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Equation 140 · Part 12 · The Bend an Elevator Cannot Fake

Numerator: 1.414 × 10^-18

h=2 σyϕ,zz=1.414×10−183.43×10−23 m−2≈203 m.h = \sqrt{\frac{\sqrt2\,\sigma_y}{\phi_{,zz}}} = \sqrt{\frac{1.414\times10^{-18}}{3.43\times10^{-23}\,\mathrm{m^{-2}}}} \approx 203\,\mathrm m.
1.414×10−181.414\times10^{-18}

What this part means

The complete quantity above the fraction bar.

Its job in the formula

1.414 × 10^-18 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

​ σ y ​ , gives h=2 σyϕ,zz=1.414×10−183.43×10−23 m−2≈203 mh = \sqrt{\frac{\sqrt2\,\sigma_y}{\phi_{,zz}}} = \sqrt{\frac{1.414\times10^{-18}}{3.43\times10^{-23}\,\mathrm{m^{-2}}}} \approx 203\,\mathrm m. The fourth-derivative Taylor remainder at this height is smaller than the leading term by a factor of roughly 10^{-8} , confirming that the quadratic approximation used above is, for this purpose, exact. Two hundred metres of half-spacing — roughly four hundred metres end to end — is not a laboratory bench, but it is not exotic either: it is close to the height difference a transportable optical lattice clock actually used, linked to a second clock at the base of the Tokyo Skytree tower by fibre, to measure the ordinary first-derivative geopotential difference directly against independent levelling [ 9 ] . Comparable transportable-clock campaigns have…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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