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How does one get the value of acceleration of gravitation on earth accurately to 5 significant digits by experiment without electronic device?

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g varies by one part in $10^5$ with about a 25 m change in height. g varies by 500 parts in $10^5$ as you move around on the surface of the earth, especially from equator to poles. Is this just idle curiosity on your part? – mwengler Apr 26 '12 at 0:21
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I think he is asking how you would have measured this before we had fancy electronics. – Colin K Apr 26 '12 at 0:25
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Pendulums are accurate enough to see the pole-equator variation for sure, since this was a famous 18th century experiment that validated Newton's oblate model of the Earth. Five significant figures seems to require either a 10m arm, or a meter arm and $10\mu m$ precision machining, and neither seems easy using the metrology and tools of the 18th century. – Ron Maimon Apr 26 '12 at 5:02

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A pendulum of (long) length L will tick with a period of $2\pi\sqrt{L\over g}$, and air resistance can be made negligible for a mm-sized oscillation of a heavy object on a several-meter long rigid arm. You need to determine the location of the center of mass accurately to know the effective value of $L$, but this can be done arbitrarily accurately by balancing the arm and weight on a fulcrum (or by accurately finding the CM of each of the parts and measuring the configuration of the parts accurately). Then you just have to count the number of oscillations over a long enough period of time. This is a practical method that allows the determination of g to 5 significant figures (assuming the error on L is the signifcant one, the lever is 10m, and the position measurements are at the .1mm scale) with no technology.

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+1 Can't beat simplicity of that one. Also it is important to keep oscillation angle as small as possible. – Pygmalion Apr 26 '12 at 6:01
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You can make very very sensitive mechanical balances without any electronics. And with a torsion bar and an optical measurement of the deflection you can probably do a reasonably accurate relative measurement

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