A Precision Gravity Pendulum

[S KParticipant @sk20060]

Posted by SillyOldDuffer on 25/02/2023 16:36:22:

Posted by S K on 25/02/2023 14:39:41:

…

I'm now within 0.008% of the predicted value of g at my location. 😀

What's the value? As a check I can tell my scatter-gram to highlight all the points in the UK within 0.008% of it. Be interesting to see how much of the country lights up. Wonderful if only one dot comes on, and it marks your neighbourhood.

Dave

Haha. I'm not in the U.K., but would you like my social security number and my mother's maiden name too? 😉

I'm working from an extremely fine-grained estimate. If I look at another house close to me, or change the elevation by the height of my workbench, the estimated number changes. So there would be an essentially infinite number of matching estimates, though with no matches at all in many areas.

I'd say 0.008% is pretty darned good, but I've got a few more digits of precision to go even to match Kater's claimed resolution.

As of now, I'm also 0.008% off the closest measured value that I know of, but that's about 60 miles away (I don't know the precise location). I expect I could find a spot where a high resolution measurement was made if I look a little harder.

[S KParticipant @sk20060]

Posted by John Haine on 25/02/2023 15:53:29:

How accurately can you weigh something?

You prompted some thinking: If I put it on a knife-edge, but had one end resting on a scale, and I measured the weight as I noted the position, when the weight just reached zero I'm in principle at the COG. If I then did the same in the other direction, I could plot the two lines and find an intercept. Interesting. A gram scale is probably not good enough, though.

[S KParticipant @sk20060]

I developed a nagging suspicion that my last measurement of within 0.008% of the predicted g was a little too good, and too easily obtained to be that good.

After adding a base for the pendulum's support structure, I balanced the pendulum such that the normal and reverse periods were identical to better than a part in 1000. This reduces the contribution of the center of gravity measurement to a level making it nearly irrelevant. Note that when balanced in this way, the centers of oscillation of the pendulum are coincident with the pivots. Therefore, L in the ideal pendulum formula becomes the distance between pivots. And … my new, more precise measurements did not improve the accuracy of g.

I then performed a basic error propagation analysis and concluded that my earlier finding was indeed more down to luck than actual precision. The problem is the measurement of L. For this, I used an engineer's scale (3 feet) with graduations of 0.01". How much precision can I get from this for measurements of about 2 feet?

For example, what value would you accord this example shot (just taken with my phone camera):

In this case, it seems clear: the anvil lies pretty much right on 20.83". Pixels are hard to see, but it's within one pixel of that, I believe. If it lay somewhere else, well, trust me when I say that interpolation is not easy, especially when trying to find the edge of a transparent surface. In this case, it's easier to see since it lies on a black line.

I also did tests that calibrated a much better camera and macro lens at a per-pixel level, to about 0.00045" per pixel, and I found that I could interpolate to about 0.002" with decent confidence (i.e. +/- 0.001" ). But this is almost certainly encroaching on the precision of the ruler itself (which I separately estimated to be within about 0.0013" over 35" ).

I did a basic error propagation analysis that showed that I'm almost completely limited by this measurement (the period T is measured to far, far greater precision). I've thus concluded that this project has measured g to all the precision that is currently possible for me to obtain:

g = 9.79 +/- 0.02 m/s^2.

This easily matches the estimate for g within those limits as well as g for the nearest known measured value.

I've thought about building a traveling microscope + glass scale that would likely do a little better. But it would be best if it's hung vertically with the pendulum rather than horizontally. Other than that or bringing the pendulum to NIST, that's all I can get.

(As an aside: With all due respect to Henry Kater, I have developed doubts about his claims, including that he measured L, also with just a ruler, albeit with a microscope rather than a macro setup, to a precision 100 times greater than the above.)

It was a fun project. 😀

Edited By S K on 02/03/2023 18:36:40

[S KParticipant @sk20060]

OK, so I made one of the all-time most notorious mistakes in my calculations: I accidentally mixed metric and English units! 🙄

It didn't change the result much, but it definitely changed the error propagation calculation. After another trial with more care and a bit of other tuning, and of course correcting the error, I found:

g = 9.7964 +/- 9×10^-4 m/s^2

The RMS noise on my period measurements also dropped to 5.5 us.

So I gained 2 significant digits to 5 digits (by the skin of my teeth), mostly by correcting the error. The measurement remains identical to the National Geodetic Survey's computed estimate within my error bars.

I think my next step is to calculate how much altitude I'd have to gain before I could find a repeatable change in period. That way I can see how good it is as an "invariable" pendulum – one used for relative g measurements (e.g. after calibration at a site where g is know to high precision), rather than absolute g measurements.

Edited By S K on 06/03/2023 22:00:26

[Author]

Posts