The String That Weighs the Earth

The published anchor, before the print button

Moritz prints γ(φ) = γe(1 + k sin²φ) / √(1 − e² sin²φ). The page uses the following table values directly, then recomputes the endpoints and Melbourne example rather than copying their outputs.

g_n = 9.80665 m/s² is shown only for contrast. JCGM calls it a conventional quantity value. It is not a measured local value and is not substituted into this model.

Error budget, before you measure

The mathematics is not the dominant uncertainty. The values below are explicit planning assumptions for this build, not measured household-performance facts. An irregular bob has no defensible result until its centre of mass and moment of inertia are independently established.

Printed and assembled length: 4.0 mm minimum standard allowance

Four accepted 200 mm modules can each retain a ±0.5 mm reading floor, before cutting and joints. The calculator therefore refuses u(L) below 0.0040 m.

Bob model and other systematics: 0.10% minimum standard allowance

This is a declared screening floor, not proof that the build is accurate to 0.10%. It does not rescue an unknown irregular-bob inertia; that model is refused outright.

Start and stop: 0.283 s minimum standard allowance

This follows the stated model of independent 0.20 s start and stop uncertainties: √2 × 0.20 s. Repetition reduces random scatter, not scale, bob, drag, pivot or counting errors.

Amplitude history: 0.05% minimum relative standard allowance in C

The elliptic-integral factor is exact only for an ideal plane pendulum held at one amplitude. A real damped run needs an amplitude-history allowance; the calculator will not silently set it to zero.

Named, not corrected

String mass and stretch, pivot radius or slip, air buoyancy and drag, drafts, elliptic motion, paper humidity, cutting, butt joints and tape creep. Height changes normal gravity by roughly 0.003% per 100 m; terrain, tides, groundwater and local density anomalies are absent from GRS80 here.

What the paper does

1. On every sheet, measure the 100.0 mm horizontal line, 100.0 mm vertical line and 50.0 × 50.0 mm square. Reject and reprint if either axis is outside 99.5 to 100.5 mm. Do not average the axes.
2. Cut each narrow module on its solid boundary. Butt module 1’s 200 datum against module 2’s zero datum on a flat surface. Tape across the back. Repeat without overlapping any calibrated span.
3. Place the assembled zero cross at the actual support point. The four spans now locate an 800.0 mm target. This is not an instruction to cut 800 mm of free string.
4. Tie a compact symmetric bob and adjust until its measured centre of mass aligns with the target. For a solid sphere, use the calculator’s inertia correction. Do not apply that sphere formula to nuts, coins, keys or hollow balls.
5. Use the printed angle wedge. Release without pushing. One oscillation means return to the same side moving in the same direction. A centre crossing is half an oscillation.
6. Time 100 complete oscillations, expected near 179.5 seconds, and repeat at least five times. Record every run. Then repeat at 10°, 15° and 20° to expose the amplitude drift.

Make the school-lab approximation answer back

The ideal plane pendulum’s period is multiplied by C(θ) = (2/π)K(sin(θ/2)). The calculator evaluates the complete elliptic integral, not merely the small-angle rule. It also distinguishes the pivot-to-centre distance h from equivalent simple-pendulum length.

g = 4π² L_eff C² / (t/N)²   |   L_eff,sphere = h + 2r²/(5h)   |   (u_g/g)² = (u_L/L)² + (2u_t/t)² + (2u_C/C)² + u_model²

Let non-detection be a result

The result card may say whether a GRS80 target is numerically inside or outside the declared screening interval, but it makes no detection or latitude inference. The interval is incomplete, normal gravity is not site gravity, north and south are symmetric, and at 0° or 90° an endpoint match is true by construction.

At Melbourne’s latitude, the GRS80 value is only 0.19876% above the equator. The full pole-to-equator span is 0.53024%. One hundred swings divide a fixed start-stop error across many periods; they do not divide away an incorrect length.

The check, visible

• Four modules × 200.0 mm = 800.0 mm, with end datums taken from the SVG geometry.
• Each sheet repeats both 100.0 mm axes and a 50.0 mm square. The acceptance interval is 99.5 to 100.5 mm on each axis separately.
• Content extent: checking the selected paper.
• Finite amplitude always lengthens the ideal plane-pendulum period. At 15° its correction is already larger than the equator-to-pole period signal.
• The verifier independently integrates the elliptic integral and inspects the reader-visible refusal language and printed SVG coordinates.