# Kater's Gravity Pendulum Metrology Project



## zondar (Apr 12, 2022)

*Kater’s Gravity Pendulum Project

Introduction*

For centuries, a pendulum was the highest-precision way of measuring time by far, except for the motion of the stars themselves, which pendulums (in clocks) were calibrated against for scientific use. But if you moved a pendulum clock from London to France, for example, it would keep different time, because the accelerating force of gravity on Earth varied depending on where you were due to local changes in latitude, altitude, etc.

The formula for an ideal pendulum’s motion is T=2*Pi*SQRT(L/g), where T is the period (the time it takes for one back-and-forth swing), L is the length of the pendulum, and g is the local acceleration due to gravity. This formula thus allows for the measure of the acceleration of gravity on Earth.

No physical pendulum is ideal, though, and for a hundred years, scientists tried but failed to use simple pendulums to measure the local acceleration of gravity. That was until 1817, when Captain Henry Kater created a reversible pendulum: one that can be hung with the heavy side down or the heavy side up. By adjusting the pendulum until the period was the same when hung either way, the distance between pivots was found to be equivalent to L in the ideal-pendulum formula, and an accurate calculation of the local acceleration of gravity on Earth, “g”, could finally be made. His pendulums, and ones like them, were used around the world up until the 1950’s to measure local gravity and hence local underground geological features, to determine the shape of the Earth, to discover that the core of the Earth is iron, and so on.

This project attempts to recreate a Kater’s “gravity pendulum” as a learning exercise for machining basics (I am a newbie) as well as for the exercise in metrology (the measurement is quite difficult to make accurately), and just for the fun of recreating a historical scientific experiment.

*Kater’s Pendulum*

A drawing of Kater’s pendulum from 1817 is shown below (horizontally, without the mount), in front and side views (from Wikipedia, see "Kater's Pendulum"). Object “d” is a heavy brass bob, objects “a” on the left and right are triangular knife edges used as pivots to balance the pendulum from either end, objects “e” are “flags” used to time the pendulum’s period, and objects “b,c” are adjustment weights incorporating a fine screw adjustment. The main shaft between the pivots “a” is also heavy brass, and about 2 meters long.


The triangular knife-edge pivots, made from “Wootz” steel, rocked on hard agate plates for rigidity and low friction. Using knives as low-friction pivots is a technique from the clock world, used only in the most precise reference clocks, including in national standard clocks right up until the atomic-clock age.

The period of the Kater’s pendulum’s swing was measured by comparing it to the swing of a pendulum clock (calibrated against the motion of stars) via a “method of coincidences.” If the two periods are slightly different, then watching for coincidences of the two allows a vernier-like measurement. So, for example, if the two coincide once out of every hundred swings, then the two periods differ by (approximately) a part in 100.

Kater’s original measurement, made in London, was g=9.81158 m/s^2, yielding six digits of precision, though the 6th digit was likely dubious. The error on his measurement of the length of the pendulum used in this calculation (of a so-called “seconds pendulum” - one whose half-period equals 1 second) was no more than 7.1 micrometers (0.00028 inches). Thus you can see the level of precision he achieved, and this is actually required!

*A 3-D Printed Gravity Pendulum Mock-Up*

I first made a 3D-printed mock-up of a gravity pendulum to test the basic idea (photos attached), and I made a rather coarse measurements of g with it. This mock-up is a hybrid of Kater’s design and the subsequent Repsold-Bessel design (see Wikipedia). I placed the “agate plate” equivalent on the pendulum and the knives on the mount, rather than the opposite as seen in the figure. This means that only one set of knives are needed instead of two, eliminating one source of error, as knives may have different radii on their edges, which would subtly affect the period.




The shaft is 0.25” by 36”, made out of Invar, which is a nickel-iron alloy with an unusually-small coefficient of thermal expansion (the inventor of Invar received the Nobel Prize for it). The use of Invar substantially alleviates temperature as a factor, otherwise corrections or re-measurements would be needed if the temperature changes, since the length of the pendulum would change with it. This is a relatively expensive material that was purchased for the “real” version, but used in the mockup too.

The cylindrical feature is the bob, filled with lead shot and weighing about 1.25 lbs. Each end of the pendulum also has plastic anvils that ride on the plastic knife edges at the pivot. Because of the use of plastic knives and anvils, I expected friction to be high and hence the measurement quality to be low. However, I was surprised with how well the pendulum operated. Starting from a swing of about 5 degrees, I found that it oscillated for at least a half hour.




To calculate g, one needs the distance between the anvils (the equivalent length of the pendulum), the period of oscillation, and the distances between the anvils and the center of gravity (again see the Kater's pendulum page on Wikipedia). Using very rough methods, including measuring the period by eye and a stop-watch, and the distances with a simple scale to about 1/32”, I obtained a value for g that was within 1% of g for the nearest large city. Not great, and very far from Kater’s accuracy, but within expectations given the mockup’s limitations.

*The Gravity Pendulum*

After making the mock-up, I began work on the “real” pendulum. It is planned to consist of:

An Invar shaft, 0.25” by 36” (acquired)
A fixed brass bob about 0.5 by 2.75”, weighing about 1.5 lbs (seen in an attached photo).
Sapphire anvils attached directly to the bob.
A brass bar holding anvils on the “light” side of the pendulum that can be positioned along the shaft to adjust the period.
Sapphire anvils attached to the positionable brass bar.
A positionable weight for fine center-of-gravity adjustments.
The platform on which the pendulum is hung will include two knives made from square tool steel blanks, accurately mounted in-line with each other. It’s imperative that all features be extremely rigid and square, and that the pendulum hangs perfectly perpendicular to the ground, or the pendulum will gyrate or walk on its pivots, and the measurements will be useless.

*Goals*

So what can be achieved? It’s not going to be possible to accurately measure local g to the degree of precision that modern instruments can. These days, that is done by dropping an object in a vacuum, and measuring the acceleration at which it falls (a direct measurement of g) via laser interferometry. I think I should be able to get another factor of 10-100 improvement over my mock-up’s value. However, I don’t have an accurate measurement of g for my immediate location to compare with, so its accuracy compared to reality could remain somewhat uncertain. (Kater didn't have anything to compare his measurement to either.)

What may be a better goal is to reach enough sensitivity to detect relatively small changes in g, rather than produce an absolute measurement of g. So, for example, measuring g multiple times with changes in altitude between measurements, perhaps between the bottom and top of a tall building. A conventional (non-reversible) pendulum actually suffices for relative measurements, but I’ll proceed with a reversible one and measure absolute g anyway.

*The Bob*

The bob (seen in an attached photo) is a relatively thin but wide brass disk (about 1/2 by 2.75”), and it’s intended to be mounted with the thin side along the shaft rather than the long side as in the Kater's pendulum figure above (in this case via a shrink or pressed fit). This places the weight as close to the pivot as reasonably possible, improving the pendulum’s performance. This is based on the Repsold-Bessel approach, which was an improvement on Kater’s pendulum.




*The Anvils*

I purchased some small pieces of flat borosilicate glass to use as anvils, but I’ve concluded that these are not going to work out, being too thick and too hard to cut to size. Instead, I intend to purchase small pieces of thin sapphire, pre-cut in a suitable size (pricey!). Meanwhile, I’ll use the glass to test methods of attachment, e.g. a trial with very thin CA glue.

*Support Structure*

I’ve also been considering concepts for the pendulum’s support structure. I’ve settled on using 1/4” square tool steel blanks as the knives, after polishing two adjoining sides. These would be mounted in angle blocks and possibly using 123-blocks to fix the angle-block’s mutual alignment. Unfortunately, there are incompatible mis-matches in hole geometries in the various blocks I acquired, and I haven’t yet figured out a satisfactory way to assemble it all. I may have to fabricate something instead of using off-the-shelf components.

It’s also necessary to consider how the assembly will be carried. It has to be rigidly suspended about a meter in the air. High rigidity is needed since any reacting movement during swings will sap energy from the pendulum plus corrupt the measurements. Indeed, Kater’s original mount was found to move by about a mm with each swing, and this had to be corrected. I was thinking of using heavy aluminum profile for supports, maybe with a cheap granite surface plate as a stable base, but I haven’t gotten far with this yet.

*Time Measurement*

Instead of Kater’s “method of coincidences”, I intend to use modern time-measurement techniques using an optical sensor and either an oscilloscope or a high-frequency counter. I thus purchased and tested an inexpensive optical sensor assembly, which I found to be good for better than 1 microsecond resolution for a single period, or better than a part in 10^6. Thus, the measurement of a single swing is already likely better than what Kater was able to achieve after his many days of exhaustive measurements. It can be improved further by averaging measurements, just as Kater did.

*Length Measurement*

The biggest issue is how to measure the the distances between the anvils, measure the center of gravity, and to measure from both anvils to the center of gravity of the pendulum. These need near micrometer precision, especially the anvil-to-anvil measurement. I only have a 36” machinist’s scale to work with for now, and I can use a camera with a macro lens aimed at the anvils to help. Still, I'm limited to the accuracy that the scale can provide, which is not great: within 0.01” can be hoped for with a machinist’s scale over a distance of 36”.

Kater also only had scales available, and he used 3 to compare measurements between them. His scales were the most accurate available at the time, and he used a microscope to help make the measurements. But such scales were hand-made, and the definition of a meter wasn't even completely fixed back then. Indeed, the definition of a meter was even set as equal to the length of Kater’s seconds pendulum measurement for a while, in case the ultimate physical reference meter was destroyed. It’s not an accident that the length of a second’s pendulum is very close to 1 meter!

A high-resolution glass scale, e.g. a DRO mounted to a long-bed mill, and using a head-mounted microscope or macro camera to locate the anvils, would do much better. And if I feel like a 5 significant digit measurement of g is within reach, I could also contact a metrology lab to see if I can get a traceable measurement.

*To Be Continued*

The project will take a while, but I’ll come here to ask questions (remember, I’m an absolute newbie at machining) and provide updates. In the mean-time, here is a link to an early question and answer thread discussing this project, from which many great ideas by the community emerged:

Micron-level accuracy over a meter.

Thank you.


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## DavidR8 (Apr 12, 2022)

That is uber-cool!!!


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## homebrewed (Apr 13, 2022)

Out of curiosity I did some searching regarding invar and the reason for its low coefficient of expansion.  It turns out that it probably is a quantum-mechanical effect, depending on the populations of two different spin states in iron.  Thw volume of iron in its solid state depends on the populations of the two spin states, and if the concentration of iron in the nickel alloy is right that change (which itself depends on temperature)  balances out thermal expansion of the alloy.

I actually was looking for some alternative way to measure the length of your invar rod and thought it might be done by measuring the time it took sound to travel down it.  See here.  Not sure if that's feasible or not, especially if it would require equipment beyond the means of a hobbyist -- it looks like it takes about 1.3us to travel 1mm (2.6us round-trip) so it would take about 2.6ms to travel down and reflect back from the end of your 1 meter length of invar.  If you want to turn that into ~1um accuracy you'd need to get down to ~100nS time accuracy.  That actually doesn't sound too bad <pun> but you'd be looking at a wave packet whose period likely is far longer than that.  Oh, well...

So in addition to using classical physics to perform a fundamental measure of gravity (which origin still is not fully known), you are using a quantum-mechanical effect to facilitate that measurement.  Cool, indeed!


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## zondar (Apr 13, 2022)

Very interesting! A physicist friend also suggested measuring it with sound waves.

But I don't need to measure the length of the rod, I need to measure the distance between the two pivots (the anvils in the above description), which are spaced somewhere along the rod. This could make the measurement more awkward, since the bob, etc., could get in the way of whatever instrument might be used.

Anyway, I plan on using my 36" ruler with its 0.01" scale, at least initially. I have a meter scale too, but that has a slightly lower resolution graduations of 0.5mm.

The 36" ruler should be compliant with gov. standard GGG-R-791H. It's an interesting document to browse through, actually, and telling of what a control-freak the government can be! But it's a good standard that goes into details about the finish, end-squareness, various accuracy factors, etc., for all kinds of different rules, including machinist's scales. It's apparently a defunct standard, but makers still claim compliance to it.

Assuming my ruler complies, I at least can say that the scale graduation error over the length should be between +0.007" to -0.0035".

Looking at the pertinent equation (e.g. Equation 2 in the Kater's Pendulum wikipedia page, under Repsold-Bessel), only the second term has distance in it, and luckily it should be significantly less important than the first term, since if T1 and T2 (the periods when hung one way and the other) are nearly identical, that whole term should be close to zero.

Edit: I'm doing my own full error analysis and should have that data soon.


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## homebrewed (Apr 14, 2022)

I don't think it would be necessary to measure the distance between the knife edges with all the other stuff attached, but, still, the requirement to measure that distance pretty much eliminates the acoustic-propagation-time approach.  A LIDAR approach with mirrors, well...a round trip over 1 meter would take about 2/3e8 seconds or about 6.66e-9 seconds.  Resolving that with an error of 2e-6 (remember it's a round trip!)  means you'd need to measure time differences on the order of femtoseconds.  Sorry, technology isn't there yet!

How about this for a devious workaround.  Let's presume you can find an accurate value for g at a location somewhere near you.  Talk someone near that location into letting you set up your apparatus so you measure the value of g you get with your setup.  Correct the length of your rod based on your equation.  Now you're good to go (anywhere).


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## zondar (Apr 14, 2022)

I think it needs to be measured with the intact apparatus because the bob will be fixed in position anyway (e.g, via a shrink fit), and removing the other components means more possible errors. For example, I'd have to mark the shaft where the component was, and then measure the mark, instead of measuring the anvil directly. And I wouldn't be able to get it back in the exact same spot without yet another small error creeping in. Furthermore, I have to find the center of gravity of the whole pendulum and factor those distances into the equation too. It's best to leave it intact once the periods are well adjusted.

Cheap laser range-finders and lidar sensors are available, but those are definitely not good enough. Doing a proper job with lasers (I assume one would be measuring phase differences rather than time of flight) would probably require more money and time than I can invest. I think a glass scale and traveling microscope would be the next most viable option if I want to build something better. At least I know in principle how to do that.

I did a little research looking for gravity maps, etc. There are specific, precise locations around the country where very high resolution gravity measurements have been made, some in labs and some in the field. Finding one of those would be ideal. But since most of this work was done decades ago, information on them is hard to find on-line. Some areas such as those of interest for natural resources, or active earthquake zones, have been contour mapped. I could probably find a spot somewhere.

It's not clear to me how I can correct the pendulum if it produces an incorrect value compared to a single known-good value. To do so would probably require collecting observations at several different locations, to create a scatter plot of g_known vs. g_measured.

I'm most of the way through an error analysis for the calculation of g from the pendulum inputs (what a major pain!). With that, I'll know if my measurement +/- the error bars plausibly fits a known value for g, and where I have to focus efforts to reduce error.


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## zondar (Apr 14, 2022)

I performed an error analysis on the calculation for g. What a pain those calculations are! I'm not 100% sure that I didn't make a mistake in such a long chain of calculations, but the numbers do look reasonable.

When I load in the numbers I had for my mock-up, I did find that the error bars (+/-) encompass the value for g that I found for the nearest largest city. 

When I put in some guesses for what I might achieve using my own tools, e.g. distance measurements to 0.01", timing measurements to 1x10^-6 (one microsecond), and with more careful tuning of the pendulum (I didn't try very hard earlier), I get an error of about 0.01 m/s^2. So, if that's all realized, I should get about 3 significant digits. 

The tuning is very important, as if that's done well (i.e., close to zero difference between the periods for both orientations), then the error on distance measurements becomes much less important. 

Kater got 5-6 significant digits. So how is it that I can hope for 3 at best despite, for example, having modern microsecond-level measurements? I have to study the numbers some more to see what dominates the error and if and how 4-5 significant digits might be doable.


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## zondar (Apr 15, 2022)

*A Tale of Two Rulers*

As I intend to use a ruler, at least initially, to measure the anvils, etc., I thought I'd see how accurate they might be.

I have both a meter stick and a yard stick, so I thought I'd compare them against each other. If two different scales, from two different companies, one made in Japan (Sanwa) and one in the U.S. (Product Engineering), in two different measurement systems match well, I would have some confidence that either will be accurate on its own.

The meter stick has 0.5mm graduations on its finest scale, while the yard stick has 0.01 inch graduations, which is about twice as fine. I laid them next to each other, butted up against a straight-edge, and took high resolution photos of both ends of the inch ruler with a macro lens.

I thought of measuring where 36" is against the meter stick, which should read 914.4 mm. However, that can be a problem because it includes the ends of both rulers, which may not reflect "0" on either. But if I moved in from the ends, say by looking at 1/2" vs. somewhere on the meter stick, the comparisons would need to use interpolation at either or both ends, since the scales won't line up exactly.

Instead, I used a vernier-like approach. Counting from either end, I found the first graduations that lined up with each other as perfectly as possible. I circled my choices for that in red.








On the left, I'm reading 13.5 mm and 35.47 inches (the rulers are reversed compared to each other to line up the finest scales).

On the right I'm reading 900.5 mm and 0.55 inches.

Subtracting them, I have 887 mm versus 34.92 inches in total.

An inch is defined as exactly 25.4mm. So 34.92 inches converts to 886.968 mm exactly. *So the two, over a span of about 35 inches, match within 0.032 mm, or about 0.0013 inches.*

Not bad at all!


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## zondar (Apr 16, 2022)

*A Tale of Two Rulers, Part II. *

Unfortunately, with a simple ruler one cannot use the vernier approach to measure something fixed in size. Instead, one needs to interpolate, and this will reduce accuracy. So how much resolution is practical if one is interpolating twice - once on each end of the measurement?

I use the same technique of comparing the two rulers, but this time I have to measure the distance between my plastic anvils on the shaft. Interpolation takes place twice, once at each anvil. Note that the plastic anvils have a distorted corners, so I am looking somewhat in from the edge. First the inch scales:






There is obviously some guess-work when interpolating by eye, and a temptation to adjust guesses after the fact (but one must resist!). For these measurements, both anvils happened to fall almost exactly on divisions, so it's pretty easy. I get 2.15" and 35.16", for a net of 33.01".

Next the mm scales:





There's more guess-work here, but I come up with about 153.4mm and 991.7mm, for a net of 838.3mm.

Converting, *there's a difference of about 0.15mm, or about 0.006"*

This is worse than the vernier measurement, but considering that there's 4 interpolations involved, each with error bars of maybe +/-0.002" or so, it's a quite reasonable and good result. If only doing one measurement (two interpolations), i.e. without subtracting values from two rulers, *the error would presumably be root-2 lower, or perhaps about 0.004"*.

These measurements really should be repeated a number of times, removing and replacing the scales or anvils each time. With repeated measurements, one can close in on more accurate estimates of distance and error. But I'm pleased with what can be accomplished with a simple ruler.

The next thing I might investigate is how to do more accurate interpolation by counting pixels, but because of the distorted plastic, that needs to wait until sharp-edged metal, glass or sapphire anvils can be photographed.


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## zondar (Apr 17, 2022)

*A Tale of Two Rulers, Part III.*

I examined if interpolation by counting pixels is beneficial. It certainly appears that it is, but I encountered some new problems.

Since I was intending to use glass or sapphire anvils, I placed some glass against the scales and photographed that. I saw, unfortunately, that it is very difficult to precisely detect the edge of the glass. This throws my plans into question, and I may likely have to go back to placing the knives on the pendulum instead of the anvils in order to better measure the critical distance between pivots. Here's an example shot, in close-up. You can see a partial reflection in the glass, but the edge is very difficult to locate:




I then tried placing knives (actually just a 3/8" tool blank) against the scales and photographed that. Here's a close up of this:




I tried to locate the edge at the intersection of the two rulers in case the knife was tilted at all (it was, slightly, in this image), and to avoid the slight shadow on the left side (that ruler is slightly thinner and so sat back a bit).

It's surprisingly difficult to precisely locate the edge, and it looks even worse as you zoom in closer. The scales, by comparison, are very easy and confident to work with at the pixel level.

I calculated the pixels per inch/mm independently for each side, but am reporting the average here:

Pixels per inch: 2216
Pixels per mm: 87.24

This is a good resolution, resulting in 0.00045" and 0.011 mm per pixel, about 20-50 times greater resolution than the markings on the rulers.

I proceeded to calculate the distances between knife-edges using pixel-level interpolation:

Length between anvils, inches: 34.094"
Length between anvils, mm: 865.94 mm

Discrepancy in inches: 0.0019"
Discrepancy in mm: 0.049 mm

So this is only slightly worse than the vernier measurements shown in a prior post, and about 3 times better than the previous measurement. Excellent!

The biggest problem found is that it's not easy to precisely locate the edge of the knives at the pixel level. Glass seems to be almost impossible, though perhaps they can be marked in some way to help. Polishing and keeping steel knives very clean could help, but it's also clear that the knives and the ruler must be held in contact. Serious planning must go into how the fully constructed pendulum will be measured.


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## Boswell (Apr 17, 2022)

if you frost the edge of the glass then shine a light into the glass from another edge it might allow the edge to show up clearer.


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## zondar (Apr 17, 2022)

Yes, I've been thinking about ways to make it more visible. That's an idea, or possibly just some black paint along the side might do. I can't disturb the main face, though, since that has to be as flat and friction-free as possible.

I'll probably go back to mounting knives on the pendulum, though. The main reason is so I can let the knives extend slightly past the bob for easier measurement. For that, I have to cut a triangular grove across the bob (as seen in the first post), plus an additional shallow cut to allow the knives to sit in it square in the bottom of the groove. Or else maybe grind some off the back edge of the knives.

I also have to cut the tool-steel blanks I bought for knives in half or smaller so they will fit. I bought some cheap diamond-dust cut-off wheels which might work, but I'm skeptical.

If anyone can point to the most appropriate Dremel-scale cut-off wheel for cutting tool steel, please do so. Thanks.


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## whitmore (Apr 17, 2022)

zondar said:


> Serious planning must go into how the fully constructed pendulum will be measured.


The usual approach is by stacking blocks (Johanson blocks for machinists, etalons for science labs).
While invar is a nice low-tempco material, it does have mechanical properties; it will
stretch when the brass is down, but shrink when the brass is above the support.   
Young's modulus for invar is 137 x 10**6 psi   ( convert to bar, * 0.068947573)
and you want its length while it's hanging...
This is the province of granite surface plates, differential measure, height gages..
calibration of cheap 1-2-3 blocks would be useful, and a 68 F/20 C measurement room.
Do you have some white gloves?


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## RJSakowski (Apr 17, 2022)

zondar said:


> If anyone can point to the most appropriate Dremel-scale cut-off wheel for cutting tool steel, please do so. Thanks.



I use the abrasive cutoff disks for the Dremel for cutting steel.


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## zondar (Apr 17, 2022)

whitmore said:


> Do you have some white gloves?


No, nor a national lab's facilities at my disposal. Sad. 

My current idea for the over-all physical construction looks more like this:

* Two glass anvils rest on a squared-up 123 block assembly.
* The assembly is clamped or bolted to a fixture plate. 
* The fixture plate is bolted to a heavy aluminum profile stand.
* The stand is mounted to a granite surface plate or similar.
* Dampening and leveling feet are mounted under the surface plate.
* Finally, an optical reader is positioned via a flexible arm (e.g. Noga-style).

Unfortunately, work on this project will be interrupted for a while. SAD!


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