Is square inverse force orbit possible in reality, and mathematically?

Roughly speaking the components of the basis vectors in polar coordinates change depending on where you are so we, they are not like coordinates such as Cartesian that are constant. So they are taken into account when you take time derivatives. Do you understand this?

This is a good demonstration of what I mean as I don't want to fill the thread with calculations.
http://www.iitg.ac.in/physics/fac/saurabh/ph101/Lecture2-SBasu.pdf

When taking time derivatives of vector valued functions in polar coordinates (or any other coordinate system) you need to take time derivatives of the basis unit vectors and apply the chain rule, the time derivative of the theta unit vector being the negative of the radial unit vector times theta dot and the derivative of the radial unit vector being the positive theta unit vector times theta dot.  The author of the paper correctly took said time derivatives.

I have stated the definition of uniform circular motion, which you stated your system to be undergoing.

The point is that a derivation of the correct equations for a uniform circular (or elliptical) orbit are not supported by a single gravity force field.

That is a simply lack of understanding of newtons second law, coordinate systems and rates of change so I have attached further readings for you.

The further readings do not bear out anything that you have claimed, to my estimation.  What do you believe those readings say that contradict the derivations the paper which you claimed to be incorrect?

You are talking about derivatives of newtons second law, which is in fact talking about newtons second law...

If you want to be pedantic, talking about something that is related to something else is not really the same as talking about the thing that the thing you are talking about is related to. 

Oh? And if the force does vary with time, I suppose the object will what, move away from it? move crosswise? not move at all?
Seriously, you're causing me to question my sanity and your sanity. One of them is breached and I can't tell which one. But I got a hunch it's not entirely mine.

It will move in the direction the equation says it will.

Now wait a second. If you say that it's distance and mass dependent force, fine.
But just because a force changes depending on other parameters doesn't mean it's not a force.
Even the force of a spring changes with distance. Lots of forces depend on other things for their exact value and direction.

I don't think you understand what is at issue here.

Your blunders: First calculating acceleration for the heavy mass. And telling *ME* I was all wet. Doh? The wrong mass? That's like this guy that is hired to drive a thousand miles. When he arrives, he's late. It took him 3 times as long as it should have. Boss says "What happend?" He says "Oh not much, I got a little lost."  Going 1000 miles in the wrong direction then having to drive 2000 miles back to the correct destination is not a small blunder.

And then I'd asked you what the velocity should be after 10 seconds, and instead you calculated the distance after first 6 minutes, but didn't tell me what the units for the number was.
Then you assumed some absurd radius - which you can't pin on me because you knew the info was missing - don't you know that if you're missing an input parameter you need to ask? You can't just guess at it! That's a blunder for sure. Especially an input which affects it by the square of the distance!

And now you're stuck on trying to contort gravity into something other than a force.

Not to mention that you told me my experiment looked wrong before you even did any math at all.

Can you see why I would not be inclined to place a strong confidence in information you provide to me?

I will admit to rushing my calculation:  I have things to do other than responding to forum posts, but I take responsibility for the errors I made plugging in the values into the equations.  I have no interest in further discussing this tedious matter.

All assertions made without evidence or rational argumentation will be ignored.

Do you ignore yourself?

I don't think you understand what is at issue here.

One of us doesn't. At least one of us doesn't. Maybe both of us doesn't. 

I will admit to rushing my calculation:  I have things to do other than responding to forum posts,

That of course is your choice. You were also the one that chose to dismiss my experiment without even doing calculations and then with doing them with absurd assumed input values.

but I take responsibility for the errors I made plugging in the values into the equations.

Fair enough. And everybody makes blunders. But you asked, and I explained why I cannot place a high level of confidence in your assertions.
I know you meant no harm. But still, you made the choice to respond and dismiss my experiment's results without actually doing the math to show it.
And you chose to do so when you didn't have time. Which are all your choices to make.
However, knowing all these things, I must be aware that when you tell me that I'm wrong, I must consider that you may not have had time to actually do the math yet are the kind of person to call someone out without actually proving first that you're right.
So yes, it's all your choices to make, but it's my choice to consider you a low quality source of data as a result of your choices to provide low quality data.
I'm sure you're a wonderful, kind, generous person - but that doesn't mean you're a source of high quality information.

I have no interest in further discussing this tedious matter.

Fair enough. Just so happens I did have just such an interest. The accuracy of information coming to me is pretty much as important as the information itself.

So now, I guess all that's left is to figure out a way to test an orbit in a controlled environment that is based off of a square inverse attraction.
Then maybe we can see if math prevents it from working.

Quote from: Tom Foolery on March 17, 2019, 08:38:18 PM

Had another idea:
Take a large pane of glass, put a large magnet under it, and a large weight with a strong magnet on top, with ice cube feet.
Since magnetic attractive force also reduces with the square of the distance, and ice cubes slide real nicely on wet glass, we should be able to see of the free moving weight can orbit on any of the traditional orbital paths.

That should essentially be good, right? I mean there's mass, there's square inverse attractive relationship, and there's freedom of movement.

It should be hypothetically possible to set it up, but I'm not sure on how stable it is, because magnets, while you can approximate them as a collection of magnetic monopoles, don't actually follow the inverse square law as they are dipoles. There is also the issue of the magnet trying to tip over.

Let's say we use magnets shaped like hockey pucks with north on top and south on the bottom. Except bigger, taller, and meaner.

The "sun" would have north up, and the "earth" would have north down,  so they would have essentially the attractive force all the way around.
They would also not want to flip over particularly.

Actually, we could even test with moon and earth. Have them both riding around on ice cubes.

By having the weights identical, we could simulate a binary star.

The point is that a derivation of the correct equations for a uniform circular (or elliptical) orbit are not supported by a single gravity force field.

Again, you have repeatedly asserted this, but have provided nothing to back it up.
Are you planning on backing it up any time soon, or telling us what kind of motion it will cause, preferably with the math backing it up?

So far all you have done is repeatedly asserted that it can't happen, and pathetic assertions like that wont convince anyone.

It will move in the direction the equation says it will.

So it will move in a circular path as the equations indicate?
So your entire argument is pure BS?

If you are going to claim otherwise, then show just what you think the equations will indicate.
Provide a set of parametric equations to describe the position of the object over time.
Can you do that?

If not, you should start ignoring yourself, because so far all you have done is made assertions without any evidence or rational arguments to back them up.

Let's say we use magnets shaped like hockey pucks with north on top and south on the bottom. Except bigger, taller, and meaner.
The "sun" would have north up, and the "earth" would have north down,  so they would have essentially the attractive force all the way around.
They would also not want to flip over particularly.

I was thinking along those lines, with one being under a sheet of glass (perhaps a glass table) and the other above, but your way is better, with them both above as that should negate the turning force.
However, I'm still not sure exactly what the force will be (i.e. what function) or how difficult it will be to set up a stable orbit.

It’s not a analogy.

You are talking about these higher orders fundamentally as a result of the circular motion, so it must hold for all rotating systems.

In a rigid system, any additional forces would introduce coresponding stresses and strains.

All of which would be a surprise to tens if not hundreds of thousands of engineers who carefully design such systems.

Let's consider a rigidly rotating object and ignore any friction that might be acting upon the object.  Due to conservation of angular momentum the body will spin with a constant frequency forever.  This is not the same as the proposed two-body orbit:  in the rigidly rotating body all of the internal forces that keep a given particle at the extremities are rotating along with it due to the initial torque imparted to the system.  In the two-body system there are no internal forces keeping a planet and its satellite connected together:  there is a conservative, non-rotating vector field that is pulling the body towards the center and only the center.  To quote from the paper I quoted from earlier:

Hence, the requirement of higher derivatives of circular motion is not mathematical alone, but has a clear physical
application. It necessarily means that to generate circular motion, one has to generate not only a central acceleration,
but also all the rotational forces. The track in case of a train, the friction in case of a road, the hand in case of
slingshot or the axle in case of a wheel hence bear the combined effect of the acceleration and higher derivatives,
acting both toward and away from the center, along and opposite to the velocity.
How about the simple experiment done in all physics labs across the world to reproduce Huygens’ relation F = mg =Mv2/r? The diagrams usually show something like this: 

"Does this show that rotational motion involves only acceleration, since it is balanced by acceleration due to gravity?  On the contrary, it merely shows that gravity is able to balance one of the components of rotational motion, while the axis (hand or motor) generates all the other components. It is the component one chooses to measure that oneends up measuring in the experiment. If other derivatives are generated, by hanging variable weights and so on, thenthe values of other rotational forces can also be demonstrated – something not pursued in the teaching field.  Hence, the existence of rotational forces – some source of the entire rotation – is essential for a full description of circular motion even under normal terrestrial conditions. Any error in their application for terrestrial phenomena prevents them from predicting celestial movements"

https://reciprocalsystem.org/PDFa/Celestial%20Dynamics%20and%20Rotational%20Forces%20(Vijaya,%20Gopi%20Krishna).pdf

Oh, right.

So your paper does say it applies to rotating machinery. Only that everyone is doing it wrong.

As I say, this should be news to anyone designing things like turbine blades, where forces are very carefully calculated.

Oh, right.

So your paper does say it applies to rotating machinery. Only that everyone is doing it wrong.

As I say, this should be news to anyone designing things like turbine blades, where forces are very carefully calculated.

Those higher order forces in a rigidly rotating do not need to be considered because the entire object is rotating together.  If you were to isolate a single particle and abstract away the internal forces keeping the body together then you would need to consider the higher order forces but throughout the whole body all of those higher order forces cancel out which is why a body rotating at a constant angular velocity is said to be in equilibrium.  If the body is not sufficiently rigid then spinning it around will cause it to fall apart because the internal forces are not sufficient to balance the higher order forces due to rotation.

Quote from: Unconvinced on March 19, 2019, 09:29:17 AM

Oh, right.

So your paper does say it applies to rotating machinery. Only that everyone is doing it wrong.

As I say, this should be news to anyone designing things like turbine blades, where forces are very carefully calculated.

Those higher order forces in a rigidly rotating do not need to be considered because the entire object is rotating together.  If you were to isolate a single particle and abstract away the internal forces keeping the body together then you would need to consider the higher order forces but throughout the whole body all of those higher order forces cancel out which is why a body rotating at a constant angular velocity is said to be in equilibrium.  If the body is not sufficiently rigid then spinning it around will cause it to fall apart because the internal forces are not sufficient to balance the higher order forces due to rotation.

Abstracting the forces at every point is literally what engineers do.  Because we work out what the stresses and strains are everywhere (or at least the highest/most critical points).

This is not the same as the proposed two-body orbit:  in the rigidly rotating body all of the internal forces that keep a given particle at the extremities are rotating along with it due to the initial torque imparted to the system.

Nope. Nothing to do with any torque. Assuming it is rotating at a constant speed, (to make it an analogy to the orbit) then the reason the forces acting on any particular point changes is because the system moves.
There are no magical higher order "forces" magically holding it together.

To quote from the paper I quoted from earlier:

Stop quoting the blog. It isn't helping your case. It is just repeating the same false baseless assertions.

Now can you do what has repeatedly been asked of you?

You claim that this kind of force cannot produce uniform circular motion.
If it doesn't WHAT DOES IT PRODUCE?

Can you provide a set of parametric equations to show the position of the object at a given time?
So far all the evidence indicates that the object should follow a circular path.

So far all we have is your baseless assertion that some magical, non-existent higher order "forces" are magically required.

Quote

Are you saying it would be impossible for something to orbit without either crashing into the center mass or flying off away from it

Yes, I am saying that.

So how much could something orbit? I mean it could orbit for what, a few circles? half a circle? A thousand circles?

What do you think of the funnel wishing wells?

Do the pennies suffer from a roughly square inverse force?

Obviously they do have drag, and the orbit eventually decays, but I still think it's relevant.

Can you see why I would not be inclined to place a strong confidence in information you provide to me?

yep

Quote from: GDg43SA on March 18, 2019, 09:57:42 AM

Quote

Are you saying it would be impossible for something to orbit without either crashing into the center mass or flying off away from it

Yes, I am saying that.

So how much could something orbit? I mean it could orbit for what, a few circles? half a circle? A thousand circles?

It depends upon the initial tangential velocity imparted upon the mass.

What do you think of the funnel wishing wells?

Do the pennies suffer from a roughly square inverse force?

No, the force field due to the rotation of the water has a curl.  Inverse-square fields are conservative and thus irrotational.

Obviously they do have drag, and the orbit eventually decays, but I still think it's relevant.

It isn't relevant at all because it isn't an inverse square field, nor does it demonstrate perpetual orbits since the radii of the circles described by the pennies continuously declines. 

Quote from: Tom Foolery on March 19, 2019, 10:16:46 PM

Quote from: GDg43SA on March 18, 2019, 09:57:42 AM

Quote

Are you saying it would be impossible for something to orbit without either crashing into the center mass or flying off away from it

Yes, I am saying that.

So how much could something orbit? I mean it could orbit for what, a few circles? half a circle? A thousand circles?

It depends upon the initial tangential velocity imparted upon the mass.

You're still side-stepping my question. I asked about orbits existing and you're talking about whether they start.
Are you saying that orbits could exist if they were started, but they just can't be started?
Will you grant that if started correctly, an orbit, if it had significantly small energy losses, could run for a thousand orbits? What about 10,000?

Quote

What do you think of the funnel wishing wells?

Do the pennies suffer from a roughly square inverse force?

No, the force field due to the rotation of the water has a curl.  Inverse-square fields are conservative and thus irrotational.

Umm, I don't believe there is any water in these funnel wishing wells. A wishing well is something you throw money into when you make a wish, maybe it has water, maybe it hasn't.
The funnel wishing wells into which you roll coins would not have water in the funnel because that would slow down the coins and ruin the show.

Quote

Obviously they do have drag, and the orbit eventually decays, but I still think it's relevant.

It isn't relevant at all because it isn't an inverse square

Well now your confidence is charming. But please think about it. Thinking kind of hurts the first time you do it but once you get the hang of it there's fun to be had.
Remember that the coin has to stay perpendicular to the surface of the funnel or it will fall over.
Notice how it's nearly flat on it's side as it goes down the hole?
That proves that the outward force it's experiencing is many many times 1g.
The force clearly has a non-linear relationship.
Maybe someone more math skills can calculate the outward force as it relates to the diameter of the orbit.
But it is extremely clear that the force doesn't go up just linearly. It goes up much faster than that.

field,

I didn't say a field, I said FORCE. Do you know the difference between a field and a force? Look it up. It's important to this discussion.

nor does it demonstrate perpetual orbits since the radii of the circles described by the pennies continuously declines.

Yes but do you know *why* the radii of the circles is continuously declining?  Because it's not in a vacuum, and energy is lost as vibration and heat as the pennies move.
If it was in a vacuum and the pennies had perfectly smooth edges and the funnel was perfectly solid so no energy was lost as they went around, they would orbit in the same place forever.

I did a little googling, please check out the following link. They provide the math to make an ideal vortex funnel, etc.

They also show that by moving the vortex funnel to pump energy into the coin you can even increase the diameter of the orbit.

https://www.spiralwishingwells.com/guide/physics.html

I'm thinking that maybe a funnel of the right shape might also be a quite meaningful method to prove whether an orbit can happen.

If the surface angle of the funnel is correct, the outward force on the coin (or ball) would in fact be a function of the inverse of the square of the distance.

Umm, I don't believe there is any water in these funnel wishing wells. A wishing well is something you throw money into when you make a wish, maybe it has water, maybe it hasn't.
The funnel wishing wells into which you roll coins would not have water in the funnel because that would slow down the coins and ruin the show.

You're right, my apologies.  I didn't watch the video, I assumed it was some kind of whirlpool.  This still is not an example of an orbit (as mentioned it isn't really an orbit at all because of the constantly diminishing radius) due to an inverse square field, the circular motion is due to the curvature of the surface.  The fact that you see this as analogous to Newtonian orbits kind of destroys any credibility you might have had.

As for the rest of your post, you have a lot of reading up to do on math and physics before we can continue this conversation, your ignorance is kind of astounding.  I suggest enrolling in a community college and taking some introductory calculus and physics classes because I don't have time to provide you an education.  You also come across as a passive aggressive tool, so I think I'll bid you a fond adieu.

I was thinking along those lines, with one being under a sheet of glass (perhaps a glass table) and the other above, but your way is better, with them both above as that should negate the turning force.
However, I'm still not sure exactly what the force will be (i.e. what function) or how difficult it will be to set up a stable orbit.

Yeah. But I did a quick google search, and it indicted that magnetic force was square inverse. And it sure feels like it when you play with magnets.
It would definitely be interesting to try. And with calipers and a scale the force to distance relationship could be measured.

I'm also really interested in the vortex funnel shape. Or whatever shape it takes to be inverse square.

And I guess force will be a direct function of angle, and so if the angle at any given distance from center is correct, it would have to be a square inverse force.

Looking at the chart here: https://www.spiralwishingwells.com/guide/physics.html

It looks like the outward force will basically be g/x^2.

So at a distance of 1, it's experiencing a 1g outward force.
At a distance of 2, it's g/4 force.
At distance 1/2, i's g*4.

Blah, I just looked at the PDF linked on the above mentioned page, and yes, it looks like this vortex funnel is exactly how they model gravitational orbits:

https://www.spiralwishingwells.com/guide/Gravity_Wells_Mirenberg.pdf

Well I guess orbits are possible. 

I still want to try the magnets on ice on glass, because it would allow multiple objects to orbit eachother.

You could literally have a moon orbiting an earth, and you could pump the moon by pumping the earth.

Quote

Umm, I don't believe there is any water in these funnel wishing wells. A wishing well is something you throw money into when you make a wish, maybe it has water, maybe it hasn't.
The funnel wishing wells into which you roll coins would not have water in the funnel because that would slow down the coins and ruin the show.

You're right, my apologies.  I didn't watch the video, I assumed it was some kind of whirlpool.

See? You dismissed my argument without even watching the video, and as a result, gave an absolutely absurd and useless reply. "The water, he says." Another perfect case in point why you're a low quality information source.

(as mentioned it isn't really an orbit at all because of the constantly diminishing radius)

Do you really believe that? Are you not aware of the fact that the constantly diminishing radius is *only* because of the loss of energy?
Do you really believe that the radius would continue to shrink if there was no loss of energy?
Watch this video and you can see that when energy is added, the radius increases. It stands to reason that with no loss in energy, the radius would not change.
You're seriously confused if you think there's no loss of energy in these examples.

This still is not an example of an orbit due to an inverse square field

As best as I can tell, it's a matter of simple math to see that the outward force experienced by the coin is equal to g/x^2 -- which is square inverse.
But if you think I'm wrong about that, then please show me.

The fact that you see this as analogous to Newtonian orbits kind of destroys any credibility you might have had.

Well I think the cows are still out on that one.
Check out this paper: it turns out I'm not the first one to see this analogue: https://www.spiralwishingwells.com/guide/Gravity_Wells_Mirenberg.pdf
If it turns out that the angle of the surface is such that it produces on the coin a force equal to (or very close to) g/x^2, then I'm quite right to see this as analogous to Newtonian orbits.
But I'm so glad you pointed this out because if it turns out that the force on the coin is analogous Newtonion orbits, then we will know you never had the credibility you thought.

As for the rest of your post, you have a lot of reading up to do on math and physics before we can continue this conversation,

We both might, my friend. You're digging yourself in deeper every post you make.

your ignorance is kind of astounding.

Again, thank you for saying that. Sometimes I wonder about it myself. But I think you and I make good company in that regard in this instance.
I cannot believe you're not acknowledging that the coin's radius is reducing because of loss of energy.
Totally astounding.

I suggest enrolling in a community college and taking some introductory calculus and physics classes

Wouldn't I need a HS diploma or a GED to do that? I haven't got either of those.

because I don't have time to provide you an education.

I constantly wonder why you're always whining about your lack of time. Did anybody make you read my post? Did anyone make you respond to my thread? Nobody.
What you do with your time is entirely your choice. Unless you're a paid actor hired to come here and support flat earth, which, come to think of it, would explain why you make so many embarrassingly obvious blunders and yet you're so persistent.
Why go on complaining to others about the choices you make for yourself?

You also come across as a passive aggressive tool, so I think I'll bid you a fond adieu.

It's so nice of you to introduce yourself. I feel like I already know you.

It sure looks to me like Newtonian orbits can work, and that the vortex funnel is a fine way to demonstrate it.

It depends upon the initial tangential velocity imparted upon the mass.

So you are saying it can remain in orbit indefinitely?

Like I said, provide a set of parametric equations to describe its motion.
If you can't do that, don't claim it can't provide an orbit, as that is what the math shows it can and will do.

This still is not an example of an orbit (as mentioned it isn't really an orbit at all because of the constantly diminishing radius) due to an inverse square field, the circular motion is due to the curvature of the surface.

The field aspect doesn't matter.
What is important is the force.
What kind of force do you think it imparts?

As for the rest of your post, you have a lot of reading up to do on math and physics before we can continue this conversation, your ignorance is kind of astounding.  I suggest enrolling in a community college and taking some introductory calculus and physics classes because I don't have time to provide you an education.  You also come across as a passive aggressive tool, so I think I'll bid you a fond adieu.

You are the one who has come in and made a bunch of ignorant, baseless assertions showing a complete lack of knowledge regarding physics, even very simple physics. When pushed you then insult people and ignore them.
Perhaps you should follow your own advice and enrol, but I would recommend starting with high school as you clearly lack basic high school physics knowledge.
One thing you will learn there is how Newtonian mechanics work and how an inwards pointing force can produce circular motion, even an inverse square one.

If you can't defend your claims don't make them.

As for the rest of your post, you have a lot of reading up to do on math and physics before we can continue this conversation, your ignorance is kind of astounding.

As is yours and in case you hadn't heard the good news:
Any attractive central force that varies as k.Rⁿ can support stable orbits for n ≥ -2, where stability is taken in the sense that any small perturbation will cause only a bounded change in the orbit.

Some particular initial conditions can lead to non-periodic orbits, such as the hyperbolic orbits of non-periodic comets and "space-objects" that enter then leave the solar system.

Yeah. But I did a quick google search, and it indicted that magnetic force was square inverse. And it sure feels like it when you play with magnets.
It would definitely be interesting to try. And with calipers and a scale the force to distance relationship could be measured.

A magnetic monopole would produce an inverse square relationship, but magnets are dipoles, which means it is more complicated.
A simple approach is a combination of 4 monopoles, but I haven't done the math for them sideways. I know that if they are end on it is inversely proportional to the distance to the 4th power (for large distances). I am assuming it would be similar for sideways interactions but might be wrong.
The "unsimplified" version, assuming an infinitely thin rod, would have a force proportional to 1/r²-r/((r²+l²)^3/2), assuming l<<r this simplifies to being proportional to 1/r⁴

I'm also really interested in the vortex funnel shape. Or whatever shape it takes to be inverse square.

Well there would be 2 approaches. One is guess and check, the other is to try and work backwards.
So lets try going backwards.

In order to not have it fall (i.e. if no energy is lost and it remains in a stable position) and assuming perfect rolling (i.e. no sliding) then the normal force will provide a force to counter gravity and a force to push it in.
This works just like a right angle triangle.
The inwards force we want to be proportional to 1/x², while the upwards force remains as constant, and thus is proportional to 1.
The direction of these forces produces a slope of -x²/1. (for positive x)
This is normal (perpendicular) to the slope we want.
The product of the slopes of 2 perpendicular lines is -1.
That means the slope of our line, multiplied by -x²/1 must equal -1, which means our line must have a slope of 1/x²
To find the actual function we just need to integrate this, which gives us -1/x. As this needs to be symmetrical we take the absolute value of x, so our function to describe this vortex has a height that is proportional to -1/|x|.

Any attractive central force that varies as k.Rⁿ can support stable orbits for n ≥ -2, where stability is taken in the sense that any small perturbation will cause only a bounded change in the orbit.

Just to confirm, this means things like F=-r¹⁰ will work by -1/r¹⁰ wont?

A magnetic monopole would produce an inverse square relationship, but magnets are dipoles, which means it is more complicated.
A simple approach is a combination of 4 monopoles, but I haven't done the math for them sideways. I know that if they are end on it is inversely proportional to the distance to the 4th power (for large distances). I am assuming it would be similar for sideways interactions but might be wrong.
The "unsimplified" version, assuming an infinitely thin rod, would have a force proportional to 1/r²-r/((r²+l²)^3/2), assuming l<<r this simplifies to being proportional to 1/r⁴

Interesting... but I'm a tad confused about what a monopole magnet is, and what the infinitely thing rod refers to, unless perhaps that's the shape of the magnets needed in order to provide the ideal force as a function of distance.

Quote from: Tom Foolery on March 20, 2019, 12:04:07 PM

I'm also really interested in the vortex funnel shape. Or whatever shape it takes to be inverse square.

Well there would be 2 approaches. One is guess and check, the other is to try and work backwards.
So lets try going backwards.

In order to not have it fall (i.e. if no energy is lost and it remains in a stable position) and assuming perfect rolling (i.e. no sliding) then the normal force will provide a force to counter gravity and a force to push it in.
This works just like a right angle triangle.
The inwards force we want to be proportional to 1/x², while the upwards force remains as constant, and thus is proportional to 1.
The direction of these forces produces a slope of -x²/1. (for positive x)
This is normal (perpendicular) to the slope we want.
The product of the slopes of 2 perpendicular lines is -1.
That means the slope of our line, multiplied by -x²/1 must equal -1, which means our line must have a slope of 1/x²
To find the actual function we just need to integrate this, which gives us -1/x. As this needs to be symmetrical we take the absolute value of x, so our function to describe this vortex has a height that is proportional to -1/|x|.

Thanks!
If I understand that all correctly (Math is not my strong suit)  the function needed to describe an ideal gravity well would be -1/x which looks to be just the same as these people use to make their penny vortex funnels: https://www.spiralwishingwells.com/guide/physics.html

Now for a zero diameter  ball rolling down, it would be ideal, but the diameter of the ball would move in the center of gravity for the orbiting ball so it would be following a different virtual surface who's orbit radius would be smaller. However, the angle of the virtual surface would be identical, so maybe it's still a square inverse relationship.

But I think we are concluding that a small marble going down a large -1/x vortex funnel is in fact a very good analogue for Newtonian orbits?

Interesting... but I'm a tad confused about what a monopole magnet is, and what the infinitely thing rod refers to, unless perhaps that's the shape of the magnets needed in order to provide the ideal force as a function of distance.

As you hopefully know, all magnets are dipoles, they always have a north pole and a south pole. However modelling them can be tricky. One way to do so is model them as 2 separate monopoles, so you have a north monopole and a south monopole.
A monopole works in a similar way to an electric charge. They produce a magnetic field that is inversely proportional to the distance squared. Two north monopoles repel each other with a force inversely proportional to the distance squared and a north and south monopole would attract each other with a force inversely proportional to the distance squared.

Assuming the magnet is a cylinder, we can treat it as a bunch of north monopoles over the north end, and a bunch of south monopoles over the south end.
But this then requires integrating over a disc which is a pain. So we can simplify by grouping it all into 1 monopole for each end, which would correspond to an infinitely thin rod, and would be a reasonably approximation for when the radius is negligible compared to the distances involved.

So yes, the infinitely thin rod and monopoles are to get an ideal force as a function of distance.

Now for a zero diameter  ball rolling down, it would be ideal, but the diameter of the ball would move in the center of gravity for the orbiting ball so it would be following a different virtual surface who's orbit radius would be smaller. However, the angle of the virtual surface would be identical, so maybe it's still a square inverse relationship.

But I think we are concluding that a small marble going down a large -1/x vortex funnel is in fact a very good analogue for Newtonian orbits?

Yes, it would still be an inverse square relationship, but the distance is to the point of contact, not to the centre of mass.
The big issue for their approximation is friction, otherwise it would work wonderfully and that kind of vortex can be used to represent the field from such an inverse square force, not the field directly, but the effect it has on objects.

Good friend GDg43SA,

Regarding vortex funnels:

(as mentioned it isn't really an orbit at all because of the constantly diminishing radius) ...

I would still like to hear you explain yourself here. It sure looks like you're saying that the coins go around with a diminishing radius and therefore it's not really an orbit at all, even though the video shows that they go down due to loss of energy and that by adding energy they go back up.

Are you not aware that the coins go down because they are losing energy, and that by adding energy they go back up, or by maintaining energy, they maintain radius?

The fact that you see this as analogous to Newtonian orbits kind of destroys any credibility you might have had.

No, my friend, the fact that you didn't see this as analogous to Newtonian orbits completely destroyed any credibility you would have had if you hadn't created an alt just to say it.

(Hey. I don't have anything against alts. I'm not complaining. But look. You signed up the very day I created the topic "I proved gravity."
Up to this point, you only publicly posted there and in this gravity related thread. Noplace else.
Then a couple days ago you says "bye" and stops posting publicly entirely.
You had repeatedly threatened to permban people, you insulted people's intelligence, and made numerous factual blunders yourself.
That's all easy to do from a temporary alternate account purpose created when you want to make arguments and you don't have the credibility to do it from your main account.
I mention this because I find it all very relevant to our discussion - it tells me that you already knew going in that it was going to be a losing proposition for you.
When my opponent in a debate doesn't even believe he has a good case, then I should not be surprised when that's what follows - he makes a terrible case!)

Anyway, since I'm sure you're still around and still reading this and may even reply (Perhaps as your real noble self!), I'm hoping you can see that it was in this case you who lost all credibility you might have had by not realizing that the vortex penny funnel is a very meaningful analogue of Newtonian orbits, and that the coins have a shrinking orbit because (and only when) they are losing energy. 
Just like real planets orbiting!

your ignorance is kind of astounding.

Yeah, yeah, we know. But I think you can take home the gold medal on that one today.

But thank you for helping me to see that not only do Newtonian orbits work, they are demonstrated every day by common science fair demonstrations and money raising machines!
(Can you call a single piece of plastic with no moving parts a "machine?")

...
A monopole works in a similar way to an electric charge. They produce a magnetic field that is inversely proportional to the distance squared. Two north monopoles repel each other with a force inversely proportional to the distance squared and a north and south monopole would attract each other with a force inversely proportional to the distance squared.
...

JackBlack,

So I've been thinking and doing a little more reading (but not much LOL).
Am I understanding correctly that two magnets with one's north pointing at the other's south will have an inverse square force to distance relationship while if they are pointing one north up and the other north down and parallel, the relationship will be the inverse of the 4th power?

If that is the case, could we have the magnets oriented such that they were pointing north to south axially aligned, but have them orbiting in double tidal lock mode, so we have the square inverse relationship again?

of course that would preclude a 3 body orbit. But if it allowed two masses to orbit eachother with a square inverse force relationship it would be very interesting.

Am I understanding correctly that two magnets with one's north pointing at the other's south will have an inverse square force to distance relationship while if they are pointing one north up and the other north down and parallel, the relationship will be the inverse of the 4th power?

The individual poles will have an inverse square relationship, however the overall force will be complex, but can be approximated as 1/x⁴. I haven't checked for all cases, but this holds for 2 axially aligned magnets (i.e. their poles pointing towards each other), and for 2 radially aligned magnets (i.e. their poles both pointing up and down, with them at the same height).

I'm pretty sure that because they are dipoles they will always hold to the 1/x⁴ relationship as an approximation.

Quote from: Tom Foolery on March 24, 2019, 08:44:24 AM

Am I understanding correctly that two magnets with one's north pointing at the other's south will have an inverse square force to distance relationship while if they are pointing one north up and the other north down and parallel, the relationship will be the inverse of the 4th power?

The individual poles will have an inverse square relationship, however the overall force will be complex, but can be approximated as 1/x⁴. I haven't checked for all cases, but this holds for 2 axially aligned magnets (i.e. their poles pointing towards each other), and for 2 radially aligned magnets (i.e. their poles both pointing up and down, with them at the same height).

I'm pretty sure that because they are dipoles they will always hold to the 1/x⁴ relationship as an approximation.

You're right again Jack Black!

I taped a small magnet to a small digital scale and used another magnet and measured the force to distance relationship.

I tried axial and radial orientation, as well as thin ring magnets and thick (4 thin ones stacked) - 4 tests.
Attraction and repulsion seemed to be the same (but naturally opposite) for any given magnet thickness and orientation.

I then put my measured results for one of the tests into a spreadsheet and graphed it along side a predicted force of 209.7152grams/inches^4 and it came out surprisingly close considering my crude method of measuring force and distance.



So having ruled out magnet to magnet attraction as functioning by the square inverse relationship, what would work by it? a magnet to steel ball force?

This also raises an interesting question, as to whether an orbit would be possible with a relation ship of the inverse of the 4th power.

If that works, then surely an orbit to the inverse square force would work also?

Frankly, I should have tested steel against magnet but my setup was a pain to use I wasn't really thinking I guess. But I could set it up again easy enough.

So having ruled out magnet to magnet attraction as functioning by the square inverse relationship, what would work by it? a magnet to steel ball force?

I was thinking electrical charges.
Find some materials that are good at getting statically charged (like perspex) and rub them enough to charge them. Then see if you can have them orbit.
The issue is that the force will likely be quite weak.

As for the magnet to steel, that will typically be worse than magnet to magnet.
For the attraction to happen, a magnetic field is established inside the steel. This would then act just like a normal magnet and attract the steel towards the magnet with a force proportional to 1/x⁴. But the induced magnet in the steel is dependent upon the magnetic field it is experiencing. A strong field results in a strong induced magnet. This lowers the force the further out you go. I'm not sure of the exact relationship though.

This also raises an interesting question, as to whether an orbit would be possible with a relation ship of the inverse of the 4th power.
If that works, then surely an orbit to the inverse square force would work also?

Hypothetically, an orbit is possible with any power. The issue is the stability.
If you have a perfectly circular orbit then almost anything can work.
However if you have any perturbation, then it can be unstable.

For a simple case of inverse square forces, within a large range the orbit is stable. If you have too great a velocity for the radius you are at, you will go out further, slowing down in the process and reach a point where you speed becomes too low for the orbit and you start falling back in, speeding up and so on. This produces an elliptical orbit.
For a 1/x⁴ law, you would need to loose much more speed to fall back in, so it might not be stable and going too fast can result in flying away while going to slow can result in crashing down. But I haven't done the math to see how stable it is.

Thanks Jack, all very interesting!

I was thinking electrical charges.

I'll be thinking about that.

Using a static charge would be tricky because static electricity has a tendency to dissipate in an erratic manner.

However I definitely have high voltage supplies so I could provide a constant  high voltage to the orbiting bodies.

In fact, maybe a weight hanging on a 50 foot long super thin wire so the gravitational centering was minimal, then use electrostatic attraction to have it orbit another weight.
Might even be able to do a two body orbit because the lower weight could even be sliding on ice cubes or an air hockey table or something, just maybe.

Electrostatic force is significant. When I was doing Cavendish, I disconnected the grounding wires between my fixed and hanging weights and did a couple control tests to see how electrostatic attraction and electrostatic repulsion affected it. The results were real nice.
I think we probably could come up with a jig to orbit with eletrostatic attractive forces.

These are the electrostatic tests I did below:

Attracting, then repelling: