Foucault's Pendulum Recreation for Physics Project

[lesaid]

Nice idea but difficult to pull off without deflecting the pendulum. I still think a simple escapement would work provided the whole unit is hung underneath a universal joint. Tricky one.

I can't imagine what kind of escapement could work here, given that the pendulum needs to be able to swing equally in any direction without any influence to favour one direction over another. Escapements (so far as I am aware) are only used where the pendulum is swinging in a single plane, such as a clock. Can you elaborate on how it would work?

Edit : just realized what you're talking about - an escapement driving the pendulum in a single plane, but the whole plane, escapement and all, rotating under a universal joint. So - it would essentially be a double pendulum - the lower one just like a clock in principle, and the upper one being a free, non-driven pendulum under a universal joint. That sounds complex to figure out !

Would an analogy be a child on a seat attached to a single rope, that can swing any way, and on a seat that can rotate in any direction. The child keeps the swing going by swinging his legs in the direction he happens to be facing in. This induces a swing in the whole system which is able to rotate any way. Is that a reasonable way of thinking of it?

Need to think about it - not a clue how that kind of system might behave!

 

[SPW]

I can't imagine what kind of escapement could work here, given that the pendulum needs to be able to swing equally in any direction without any influence to favour one direction over another. Escapements (so far as I am aware) are only used where the pendulum is swinging in a single plane, such as a clock. Can you elaborate on how it would work?

Edit : just realized what you're talking about - an escapement driving the pendulum in a single plane, but the whole plane, escapement and all, rotating under a universal joint. So - it would essentially be a double pendulum - the lower one just like a clock in principle, and the upper one being a free, non-driven pendulum under a universal joint. That sounds complex to figure out !

Would an analogy be a child on a seat attached to a single rope, that can swing any way, and on a seat that can rotate in any direction. The child keeps the swing going by swinging his legs in the direction he happens to be facing in. This induces a swing in the whole system which is able to rotate any way. Is that a reasonable way of thinking of it?

Need to think about it - not a clue how that kind of system might behave!

Don't forget that the pendulum does not deviate from a left to right plane, not in some kind of figure of 8 pattern. It's the Earth that rotates, so provided the top pivot rotates freely around 360degrees an escapement, preferably spring wound shouldn't affect the outcome. I've been thinking about the electromagnet, electrostatic or pneumatic approaches, which would potentially deviate the natural course of the pendulum. Any induction, however small, would surely have to be at the top of the system. You've really given me something interesting to ponder anyway. Good luck & let us know what transpires... PS: a grasshopper escapement causes nominal resistance.

 

[lesaid]

Don't forget that the pendulum does not deviate from a left to right plane, not in some kind of figure of 8 pattern. It's the Earth that rotates, so provided the top pivot rotates freely around 360degrees an escapement, preferably spring wound shouldn't affect the outcome. I've been thinking about the electromagnet, electrostatic or pneumatic approaches, which would potentially deviate the natural course of the pendulum. Any induction, however small, would surely have to be at the top of the system. You've really given me something interesting to ponder anyway. Good luck & let us know what transpires... PS: a grasshopper escapement causes nominal resistance.

A question - in the 'escapement' scheme - would the escapement itself form the bob of another pendulum swinging from the universal joint, or would it be held rigid, unable to move in any way other than to rotate about the vertical axis?

If the latter, what does the swinging pendulum achieve? How will the torque at the universal joint (in relation to the rotating frame of reference that is the earth) differ from what it would be if it was simply an inert weight being suspended - such as a heavy horizontal bar - under the universal joint?

If it does swing, I'm not sure how to start analysing the likely motion!

This is indeed making me think - and I'll let you know how the project develops.

 

[SPW]

A question - in the 'escapement' scheme - would the escapement itself form the bob of another pendulum swinging from the universal joint, or would it be held rigid, unable to move in any way other than to rotate about the vertical axis?

If the latter, what does the swinging pendulum achieve? How will the torque at the universal joint (in relation to the rotating frame of reference that is the earth) differ from what it would be if it was simply an inert weight being suspended - such as a heavy horizontal bar - under the universal joint?

If it does swing, I'm not sure how to start analysing the likely motion!

This is indeed making me think - and I'll let you know how the project develops.

Yes, you're spot on. The escapement assembly and pendulum would be fixed & only rotate around a vertical axis. This should not make any difference to the progression as the Earth rotates. Although it appears to draw a figure of 8 as it progresses the pendulum does actually trace a straight line. If you think about it an advance of 1 degree equates to -1 degree at the opposite end of the swing, I can jot down the equations for you if you'd like, it's all to do with the conservation of angular momentum.

 

[lesaid]

Yes, you're spot on. The escapement assembly and pendulum would be fixed & only rotate around a vertical axis. This should not make any difference to the progression as the Earth rotates. Although it appears to draw a figure of 8 as it progresses the pendulum does actually trace a straight line. If you think about it an advance of 1 degree equates to -1 degree at the opposite end of the swing, I can jot down the equations for you if you'd like, it's all to do with the conservation of angular momentum.

I'd be interested in the equations.

I have to say I'm still struggling to see, in that scenario, what difference it makes whether what is attached to the universal joint is a pendulum or a fixed bar that doesn't swing. In each case, there will be a minute torque (relative to the rotating Earth frame) applied to the universal bearing. I'm obviously missing something in my intuition!

By the way - the current version of my pendulum (which is free swinging - not driven) works well and displays the expected Foucault effect - but does not do figure of eights - it swings in a long, very thin, ellipse that is close to a straight swing, and that never changes shape, other than, when it decays to a few percent of its original amplitude after five hours or so, the ellipse gradually becomes more circular. And the 'bearing' at the top doesn't rotate. It could, easily - but it is the swing that rotates (relative to earth) - not the bearing at the top. Otherwise my bearing wouldn't be able to show more than a small amount of rotation without colliding with its supports

Fascinating!

 

[Mister T]

By the way - the current version of my pendulum (which is free swinging - not driven) works well and displays the expected Foucault effect - but does not do figure of eights - it swings in a long, very thin, ellipse that is close to a straight swing, and that never changes shape, other than, when it decays to a few percent of its original amplitude after five hours or so, the ellipse gradually becomes more circular.

How are you releasing the pendulum to get it started?

 

[lesaid]

How are you releasing the pendulum to get it started?

I am simply holding the bob in my fingers, at a point roughly half way up to minimise the initial oscillation of the bob around the hook that supports it (about four hertz as near as I can judge by guesswork), pulling it back and with ordinary care, letting go with my fingers. Usually, I get it into a pretty straight swing at the first attempt - if not, at most two or three attempts is enough. Then I let it swing for five minutes to settle down, and then rotate the base slightly to align the appropriate indicator on the 'clock' card with the initial swing direction. (won't be able to do that alignment when the pendulum is being driven though, as it would misalign the diver electrode).

I have never needed to bother with burning threads and such, or be too paranoid about precision of release. I am wondering if that is because my setup seems not to have significant coupling between different swing directions, so the initial ellipse is stable and doesn't turn into lissajous figures. But that is guesswork. Is the burning threads technique a workaround required to overcome asymmetries and/or flexing in the system?

What I do have to be careful of, is how far back I pull the bob - too far and the pivot travels slightly across the platform. This doesn't matter for a 'free swing', but would also cause the driver to misalign.

It also matters that the pivot assembly is symmetrical - otherwise when pulling the pendulum back, it tends to rotate.
edit : symmetrical vertically - so the CofG is close to the pivot point in all directions. Having a rigid pendulum rather than a cord helps with this too.

Although the driver at the moment tends to suppress the Foucault effect - it instead rotates the swing to a preferred direction - using the driver, it can be started simply by switching on the driver, and giving the pendulum a tiny nudge. It takes an hour or so, but the driver will gradually build up the amplitude itself into a good swing. Then turn off the driver and the Foucault effect takes over.

 

[lesaid]

I'd be interested in the equations.

I have to say I'm still struggling to see, in that scenario, what difference it makes whether what is attached to the universal joint is a pendulum or a fixed bar that doesn't swing. In each case, there will be a minute torque (relative to the rotating Earth frame) applied to the universal bearing. I'm obviously missing something in my intuition!

After further thought ... a suspended solid bar will only rotate (relative to the earth) if the static friction in the pivot is low enough that the torque applied by the Foucault effect can overcome it. If that were easy, nobody perhaps would bother with pendula. So - in your 'escapement' type of model - with a 2d pendulum suspended from a universal joint and constrained only to rotate around the vertical, and not itself swing - each quarter swing will provide a much higher torque than the static bar would, due to the coriolis force. Over a complete cycle, those torques would sum to exactly the same as the solid bar, but individually could be high enough to overcame the static friction in the bearing. Essentially the same idea as mentioned in an early post - adding 'noise' (or in this case, nearly cancelling coriolis forces) to reduce apparent friction. The universal joint would be constantly rotating back and forth slightly with each swing, staying in motion.

Is that a reasonable intuitive view of the reason why, in your scenario, a pendulum would do better than a solid bar?

I would like to calculate what the effective torque applied to a given pendulum due to the Foucault effect would actually be - interesting to compare it to the performance of a good universal joint (if those figures are available). Not sure how to do that though - and haven't yet tried. Another thing for the 'to-do' list !

In the design I'm working with, I think it is different. There is no torque applied by the swing to the pivot - the swing can go equally easily in any direction, regardless of the orientation of the pivot. So the pivot does not itself rotate and static friction becomes irrelevant (as regards rotation around the vertical axis).

I wonder if, for the same reason - that the universal joint approach would tend to cause the rotation to be slower than ideal, as sliding friction would also be in play, hindering the Foucault rotation.

Does that all make sense?

 

[SPW]

The pendulum would have to be a solid bar. When bolted firmly to an escapement it becomes the vertical pivot. The horizontal bearing being the other one. The oval trace will form as the pendulum exerts a rotary force on the horizontal bearing (which would probably have to be a large flat plate.

I've been looking into the equations and as the pendulum reaches the end of the swing it has Potential Energy. Then, as it begins to swing back it has Kenetic Energy. These cancel each other out. You can then use the mass of the bob, acceleration under g and the sin of the angle to calculate latitude etc. I will send you the equations but my iPad is limited at writing them, so I'll put them on paper & photograph them...

 

[lesaid]

The pendulum would have to be a solid bar. When bolted firmly to an escapement it becomes the vertical pivot. The horizontal bearing being the other one. The oval trace will form as the pendulum exerts a rotary force on the horizontal bearing (which would probably have to be a large flat plate.

I've been looking into the equations and as the pendulum reaches the end of the swing it has Potential Energy. Then, as it begins to swing back it has Kenetic Energy. These cancel each other out. You can then use the mass of the bob, acceleration under g and the sin of the angle to calculate latitude etc. I will send you the equations but my iPad is limited at writing them, so I'll put them on paper & photograph them...

By a solid bar in my previous post, I was meaning not a pendulum at all - say something like a weightlifters barbell, suspended in the middle from the universal joint. Then, in a frictionless environment, it's moment of inertia alone would cause it to exhibit rotation I think. But I think this doesn't happen because no bearing we can make will be 'that' symmetrical and frictionless.

I was thinking about how the coriolis force acting sideways on 2d 'escapement' pendulum suspended from the universal joint might just make the Foucault effect apparent. It's interesting, though it feels a very complicated way of doing it. Are you thinking of building such a device yourself?

I can solve the equations of a 2d pendulum - I'm finding a 3d one somewhat harder! My modelling so far hasn't been of the Foucault effect itself. It has been purely about getting the pendulum swing truly independent of its direction and free of Lissajous patterns. I have simply assumed that when successful, the Foucault effect will appear - and that seems to be what has happened :)

 

[Nidum]

This is getting silly . The whole point of a Foucault's pendulum is that it's swing is not greatly influenced by the Earth's rotation .

Anything added to the system which puts coupling between the pendulum action and the rotation of the Earth will cause the pendulum swing to start tracking the rotation of the Earth .

The principle of design is to minimise the coupling - ideally to zero . Not possible in practice but can certainly be reduced to a very low value .

There are only one calculation of any relevance to Foucault's pendulum design . This relates the relative influence between any coupling torque tending to rotate the plane of swing of the pendulum and the resistance of the pendulum action to any such rotation .

 

[sophiecentaur]

I was thinking about how the coriolis force acting sideways on 2d 'escapement' pendulum suspended from the universal joint might just make the Foucault effect apparent. It's interesting, though it feels a very complicated way of doing it. Are you thinking of building such a device yourself?

Isn't it the fictional Coriolis Force that causes the Foucault effect in the first place? The problem with supplying energy to maintain the pendulum is that it has to be totally 'unbiased', and any force must be directed to the centre of the swing. You have to be absolutely sure that it is not introducing any perturbation. It would be possible to introduce a force along the length of the suspension to supply energy which would satisfy this requirement (a bit like the way you can work yourself up when standing up on a swing. You do 'dips' at the appropriate time by bending your knees. The same could be achieved with a solenoid, moving the bob up and down in the right phase. A 1kg battery could probably supply enough energy for a 24+ hour operation, if the Q is as high as has been measured.

 

[lesaid]

I like that idea! though challenging perhaps for a home DIY-er to fabricate!

Pulling the bob up with a solenoid also perhaps leads to the same kind of challenges as all the other schemes, to get things adequately symmetrical - getting the solenoid force to act very precisely along the length of the pendulum for example. And in my case - brass rod bends a little and so the pendulum shaft is not absolutely straight. If the vertical stress on the shaft is changing each time the bob is 'pulled' - it might momentarily flex the pendulum shaft in the plane of its bend, introducing another asymmetry. However we approach it, the challenges are analogous I think :)

In the post you quoted, I was speculating on how SPW's idea might also show the Foucault effect, for all that it looked awkward. Perhaps I didn't word it well.

I've done a little experiment to look at the strength of the electrostatic force between chunks of brass rod of the same kind as the pivot frame is built with. Just to get some evidence to compare with calculations. I was astonished, that in a simple torsion balance made from dangling a rotating electrode made from brass bar off a length of fine harpsichord wire, very close to a matching static electrode, a 9V battery is enough to bring the two electrodes together! I am this evening, exploring the electrostatic forces in practice, using a handful of 9V batteries in series to generate different voltages, offset grounds and so on. Whereas I had expected to need hundreds or thousands of volts. Little wonder that the 6-10 kV used in the pendulum was influencing the orientation of the pivot piece! Measuring the deflection by looking at the movement in the reflection of a little LED laser off of the moving electrode, shone on to a sheet of paper.

So electrostatics demonstrations don't always need high voltages ! And pendulum pivots don't want them!

 

[SPW]

No I'm definitely not going to attemp the challenge. I'm strictly a paper & pencil physicist, but it was an interesting proposition. I am unsure what part of the Faucault effect you are trying to model? In the real world there will be slightly different outcomes due to the imperfections of whatever system you choose eg: the centre of gravity shifts depending on width of the bob. In an ideal system, with a perfect bearing, directly above the north/south,pole, in a vacuum etc, the standard equations hold true. As do the calculations for arc seconds at specific latitudes. The coriolis effect is due to your frame of reference.

In a body rotating with angular velocity

, a particular point with position vector

has a velocity

given by

In fact for any vector

(not necessarily a position vector) fixed in a rotating body,

Thus for unit vectors

,

and

, directed along the

(east),

(north) and

(upwards) axes fixed on the Earth's surface (see figure 1), we have

Figure 1: The directions of the

,

and

axes for a point on the Earth's surface with latitude

.Another more accurate way of measuring the effect in a physical system is with a gyroscope. You could bastardise a hard drive but be careful as most spin at 7,200rpm. Set in a gimbal parallel with the Earth? The effect would be small but measurable. A time lapse app would help.

As a thought experiment, imagine a man in a box traveling at a constant velocity in 0g. What would happen if a pendulum was 'hung' from the top (as judged from the man's point of view, who would feel the bottom was directly opposite from the vector of travel)?

 

[lesaid]

Thanks for those equations. I found the reference for them here https://www2.warwick.ac.uk/fac/sci/physics/intranet/pendulum/derivation/. Those equations were part of the introduction to that paper. The paper goes on to derive the rotation period of a Foucault pendulum, and which I will enjoy working through. Thank you.

I am unsure what part of the Faucault effect you are trying to model?

Not sure what you mean - the Foucault effect is what it is - 'parts' of the effect?

But I am not actually trying to model the Foucault effect itself. I am interested in what it takes to get a pendulum 'ideal enough' to be able to display the Foucault effect reasonably accurately for a 24 hour cycle on a 'desktop' scale. Which is why I started with a lump of metal hung off a piece of string (which was never going to work!), and have been analysing and solving each problem one-by-one until the Foucault effect appears. I don't want to make the 'perfect' pendulum - I want to find out what is 'good enough', and prove it in practice (hence the iterative approach rather than simply building the best I can immediately).

I would really like to model, for example, the effect of a slightly misaligned driver - where the 'kicker impulse' has a small tangential component to it. So I can find out analytically, how accurate the driver alignment needs to be and just what effect it has on the whole system. But so far, the resulting messy mathematics has defeated me. Perhaps because I haven't yet looked at Lagrangian mathematics, which, from what I have read, is useful in this kind of problem. Related to that - in practice, in my electrostatic driver, the spark jumps slightly before dead centre, and the repulsive impulse will be strictly radial - not directly in line with the swing. Will that impulse tend to widen, flatten, or not affect the shape of the ellipse? I have a hunch that it 'might' flatten it, but having a hard job working that out!

(Motivation is to practise newly learned physics and mathematics skills as well as experimental techniques - I am currently studying the subject)

the centre of gravity shifts depending on width of the bob.

does it? I don't see why (assuming the bob is hanging plumb and cylindrically symmetrical) and the bob doesn't change shape over time! The centre of gravity will shift as the pendulum swings - otherwise the thing wouldn't swing! But the restoring force due to gravity surely has to be radial, and unable on its own to introduce a torque around the vertical axis.

Another more accurate way of measuring the effect in a physical system is with a gyroscope. You could bastardise a hard drive but be careful as most spin at 7,200rpm. Set in a gimbal parallel with the Earth? The effect would be small but measurable. A time lapse app would help.

I don't think I'll attempt that - sounds too challenging for a home DIY project, to get the system precise enough. But why should it be more precise than a pendulum? I haven't looked at gyroscopes, but I'd have thought it easier (at least, outside of a professional laboratory) to set up a very precise pendulum?

As a thought experiment, imagine a man in a box traveling at a constant velocity in 0g. What would happen if a pendulum was 'hung' from the top (as judged from the man's point of view, who would feel the bottom was directly opposite from the vector of travel)?

I don't think the man would 'feel' anything and the pendulum would not swing - the whole system would be in free fall with no forces of any sort acting relative to the box? The man should be unaware of any travel and should not be able to 'feel the bottom'. For that, there would have to be an external force accelerating the box. But if you meant the box was accelerating in a straight line rather than at constant velocity, then the pendulum would swing in a constant plane that would not rotate. There would be no Foucault effect.

 

[SPW]

My apologies, I did indeed say "constant velocity"! Thank goodness for peer review. I apologise for any misunderstandings. It has been many years since I studied these things. However, I have only offered ideas and opinions in an attemp to help or provoke thought. I will be careful to reference any papers in the future. The joy I find in physics is seeing the realisation in my childrens' eyes when things fall into place, no matter how small. They have been raised to ask why or how. I may have assumed you were not as advanced as you are, again I apologise. With respect, I am a little confused; you have said that it was the challenge of making a device to impulse a force on a short pendulum but have also mentioned Langrangrian mathematics and modelling the tangential effects of electrostatic impulses. I have not researched this but I can only offer an opinion. I can remember coveriing double pendulums. I admit I did not know enough to fully comprehend it at the time. Although with fiendishly complicated mathematics it can be done; in reality it remains chaotic (due to the infinity of possible variables) Your proposal, if I have understood, will produce a very, very complicated set of equations, similar to those of the double pendulum, but remain experimentally chaotic. I would most definitely need to spend many months working on such a project, and need help from someone wiser than myself. Even Einstien needed Hilbert.

All the best
S.P.W.

 

[lesaid]

No problem :) I'm no expert - I'm an undergraduate student (later in life) studying mathematics and physics. I'm feeling my way along with newly learned skills, and know exactly the joy of things falling into place that you mentioned !

I am not sure whether the maths I need is in fact that complex, if I tackle it in the right way. I have not yet studied Lagrangian mathematics, but have read that it is helpful for these kinds of problems.

I have been simplifying the maths by treating the motion of the pendulum as a horizontal ellipse with a restoring force due to gravity - otherwise ignoring the vertical components. Where I have a problem is when I add in 'radial' forces centred on a point slightly offset from the 'origin' - i.e. a misaligned driver. It would be useful to know how precise the alignment needs to be!

I am wondering whether a path integral along a section of the ellipse covering the significant part of the impulse might let me find the overall direction of the bob's acceleration due to each impulse. By comparing that with the direction of the elliptical path over the same region, I think I should be able to find whether the impulse tends to flatten or broaden the ellipse. I shouldn't need to derive specific equations of motion for this - just look at an impulse on a point mass traveling around a 'generic' ellipse. Don't know if that's a good way of doing it, but I'll explore it.

But this is now a background thing for me - my regular studies have to take precedence now term has started :)