park) reduction due to the speed traveling north... say also 1000mph. ??

 

[A.T.]

just after leaving the equator going north?

Here the Coriolis force is zero, because velocity is parallel to the frame's rotation axis. Do you understand the cross product?