# Micron-level accuracy over a meter?



## zondar (Feb 16, 2022)

Hello Forum Members,

I have a project idea, but it's one that comes with a potentially difficult metrology issue. I need to measure a distance of about a meter to micro-meter accuracy. The actual length isn't very important (it doesn't have to actually be one meter), it just has to be measured accurately. The part to be measured would likely be a carbon-fiber or Invar rod (for temperature stability) with knife-edges at both ends. The distance between the knife edges is to be measured.

I imagine that laser interferometry would be the ticket here, but I'm not aware of such devices suitable for amateur-level budgets.

About the best idea I have at present is to construct a traveling microscope mounted along a meter+ glass scale. However, despite various claims, I don't think these scales (at least typical hobby-grade ones) are good to a micrometer over the span of a meter.

Using a large mill with high quality scales and a centering microscope could substitute for a custom-made solution, but the part may need to be measured repeatedly in situ if the temperature or orientation changes, and I don't have access to such a machine anyway.

Any brilliant ideas short of asking NIST for help?

Thank you.


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## rwm (Feb 16, 2022)

.001 mm will be tough to be repeatable. .01 more easily achievable. No good idea here, but I am curious what others have to say. Many DRO scales boast accuracy to 1um but I don't know how that goes in practice with temperature changes and backlash.


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## zondar (Feb 16, 2022)

Yes, six orders of magnitude is going to be tough!

Glass DRO's boast 1 um "resolution." That is not the same as accuracy. It just means that one unit of distance to the next (e.g. on your DRO display) is 1 um.

If I look at dropros.com for an example, their "1um" glass scales show +/- 12 um accuracy over about a meter. I interpret that as being 12um standard deviation, meaning that any given scale could be off a fair amount more than that (or not).

Repeatability is claimed to be 2-3 um, but repeatability doesn't confer accuracy either, it just means your measurement is wrong by about the same amount every time.

I know that scientists were able to do this a century ago using optical means, but I don't know how. It may be that the traveling-microscope idea using a glass scale is the best that I could do as a hobbyist, but losing more than an order of magnitude vs. what people did a century ago would feel a bit frustrating.


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## rwm (Feb 16, 2022)

That's kind of what I was thinking. 
I actually misinterpreted your original post. When you said the exact length didn't matter I assumed you just wanted precise measurements, not accurate measurements. (For those who are not familiar, precision is the repeatability of a measurement i.e how often it agrees with itself. Accuracy is the deviation of the measurement from a know standard.) That will be even harder!


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## RJSakowski (Feb 16, 2022)

Many, many years ago, I measured  distance to a quarter wavelength or better (around 0.1 micron) using interferometry.  You say that the actual distance between the knife edges isn't important but I assume that an accurate measure of that distance is.   If a +/- 1 micron will work, then one of the 1 micron magnetic or glass scales will work.  I have a 300x microscope setup for my Tormach CNC that is capable of visually resolving .0001" or around 2 microns.  With a little effort, it could be extended down to 1 micron.  A mill with a 1 meter travel equipped with a 1 micron scale and such a microscope would offer the best low cost chance of achieving your objective.

You wouldn't need a mill but some means of making micro-adjustments to position would be highly desirable.  I would consider a movable stage for rough travel and a stepper driven sub stage for fine adjust.  With the right lead screw and stepper, you should be able to get controllable nanometer level adjustments.

As to material for the bar, Invar would be the choice.  I doubt that carbon fiber would have the thermal stability you require.  Even at that, temperature control would be mandated.  Ideally, you would be controlling ambient temperature to within a degree.


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## rwm (Feb 16, 2022)

RJ- How would the setup work with the microscope? I can't envision this.
R


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## zondar (Feb 16, 2022)

Yes, the distance is arbitrary (except for being as long a baseline as practical - a meter is good), but its length must be known accurately. 

I haven't found glass scales specified to yield 1 um accuracy over a meter. They are generally worse than 10 um over a meter (e.g. claimed +/-12 um in the above example). 

The information I have on lead screws is that most are horrible by comparison; probably 10 times worse, and I wouldn't rely on them for more than motion control.

I imagine a microscope mounted to a meter+ carriage, e.g. linear bearings on a pair of rods, with the scope attached to a glass scale. A lead-screw and stepper could be used for positioning (only, not measurement except maybe as a sanity check). It would be a fun project in itself. 

How did you do the interferometry?


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## zondar (Feb 16, 2022)

By the way, I was thinking about the materials involved.

My guess is that good glass scales use borosilicate glass, which (wikipedia) has a temperature coefficient of expansion that is at least twice that of good quality Invar. So really, the Invar would be sort-of "measuring" the glass scale. ;-)

Fun fact: The guy that developed Invar won a Nobel Prize for it!


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## RJSakowski (Feb 16, 2022)

rwm said:


> RJ- How would the setup work with the microscope? I can't envision this.
> R


A robust stage would be set up to travel the required distance.  A sub-stage would be mounted for fine adjustment.  Something similar to a lathe cross slide with the compound set to travel along the x axis.  There would be no need for a lead screw mechanism for coarse travel as you are measuring with the linear scale.  Sub-stage travel would be accomplished with the stepper.  The microscope would be mounted on the sub-stage so you can image your two knife edges.

Here is a link to my thread on the mill mounted microscope for your reference. 








						[Metrology] - Mill Spindle Mounted Microscope (aka Cheap Optical Comparator)
					

I had thought about building a spindle mounted microscope for the Tormach for some time.  The idea was to cannibalize and old phone or adapt a webcam camera but the optics was rather daunting some the project was shelved.  About a month ago, I saw an ad for a USB 50 - 500X microscope on eBay for...




					www.hobby-machinist.com


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## zondar (Feb 16, 2022)

Perhaps a manually-operated micrometer could be used for the short distances needed to bring the knife edges into view. Probably would need to lock down the carriage first. But it's likely simpler than going to the trouble of a stepper.

Another concern about glass scales is the possibility of missing steps, for which a stepper-controlled lead-screw could help by delivering slow, consistent, motion, but maybe that's not a big problem if you are careful about moving the microscope stage.


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## RJSakowski (Feb 16, 2022)

My micrometers travel .025" or 635 microns per revolution.  Making sub micron adjustments would be touchy.  One micron of travel would require a rotation of 0.5º.  Not to say a micrometer couldn't be used.  We did it that way 50 years ago.

Not to mention that a micrometer would need to be cannibalized for the mechanism.   A stepper driven screw would be inexpensive.  Stepper motors from computer hard drives are readily available at no cost.  Simple step and direction control is easily accomplished.  Microstepping controllers are available for a few bucks from eBay or Banggood.  You still need a a lead screw and nut of some sort but the pitch wouldn't have to be as fine with a microstepper drive.


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## homebrewed (Feb 16, 2022)

An interesting discussion, but any approach that uses mechanical components will be subject to thermal effects that will be difficult to nail down.


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## RJSakowski (Feb 16, 2022)

homebrewed said:


> An interesting discussion, but any approach that uses mechanical components will be subject to thermal effects that will be difficult to nail down.


Since absolute position would be made via the glass or magnetic scale, mechanical components wouldn't come into play.  Of consideration would be the thermal expansion coefficients of the bar mounting the knife edges and the glass or magnetic scale.

A lot depends upon the particular application.


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## zondar (Feb 16, 2022)

A cheap donor micrometer can be had for about $25. A micrometer would not be for measuring, just positioning, but I take your point about how difficult it might be to operate manually.

I priced out a 1200 mm lead-screw, etc., for stepper drive. Not too bad, but it's a rather rapid screw that would require fairly high micro-stepping, so I might have to look for a tighter one. I happen to have an unused stepper and driver at present.

The cost of a 1+ meter unsegmented "1um" glass scale is very high - about $400 delivered from a reputable source (dropros), and that's for a bare scale. If anyone has a positive experience with a less expensive source, please let me know.

If it's accepted that micrometer precision isn't going to be achieved anyway, it's possible that the "5um" scales will do - their accuracy over a meter is quoted as almost the same as the 1um scales. But that wouldn't feel right!

I also priced an Invar rod. Yikes, that stuff is expensive. About $100 for a 3 foot 3/8" rod from McMaster-Carr. I happen to have some 4 ft. carbon fiber tubes, which are purported to have a low coefficient of expansion, but it's not documented, and I'd be pretty certain it's worse than Invar.

I'm sure that mechanical issues will be important. Backlash or other movement (and there's always some) between the microscope and the scale could easily be in the 10's of micrometers range. I mentioned that Invar is actually less prone to temperature effects than the glass scale would be. So how the scale is mounted, even, could be important. For example, perhaps it should be mounted in its center, and only loosely held at the ends.

To do this right means having an error budget: so much for the scale, so much for this and for that. I think in the end getting 10 micrometer accuracy over a meter by mechanical means would be very, very difficult.

It's an interesting project. If anyone can guess what it's ultimately for, have at it.


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## RJSakowski (Feb 16, 2022)

zondar said:


> A cheap donor micrometer can be had for about $25. A micrometer would not be for measuring, just positioning, but I take your point about how difficult it might be to operate manually.
> 
> I priced out a 1200 mm lead-screw, etc., for stepper drive. Not too bad, but it's a rather rapid screw that would require fairly high micro-stepping, so I might have to look for a tighter one. I happen to have an unused stepper and driver at present.
> 
> ...


You won't need a meter long lead screw, only a short one on the sub stage.  It only need be long enough for your precision adjustment.  Move the stage to an approximate position an lock it.  Then make final adjustments with the sub stage.

The microscope would have to be rigidly mounted to the stage.  Backlash isn't important since you would be measuring position with the DRO.  That's the beauty of a DRO.  You would probably want to calibrate the scale.  Borrowing some gage blocks would be helpful.  I would want to calibrate as close to the final length as possible to eliminate any extrapolation errors.


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## homebrewed (Feb 16, 2022)

zondar said:


> A cheap donor micrometer can be had for about $25. A micrometer would not be for measuring, just positioning, but I take your point about how difficult it might be to operate manually.
> 
> I priced out a 1200 mm lead-screw, etc., for stepper drive. Not too bad, but it's a rather rapid screw that would require fairly high micro-stepping, so I might have to look for a tighter one. I happen to have an unused stepper and driver at present.
> 
> ...


In your OP you said 


zondar said:


> The distance between the knife edges is to be measured.



Therefore I assume that the invar/carbon rod is used as a reference for something the knife edges are mounted on.  You're not looking at fast changes if a DRO solution is acceptable, so it could be something like creep -- ground movement -- or slow changes in length due to applied stress, aging, curing etc.  I don't think you are trying to detect gravity waves .

It appears that you want an accurate measurement in terms of PPM so you want to know the distance between the knife edges, along with the relatively small displacement of the material the knife edges are attached to.  You haven't mentioned any kind of automated measurement scheme, nor discounted the use of a microscope, so perhaps you want to measure an effect due to an applied stress (internal or external) of some kind, over a relatively short range of time.

I still think that environmental effects could well swamp out the variation you are trying to measure, unless the time frame is short enough that your environment changes less over time than the stress you're applying or creating.

A microwave interferometer setup might work.  300MHz in free space has a wavelength of 1 meter.  A relatively simple transmitter/receiver setup using the same master oscillator and a high-resolution (24-bit) DVM would provide enough resolution.  The output of the receiver would be proportional to the phase shift between the transmitter and receiver, which depends on the distance between them.  Since the wavelength is the same as your measurement baseline there wouldn't be any ambiguity in the position.  To avoid multipath effects you'd want the T/R antennae to be highly directional.  

In this context you wouldn't need an invar rod, sorry about that   The speed of light is the equivalent to your invar rod....


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## zondar (Feb 16, 2022)

RJSakowski said:


> You won't need a meter long lead screw, only a short one on the sub stage.  It only need be long enough for your precision adjustment.  Move the stage to an approximate position an lock it.  Then make final adjustments with the sub stage.
> 
> The microscope would have to be rigidly mounted to the stage.  Backlash isn't important since you would be measuring position with the DRO.  That's the beauty of a DRO.  You would probably want to calibrate the scale.  Borrowing some gage blocks would be helpful.  I would want to calibrate as close to the final length as possible to eliminate any extrapolation errors.


The "meter long" lead-screw, if used, would just be to move the stage over the full length at a steady pace via a stepper rather than by hand, out of fear of missing steps in the scale. But as you say a small lead screw could be used as long as it can be slid into position, hence my comments about maybe just using a micrometer for those last few mm.

By "backlash" (probably not the most appropriate term), I meant any potential movement in the mechanism between the microscope and the scale, like if a force is applied during use. Some is inevitable no matter how rigid (and I don't expect to be able to put in the time and expense to achieve lathe-like rigidity), and almost any would be in the micrometer range.

I can't calibrate the scale without some other scale that is also about a meter long. Ringing together a whole bunch of gauge blocks would probably accumulate small errors, but it's a thought. I don't know what other physical reference could be used. NIST probably has an actual, physical, meter-long meter in a vault, if only for historical reasons, but I doubt they will let me borrow it. ;-) Calibration of a scale is presumably done at the factory via optical means (e.g. laser interferometry), or at least one hopes, but with cheap scales, probably not?


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## zondar (Feb 16, 2022)

homebrewed said:


> I don't think you are trying to detect gravity waves .
> 
> It appears that you want an accurate measurement in terms of PPM so you want to know the distance between the knife edges, along with the relatively small displacement of the material the knife edges are attached to.  You haven't mentioned any kind of automated measurement scheme, nor discounted the use of a microscope, so perhaps you want to measure an effect due to an applied stress (internal or external) of some kind, over a relatively short range of time.
> 
> ...


The Invar rod would be to hold and space-apart two knife edges at each end.

As you say, interferometry is the real solution, but I don't have the equipment for that.

The reason I can't borrow someone's huge DRO'd lathe or mill table for a one-time measurement is exactly because of the environmental issues. I'd preferably be able to measure it in-situ at current temperatures, etc.

But as I mused above, the scales are almost certainly more sensitive to temperature than a good Invar rod is. So now I'm thinking that in-situ measurements might not be as useful as I thought. Maybe borrowing the use of someone's big DRO'ed lathe or mill, with a microscope mounted to it, is actually the most all-around practical solution after all.

And yes, I want to measure gravity waves. Well not really, lol, but that would be super cool!


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## rwm (Feb 16, 2022)

Gravity is a particle.


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## zondar (Feb 16, 2022)

rwm said:


> Gravity is a particle.


Particles are actually perturbations in fields. 

Or maybe the resolution of probabilities in fields once observed. Something like that!


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## RJSakowski (Feb 16, 2022)

zondar said:


> The "meter long" lead-screw, if used, would just be to move the stage over the full length at a steady pace via a stepper rather than by hand, out of fear of missing steps in the scale. But as you say a small lead screw could be used as long as it can be slid into position, hence my comments about maybe just using a micrometer for those last few mm.
> 
> By "backlash" (probably not the most appropriate term), I meant any potential movement in the mechanism between the microscope and the scale, like if a force is applied during use. Some is inevitable no matter how rigid (and I don't expect to be able to put in the time and expense to achieve lathe-like rigidity), and almost any would be in the micrometer range.
> 
> I can't calibrate the scale without some other scale that is also about a meter long. Ringing together a whole bunch of gauge blocks would probably accumulate small errors, but it's a thought. I don't know what other physical reference could be used. NIST probably has an actual, physical, meter-long meter in a vault, if only for historical reasons, but I doubt they will let me borrow it. ;-) Calibration of a scale is presumably done at the factory via optical means (e.g. laser interferometry), or at least one hopes, but with cheap scales, probably not?


If you wanted to move the read head  at a steady pace, a length of threaded rod would be sufficient.  Thread pitch accuracy wouldn't be an issue.  The glass scales that I have used come with a calibration certificate but for best accuracy, they needed to be calibrated in situ and a scale factor  used.  I used a 6" bar to calibrate mine as it was the longest bar that I could measure with my micrometer set. 

One thought regarding your calibration.  If you made a suitable bar of arbitrary length, you could send it to a certified metrology laboratory.  A simple calibration like that would be fairly inexpensive and then you would have a NIST traceable standard in hand.  They should be able to calibrate to the same standard they use for gage blocks which should be to a microinch at worst.


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## zondar (Feb 16, 2022)

True about the threaded rod, but proper lead-screws aren't that expensive anyway. The one I looked at, with a bronze nut, was about $30. Expenses at that level are fine even if only for a touch of classiness and smooth operation.

I think the problem with scales isn't 6 inches, over which they should be quite good, it's over the ~40 inches that I'd need. But I didn't know about a requirement for calibration in-situ. Did you find that it was actually off by a significant amount?

That's a great idea about finding a metrology lab! They will be fascinated once I say it's for gravity wave detection. ;-)


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## RJSakowski (Feb 16, 2022)

zondar said:


> True about the threaded rod, but proper lead-screws aren't that expensive anyway. The one I looked at, with a bronze nut, was about $30. Expenses at that level are fine even if only for a touch of classiness and smooth operation.
> 
> I think the problem with scales isn't 6 inches, over which they should be quite good, it's over the ~40 inches that I'd need. But I didn't know about a requirement for calibration in-situ. Did you find that it was actually off by a significant amount?
> 
> That's a great idea about finding a metrology lab! They will be fascinated once I say it's for gravity wave detection. ;-)


I have some numbers from when I calibrated in my notes and it appears the scales were off between 550 ppm and 1000ppm.  I was surprised to find this much deviation.  The calibration corrected that.  

My method was to set up a straight edge on my table and sweep it to ensure it was parallel to the axis travel.  A 1-2-3 block was set on one end as a stop and and a test indicator and the DRO zeroed on the face.  A known length bare (6" parallel was then placed against the stop and the table moved to the opposite end.  the table was adjusted until the test indicator zeroed again and the DRO reading noted.  That reading was compared against the parallel length, as measured by micrometer, and the DRO scale was adjusted per the FRO manual.

Now that the cat is out of the bag, it would be great to see some details of what you are trying to do.


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## JRaut (Feb 16, 2022)

You could buy a set of these large gage blocks to get your accurate overall distance. Then use the micrometer stage strategy others have brought up.

They’re not cheap, but I’m sure the $800 price tag is a (very) small fraction of what they cost new.









						Pratt & Whitney 5" to 20" - Large Steel Gage Block Set in case  - **READ**  | eBay
					

Case is rough.



					www.ebay.com


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## Firebrick43 (Feb 16, 2022)

why not just have a shop with a zeiss machine measure it for you?


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## homebrewed (Feb 17, 2022)

rwm said:


> Gravity is a particle.


Only if you look at it using methods that can detect particles.  The wave function collapses to whatever method of observation is used.  LIGO detects waves so it "sees" waves.  The classic double-slit experiment that demonstrates interference patterns of electrons is a good example of this.

Diode lasers are an interesting way of showing this duality.  The laser color is determined by quantum mechanics -- the energy due to the recombination of holes and electrons determines the photon energy -- but the light can then be used to generate an interference pattern, most easily explained using the light-as-wave argument.


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## rwm (Feb 18, 2022)

I was kidding of course. We should not hijack this thread with the wave-particle duality discussion!
Are you really building a gravity wave detector? I assumed this was not possible on a small scale, hence LIGO.


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## zondar (Feb 18, 2022)

rwm said:


> I was kidding of course. We should not hijack this thread with the wave-particle duality discussion!
> Are you really building a gravity wave detector? I assumed this was not possible on a small scale, hence LIGO.



No, of course I'm not trying to detect gravity waves. That's not possible without having a few billion dollars to invest in building such an instrument. But the guess was actually close!

I have an interest in historic scientific experiments, and thought I'd try to replicate a few as a hobby. So one idea was to construct a "Kater's pendulum", which was likely the first device used to calculate the force of gravity (lower case g) with enough accuracy to calculate local differences.

A Kater's pendulum is a reversible pendulum. It can be hung with the heavy side down, as would be normal, or hung with the heavy side up. Sharp, hard knife edges riding on a very hard plate are used to balance the pendulum to keep friction as low as possible.

The local gravity g can be calculated from a pendulum via T=2*Pi*sqrt(L/g), where T is the period of oscillation and L is the length of the pendulum. The problem, though, is that the period of a physical rigid-bodied pendulum (a "compound pendulum") isn't the same as that of an ideal pendulum, and that equation is only accurate in the ideal case.

The trick that Kater came up with has to do with the reversibility. The Kater's pendulum is hung one way, its period measured, then hung the other way, and its period measured. This is repeated over and over while adjusting a small weight along the pendulum's shaft until the two periods are identical. At that point, with a little math (look up on Wikipedia), it can be shown that the period should now match that of an ideal pendulum of length L to a very high degree.

This reduces the problem to finding L, the distance between the two knife-edge pivots, and of course measuring the period T. The usual things one does to help measure T reliably include having a long L and a heavy weight (on one end, the other end is as light as practical), with a relatively narrow angle of oscillation. The pendulum's pivot should also be firmly anchored to a very stable foundation and the pendulum shielded from disturbances.

Kater's pendulum was constructed from brass, and had about a meter's length L between pivots. He was able to measure the distance between pivots to about 2.5 microns using a microscope comparator. I wish I knew more details about how he did that! His result included corrections for temperature, etc., and provided a figure for g to 6 significant digits.

Instead of a copy of Kater's pendulum, I'd likely construct a refined version, a Repsold-Bessel pendulum. This is a fully symmetrical version, in which both ends are physically identical. So there's a heavy brass weight on one end, and an identically-shaped very light weight (in my case, probably a 3-D printed hollow plastic "weight" in the same shape) on the other. Everything else should be made the same. This does away with the need to fine-tune the position of the small weight to make the periods equal when reversed.

So here I am more than 100 years later. I have access to Invar instead of brass for the pendulum, I can measure the period T with nanosecond precision via electronic means instead of by comparing it to a pendulum clock as he did, and yet I can't achieve close to the same accuracy of the L measurement as he did! So frustrating!


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## homebrewed (Feb 18, 2022)

In the Wikipedia entry for Kater's Pendulum it was mentioned that he used an optical comparator to determine the distance between the two knife edges.  So the simplest explanation is that he had access to a reference length that was at least close to the length of his pendulum.  Or chose its length to facilitate the measurement, slapped it into his optical comparator and wallah, as one of my co-workers used to say


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## zondar (Feb 18, 2022)

Yes, exactly. In reading an original description of Kater's work, the distance between knife edges were compared via a microscope against three different physical scales. As expected, the resulting measurements varied, but were similar to their 5th or 6th significant digits. ("A man with two watches never knows what time it is.")

I will try to do an error budget analysis to see how good an estimate of g I could expect with what's available to me. But I'm not feeling very eager to spend $500 or more to construct my own traveling microscope for the sake of a single measurement, so I'd likely still have to find someone with a large and accurate DRO'd lathe / mill or go to a lab.

Project Gutenberg has a great E-book, "Development of Gravity Pendulums in the 19th Century" by Lenzen and Multhauf that goes into a century of developments in the field. Apparently, gravity pendulums were used up to the 1950's for geological studies, as only then were better instruments finally developed. The modern approach drops weights in a vacuum for a direct measurement of acceleration due to g, and then that's used as a reference for more portable machines, e.g. a weight on a spring or even micromachined devices.

Actually, I recently built a device that measured g via a micromachined accelerometer, but that had poor accuracy for this sort of precision work (it was for a totally different purpose).


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## whitmore (Feb 18, 2022)

RJSakowski said:


> You won't need a meter long lead screw, only a short one on the sub stage.  It only need be long enough for your precision adjustment.



My shortest micrometer screw has over 20mm travel; if you can get to half a millimeter accuracy before adjustment,
an elastic linkage from the +/- 10mm down to  +/-  0.5mm would be appropriate,  and is
fairly easily arranged, using flexures like  Dan Gelbart explains.  That'd be a nice machining  project,
IMHO.


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## RJSakowski (Feb 18, 2022)

zondar said:


> Yes, exactly. In reading an original description of Kater's work, the distance between knife edges were compared via a microscope against three different physical scales. As expected, the resulting measurements varied, but were similar to their 5th or 6th significant digits. ("A man with two watches never knows what time it is.")
> 
> I will try to do an error budget analysis to see how good an estimate of g I could expect with what's available to me. But I'm not feeling very eager to spend $500 or more to construct my own traveling microscope for the sake of a single measurement, so I'd likely still have to find someone with a large and accurate DRO'd lathe / mill or go to a lab.
> 
> ...



Now it becomes interesting!  

I see now why you wish to accurately measure the distance between the two knife edges.  A question that occurred.  If the period of oscillation was determined by comparing to a pendulum clock, they would both be subject to gravity so it would seem to be a circular calibration.  The definition of a second in the early nineteenth century was in terms the length of time for the Earth to rotate around the Sun .  At that time, the most accurate and reproducible clocks were most likely pendulum based and thus dependent of gravity. 

Considering that gravity varies by location due to distance to the center of the Earth, centrifugal force, coriolis force, and localized masses, determining it becomes daunting challenge.  The question that I would ask is what is your final objective? If you are looking for an absolute value and make a determination, how do you validate that value?  Since gravity varies significantly more than the ppm precision you are looking for, what do you compare to?  

The only practical way that I can see would be a calibrated precision load cell and an accurate mass.  In practice the load cell could be calibrated by going to a location where an accurate value for the gravitational force was known and checking with your known mass,  Mass weights for balances are calibrated in metrology labs to NIST traceable standards so that shouldn't be too difficult, although you may have to take buoyancy into account.

I gather then, that you are attempting to determine an absolute value for gravity rather than looking for variations.  Hence the need for an accurate measurement of the distance between the knife edges.  If you were willing to use a shorter pendulum, A setup like mine would have the capability to measure that distance, subject of course to the accuracy of the calibration of the mill.  The Tormach mill is capable of measurements to 14" and my RF clone can measure to 18.5". I would still make an internal length standard and have that calibrated by a metrology lab.  The period will be shorter but time can be accurately measured and clocks can be easily calibrated to an accuracy of one part in 1E14. The length standard should achieve a calibration accuracy in the microinch range.  The digital microscope that I have can achieve a resolution of better than 2 microns and a glass scale has resolution to 1 or 5 microns, your choice.


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## RJSakowski (Feb 18, 2022)

Here is a calibration lab for load cells  Seriously, an interesting exploration into an allied topic.


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## zondar (Feb 18, 2022)

RJSakowski said:


> Now it becomes interesting!
> 
> I see now why you wish to accurately measure the distance between the two knife edges.  A question that occurred.  If the period of oscillation was determined by comparing to a pendulum clock, they would both be subject to gravity so it would seem to be a circular calibration.


Clocks used for scientific purposes were calibrated against the motion of stars, an independent and highly accurate standard. I presume that this was done here too. Of course it wouldn't be perfect, but time is relative, after all. ;-)

Kater's pendulum and the clock (basically the clock's pendulum) were closely, but not perfectly, synchronized. When off slightly, you can compute a period measurement of very high accuracy by measuring the beat frequency: the frequency at which the two pendulums (the clock's and the special one) coincide (the "method of coincidences"). This is similar to how a vernier caliper works.



RJSakowski said:


> Considering that gravity varies by location due to distance to the center of the Earth, centrifugal force, coriolis force, and localized masses, determining it becomes daunting challenge.  The question that I would ask is what is your final objective? If you are looking for an absolute value and make a determination, how do you validate that value?  Since gravity varies significantly more than the ppm precision you are looking for, what do you compare to?



My objective (in principle) is to do as good a job as Kater did. 

As for validation, I'd first hope to find a value that is within expectations compared to the average value of g on earth, which is 9.80665 m/s^2 (32.1740 ft/s^2), say +/- 0.3% just to check that I'm in-range. 

After that, I could probably find data (i.e. maps) that gives a better value for my particular location. Beyond that, I'd have to find a particular spot or lab at which a high resolution measurement was made, and set up there.

But it's also, and mostly, just to have fun with an interesting challenge!




RJSakowski said:


> I gather then, that you are attempting to determine an absolute value for gravity rather than looking for variations.  Hence the need for an accurate measurement of the distance between the knife edges.  If you were willing to use a shorter pendulum, A setup like mine would have the capability to measure that distance, subject of course to the accuracy of the calibration of the mill.  The Tormach mill is capable of measurements to 14" and my RF clone can measure to 18.5". I would still make an internal length standard and have that calibrated by a metrology lab.  The period will be shorter but time can be accurately measured and clocks can be easily calibrated to an accuracy of one part in 1E14. The length standard should achieve a calibration accuracy in the microinch range.  The digital microscope that I have can achieve a resolution of better than 2 microns and a glass scale has resolution to 1 or 5 microns, your choice.



Thank you for the offer. Just thinking out-loud:

A shorter pendulum is certainly possible, and it would open up more avenues for a length measurement. A meter-long pendulum swings with a period of about a second. A half-second period would be 43.5 cm or about 17 inches.

There would be a penalty somewhere, but my guess is not very much in the resolution of the period measurement given more modern techniques of measuring time.

I expect I'd measure the period with an oscilloscope (which I have) and a light sensor of some sort, which would be cut by a thin blade carried at both ends of the pendulum. I think this could get me down to under 100 ns resolution without too much trouble, and possibly under 10 or even 1 if done right. With this and some averaging, I think I could hope for 7+ digits of time resolution. If that worked out, I think the period measurement component is not the hardest part, and using a smaller pendulum (within reason) would not necessarily become the major limitation (I think L will remain that). Later pendulums were indeed smaller than Kater's, too.

But I kind of like the idea of sticking with the historical meter-long pendulum.


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## zondar (Feb 18, 2022)

whitmore said:


> My shortest micrometer screw has over 20mm travel; if you can get to half a millimeter accuracy before adjustment,
> an elastic linkage from the +/- 10mm down to  +/-  0.5mm would be appropriate,  and is
> fairly easily arranged, using flexures like  Dan Gelbart explains.  That'd be a nice machining  project,
> IMHO.


That video was positively fascinating! Thanks. (Plus I'm weirdly jealous of his Einstein-like accent. Lol.)


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## RJSakowski (Feb 18, 2022)

zondar said:


> Clocks used for scientific purposes were calibrated against the motion of stars, an independent and highly accurate standard. I presume that this was done here too. Of course it wouldn't be perfect, but time is relative, after all. ;-)
> 
> Kater's pendulum and the clock (basically the clock's pendulum) were closely, but not perfectly, synchronized. When off slightly, you can compute a period measurement of very high accuracy by measuring the beat frequency: the frequency at which the two pendulums (the clock's and the special one) coincide (the "method of coincidences"). This is similar to how a vernier caliper works.
> 
> ...



I used to build timers for equestrian events.  For calibration pf the timer clock, I used an eight decade frequency counter which in turn was calibrated to WWV.   WWV carrier frequency is accurate to a part in 1E14.  You should have no trouble measuring the pendulum period.  I would use a photodetector generating a pulse every rime the pendulum broke the beam.  The pulse could trigger a counter to count the pulses and update a timer each time the beam was broken.  Count for as many passes as needed to get the precision you want.  A pendulum with a half second period and a well setup photodetector should give you ppm accuracy with less than 1,000 passes.

I installed optical homing on my Tormach 770 to improve homing repeatability some years ago.  I used an Omron OPB829DZ optointerupter, combined with regulated emitter LED current, comparison to a regulated reference voltage via an LM311 comparator, and well designed optics.   This provides consistent positioning to within +/- .0001". While I am dealing in position, you can convert tha precision to time by calculating the velocity of the pendulum as it breaks the beam.  For best accuracy, I would place the detector near the center of the swing in order to have the highest velocity.


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## zondar (Feb 18, 2022)

RJSakowski said:


> I used to build timers for equestrian events.  For calibration pf the timer clock, I used an eight decade frequency counter which in turn was calibrated to WWV.   WWV carrier frequency is accurate to a part in 1E14.  You should have no trouble measuring the pendulum period.  I would use a photodetector generating a pulse every rime the pendulum broke the beam.  The pulse could trigger a counter to count the pulses and update a timer each time the beam was broken.  Count for as many passes as needed to get the precision you want.  A pendulum with a half second period and a well setup photodetector should give you ppm accuracy with less than 1,000 passes.



Yes, you just reminded me that I have a superbly-accurate counter available that I could borrow. It would do a lot of the work for me, such as averaging periods. Can't average for too much at a time, though, since the period changes slightly with the amplitude of the pendulum. I certainly would have to average, though, since there is constant geological noise all around me from cars and trucks and such.

Only the period is needed here, but yes, I'd measure at the bottom of the swing so the time spent breaking the beam is the shortest.

Another thought about judging the value obtained: The value of g at sea level is 9.80665 m/s^2, and equations to compensate for latitude (the earth is not round) and altitude (I am a few hundred feet above sea level) are easy to find. The equations won't compensate for local geological features, but I could start with that and see how far off I am.


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## zondar (Feb 18, 2022)

RJSakowski said:


> I installed optical homing on my Tormach 770 to improve homing repeatability some years ago.  I used an Omron OPB829DZ optointerupter, combined with regulated emitter LED current, comparison to a regulated reference voltage via an LM311 comparator, and well designed optics.   This provides consistent positioning to within +/- .0001".


I looked at that device. Natively, it shows a peak-to-peak displacement distance of 0.05 inches when the flag is in the middle. I'm not sure that's good enough, but maybe I'm not interpreting it properly.

But also, the aperture is so narrow that it would be trouble with a free-swinging pendulum (hitting it would be disaster). The pendulum would be balanced on a knife edge, not in a bearing that would limit its freedom to only pass through a narrow gap.

Also, I'd be afraid the narrow gap would cause air turbulence as the blade passes it. It would be very important to allow the pendulum to swing without any unnecessary outside interference. I think I'd need to engineer something with a significantly wider gap. But something like this sure would be convenient. I'll have to investigate alternatives.

Edit: It looks to me like that device can be severed at its gap into two independent parts (transmitter and receiver). In that case, the gap could be extended arbitrarily. Used in relative darkness, I think that would be effective. I'll poke around at similar devices, but this one looks like it could be workable after all. The counter would do the discrimination.


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## RJSakowski (Feb 18, 2022)

zondar said:


> Yes, you just reminded me that I have a superbly-accurate counter available that I could borrow. It would do a lot of the work for me, such as averaging periods. Can't average for too much at a time, though, since the period changes slightly with the amplitude of the pendulum. I certainly would have to average, though, since there is constant geological noise all around me from cars and trucks and such.
> 
> Calculating the velocity would be more difficult than measuring the period, but yes, I'd measure at the bottom of the swing so the time spent breaking the beam is the shortest.
> 
> Another thought about judging the value obtained: The value of g at sea level is 9.80665 m/s^2, and equations to compensate for latitude (the earth is not round) and altitude (I am a few hundred feet above sea level) are easy to find. The equations won't fully compensate for local geological features, but I could start with that and see how far off I am.


The velocity wasn't too difficult to calculate.  The equations which are use to calculate the period of the pendulum will will give you the velocity.  
Θ = Θo cos(2π/T*t) where Θ is the angle from the vertical in radians, Θo is the maximum angle, T is the period in seconds, and t is time in seconds. The angular velocity is dΘ/dt = -2πΘo/T sin(2π/T*t). When t = 0, the cosine =1 and Θ = Θo .when t = T/4, sin(π/4) =1 and dΘ/dt = 2πΘo/T. If T is 1 second and the amplitude is 0.1 radians (5.7º), the angular velocity as the pendulum swings through center is 2π*0.1/1 =.628 radian/sec.  At a midway distance of 500 mm, the linear velocity will be 314 mm/sec 

If the photodetector can resolve position to +/- .0001" or +/-0.0025 mm, this would be equivalent to a time resolution of 8 µsec. and timing over 10 periods would give ppm accuracy.


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## zondar (Feb 19, 2022)

I found a few more sensors that have larger gaps, which I think are needed to avoid the possibility of collisions and to keep air turbulence to a minimum:

Optech OPB315WZ has a has a slot width of almost an inch.
Omron EE-SPWL311 is a split design that can read at up to a meter apart, though the gain drops dramatically over that distance. It's also quite a bit more expensive than the common slotted ones at about $100.
As pointed out, given the timing accuracy of modern electronics, the period measurement probably would not be the limiting factor.

I might start a new thread in the "projects" section. I have very limited machinist's equipment; basically just a tiny Sherline lathe and a few other common tools. (I'm also in the middle of building a small CNC mill, which will be a distraction for a while.)

The pendulum isn't complicated in principle, but I'm an almost total newbie when it comes to machining, so achieving critically-high squareness and parallelism of the knife-edges, etc., might be a bit challenging for me. For practice, I'll probably start with a mock-up using some aluminum I have around before investing in Invar, etc.

It should be a fun project involving many different aspects: historical research, materials, machining, electronics, a bit of math, etc.


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## RJSakowski (Feb 19, 2022)

I would still try to use an optoisolator similar to what I had used.  The flag can be a piece of thin opaque plastic mounted so the plane of the flag is perpendicular to a line through the pivot point.  A wider beam is OK for detecting a vehicle passing but is not very precise as far as the trigger point is concerned.  You really want a nice crisp trigger for optimum repeatability.  Considering that you will be adjusting the distance between the knife edges, the sensor position will need to be adjustable in order to accommodate the difference in distance between the flag and the knife edge. You could use two flags.  A thin piece of plastic will contribute only a insignificant amount to the center of oscillation of the pendulum. and air resistance.

Were this my project, I would start out by not worrying too much about a high degree of accuracy.  I would concentrate on making the right mechanism and on getting the timing circuitry  right.  Once I established that I could get a reasonable value for g, then I would look into what I could do to improve the precision of the number.  Some things to consider; environmental controls to deal with temperature fluctuations, isolation of the mechanism to prevent disturbance from passing vehicles (and what about earth tremors?), possibly operating in a vacuum.  

As to measuring the distance between the knife edges, I think I would apporoach it like this.  After making the adjustments to the knife edge position to equalize the periods, I would make a fixture to hold the pendulum and support a rod to fit between the knife edges. I would make a rod to fit short of the touching both knife edges and use feeler gages to make up the difference.  The thickness of the feeler gage pack could then be measured with a micrometer and added to the rod length to determine the distance between the knife edges.  I would then have the rod calibrated buy a metrology lab or any other convenient means, depending on my desired accuracy.  This should give accuracy to a few ppm.  Anither way would be to make the test rod about an inch short of the distance and make pins to make up the difference.  It wouldn't take much effort to make a set of pins, differing by .0001" and choosing the best fit.   The pins could be calibrated at the same time as the test rod.


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## zondar (Feb 19, 2022)

Thank you for your feedback.

I've thought a little about making a historical replica of Kater's pendulum, designed from photographs of an original and the historical records, with period materials, etc., if only as a cool "art" object rather than a practical one. But that raises the difficulty level considerably. In one extreme example, the blades were made of _wootz_, which was the best steel available at the time, but now is effectively a lost technology.

Instead, I'm planning on making a Repsold-Bessel pendulum. It was an improvement on Kater's pendulum in that it's designed to be perfectly symmetrical, except with a heavy weight at one end and a dummy weight at the other (as light as practical but in the exact same shape as the heavy weight for the sake of symmetry, i.e. to match air resistance). 

This pendulum does not need to have the knife edges be adjustable, at least if you aren't aiming at a given period (e.g. a precise second's pendulum), and it also doesn't need a sliding weight to match the periods when the pendulum is reversed. Kater needed his pendulum to be almost exactly a second due to the way he measured the period, but with with modern timing equipment an arbitrary period will do. 

Since the knives would be rigidly mounted in this variant, I'd mostly expect a metrology lab to be able to measure L directly, e.g. on a traveling microscope. But I don't really know what the reaction or difficulties might be.

I'm an EE and am familiar with electro-optical circuits. I have an oscilloscope and could borrow a very accurate counter. I'd expect the trigger circuits on them to be decent enough for a first try, and I can handle the rest.

For the flags, I was thinking of two thin metal rods (just for simplicity in machining them in on a lathe), mounted to both ends past the knife holders. This is similar to what Kater and Repsold used.

Having the knives and anvil (that the knives would ride on) be very straight and perpendicular to the swing would be very important or the pendulum will wobble, destroying its accuracy. Similarly, everything else needs to be symmetrical and in-line.

But I'm such a beginner at machining that I'm probably not capable of high precision! My tools are severely lacking too. Anyway, this project is about the fun of learning. I intend to start with an inexpensive prototype (i.e., not made of Invar) that I won't feel bad about messing up, and yet is still capable of testing the whole system. 

I live in suburbia and there is constant geological noise from cars and trucks driving around, etc. I'd have to rely on averaging to wash out some of that.

As for a vacuum, ha ha, I might be crazy enough to make a project out of this, but I'm not so crazy as to go that far! (But yes, Kater also used his in a partial vacuum at some point.)

Thanks again to everyone for the interesting and useful comments.


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## rwm (Feb 19, 2022)

This is fascinating. May I ask a question about this experiment? As you indicate, you can easily measure the period of the pendulum with modern electronics. How did Kater do it? Did he have a standardized clock that he carted around with him? I assume gravity would not affect a precision watch like an automatic movement? That relies on mass but not gravity.


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## RJSakowski (Feb 19, 2022)

rwm said:


> This is fascinating. May I ask a question about this experiment? As you indicate, you can easily measure the period of the pendulum with modern electronics. How did Kater do it? Did he have a standardized clock that he carted around with him? I assume gravity would not affect a precision watch like an automatic movement? That relies on mass but not gravity.











						Kater's pendulum - Wikipedia
					






					en.wikipedia.org


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## rwm (Feb 19, 2022)

Thanks RJ, I read that but it did not really answer the question or I missed it? I did see this:
"In Kater's time, the period _T_ of pendulums could be measured very precisely by timing them with precision clocks set by the passage of stars overhead. " 
What were these precision clocks? Were these like the automatic movements I have on my wrist today? I assume he transported these around with the pendulum? I am trying to understand what they used as a time base that was not also affected by gravity.


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## zondar (Feb 19, 2022)

rwm said:


> This is fascinating. May I ask a question about this experiment? As you indicate, you can easily measure the period of the pendulum with modern electronics. How did Kater do it? Did he have a standardized clock that he carted around with him? I assume gravity would not affect a precision watch like an automatic movement? That relies on mass but not gravity.


He compared a calibrated pendulum clock (calibrated against the movement of stars) against the test pendulum positioned so that they overlap visually. He observed the two overlapping pendulums via an optical telescope from across the room to avoid parallax errors.

Both were almost exactly in sync, but not quite. Then he watched for coincidences between the two pendulums (the clock's and the test pendulum). For illustration, if every hundred swings the two pendulums were in alignment, then the two periods differ by 1/100th of the calibrated clock's period (I haven't done the math, but something like that). From that, he can calculate the period of the test pendulum to very high accuracy. He called this the "method of coincidences."

This is similar to how a vernier caliper works.


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## rwm (Feb 19, 2022)

Thank you! That is part of what I don't understand. So the Kater pendulum is affected by gravity but the simple pendulum is not?


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## zondar (Feb 19, 2022)

rwm said:


> Thank you! That is part of what I don't understand. So the Kater pendulum is affected by gravity but the simple pendulum is not?


Both pendulums are affected by gravity. The reference pendulum clock is calibrated against the stars so its pendulum has an accurately-known period of oscillation.

You might ask, why not just use that calibrated pendulum to measure the force of gravity? Well, you can't, because an ordinary physical pendulum is a so-called "compound pendulum." For example, the wood or metal rod holding it has mass too, not just the heavy bob, and that messes up the calculation. Even using a very thin metal wire is too much, plus that stretches by different amounts during swings, etc.

Another reason why you can't use the clock's pendulum to measure gravity is that the clock adds energy to the pendulum's swing with each beat (or else it would stop). That turns out to mess up everything even more.

So Kater devised a special pendulum that, after adjustment by reversing it until both orientations have the same period, does act like an ideal pendulum, making it amenable to the gravity calculation. But now you have to figure out what its period is (and its length). So Kater compared its swings against the reference pendulum by the method of coincidences.


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## rwm (Feb 19, 2022)

I think something just clicked that was not directly explained. I assumed that the clock was built at the same location as the Kater's pendulum and calibrated there. That would mean they were calibrated in the same gravity and should always read the same. Now if the clock was calibrated against the stars at another specific location and then Kater's pendulum was brought to that location they could swing with different periods. Is that the missing piece?
I do understand the method of coincidences and how it is like a vernier. That makes perfect sense.


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## zondar (Feb 19, 2022)

rwm said:


> I think something just clicked that was not directly explained. I assumed that the clock was built at the same location as the Kater's pendulum and calibrated there. That would mean they were calibrated in the same gravity and should always read the same. Now if the clock was calibrated against the stars at another specific location and then Kater's pendulum was brought to that location they could swing with different periods. Is that the missing piece?
> I do understand the method of coincidences and how it is like a vernier. That makes perfect sense.


That is true: The reference clock must be calibrated (though not necessarily built) at the same location, or else the reference clock will become inaccurate once moved, since the force of gravity depends on where you are located, and that will change the period of oscillation of the reference clock.

Then the Kater's pendulum just needs to be in the same location as the stationary reference clock. He did actually use at least two different reference clocks, and you see two in that engraving, one behind the Kater's pendulum for the coincidence measurement, and one to the side.


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## rwm (Feb 19, 2022)

Yep. That's what I missed the first time. I didn't expect that calibrated clocks would be set up at the measuring locations. 
OK! So why is it important that the Kater pendulum is a "ideal pendulum?" Could you not just use a simple pendulum and check it's period against the calibrated simple pendulum? Maybe it's because a compound pendulum would be affected by gravity a little differently? That could create accuracy issue with the computation of g? Sorry to drag this out! I know you want to get to the actual machining!


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## zondar (Feb 19, 2022)

For an ideal pendulum, calculating g is easy, from T=2*Pi*sqrt(L/g), where T is the period and L is the length.

For a hundred years or more, people tried to make a conventional pendulum that was good enough for that equation. Use as frictionless of a pivot as you can (e.g. a knife edge), a long and very thin yet non-stretchable rod or wire, a very dense and heavy bob that is in a simple shape such as a sphere, and so on. But it just wasn't good enough for those 6 digits of resolution.  

Kater's pendulum doesn't magically become ideal, but it's a configuration that mimics an ideal pendulum sufficiently for a similarly-simple equation to work (see the wiki page on it).


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## RJSakowski (Feb 19, 2022)

When I worked as a student assistant in the Physics Dept. of the university I attended, one of the first assignments that I had was to make a drive for the Foucault pendulum that was a centerpiece of the new Science building.  

The pendulum was mounted on the rooftop about 40' above and had a hollow cast aluminum sphere about 20" in diameter filled with lead shot. There was a 24 hour clock dial below the bob and the bob had a small cone attached for a pointer.  It was suspended from a steel cable about 3'16" in diameter and the pivot was a 1/2" drill chuck.     The intent was to detect when the bob passed through center and fire a Helmholtz coil  which would give an impulse to a second drill chuck located about 10" below the pivot to make up for the energy lost due to drag.  

Because the Earth rotated under the pendulum, a simple photocell,  triggering when the cable passed through center couldn't be used.  I eventually made a fine wire whisker which was fastened to the end of the pointer , along with an aluminum disk at the center of the dial and insulated from the dial to detect the center crossing. I also added an iron core to the coil and we had a working pendulum drive. It was still working many years later when I paid a visit.

I don't recall that I had calculated the period from the known length but I almost certainly did.  That was 55 years ago.


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## rwm (Feb 19, 2022)

As a child, I loved the Foucault pendulum at the Smithsonian. I think they removed it about 20 years ago. Such a cool exhibit.


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