A mass ratio leaves the screen
The balance that does not care where it is
A coin is not a laboratory weight. It is enough to build a coarse comparison that can expose its own bad pivot. Print two sheets, make a paper beam, reverse the loads, and let gravity cancel itself.
Make one side win
Drag the unknown cradle. At equal arms, two nominal 10c coins and one nominal 20c coin produce equal torque. Move either load one tick and equality ends.
What the coins promise, and what they do not
These are the statutory standard weights and allowable variations in Australia’s Currency (Australian Coins) Determination 2019. They are not measurements of the worn coins in your pocket.
| coin | standard | allowed ± | legal interval |
|---|
1 × 10c = 5.65 g
Nominal difference: 0.01 g. Legal extremes can differ by as much as 1.16 g.
1 × 20c = 11.30 g
A nominal identity, not a guaranteed balance. Legal extremes can differ by 1.76 g.
The error budget comes first
The dominant absolute uncertainty is usually the unknown true mass of the 20c reference. Its statutory allowance alone is ±0.78 g, or ±6.9%, before wear or dirt. This paper build does not promise better than about ±15% absolute agreement.
Pin friction, paper flex, cradle centring and air currents are not honestly known in advance. Repeats and reversal turn some of them into visible spread, but printed geometry alone cannot guarantee even 10 mg resolution. Air buoyancy remains, so this compares apparent mass rather than claiming a true-mass calibration.
First gate: ask the printer
Print at 100% or Actual Size. Disable Fit, Shrink and Scale to page. Measure both 100.0 mm bars on sheet 1. A common scale error cancels in an arm ratio, but unequal horizontal and vertical scale, nonlinear distortion, and misfitting folds do not.
Not measured. This page cannot vouch for the printed sheet.
Turn two sheets into a balance
You need scissors, transparent tape, a straight pin or needle at least 30 mm long, two equal-height supports, a ruler, and dry objects from 4.1 to 12.9 g. Do not load food, liquids, valuables or more than 13 g.
Cut and fold the beam
Cut the 240 × 34 mm blank. Score the lines after 10, 20 and 30 mm. Fold three 10 mm faces into a triangular tube and tape the 4 mm tab. The sheet is rotated only to fit the common paper box.
Pierce and hang the pointer
Before closing the tube, pierce the two circled pivot crosses printed on opposite 10 mm faces at the 120 mm midpoint. Close the beam, pass the pin through both holes, and attach the 55 mm keel below it so the centre of mass stays below the pivot. A keel ending at the pivot is neutrally stable; one above it is unstable. Rest the pin ends on equal-height cups or books.
Make two sliding cradles
Cut and fold both shallow trays. For each tray, wrap two strips loosely around the triangular beam as closed suspension loops, one at each side of the tray; overlap and tape each strip only to itself and its tray wall, never to the beam. The paired loops must slide together to a tick without gripping it. Centre the load between them so its centre hangs vertically below the selected station.
Demand free return
With empty cradles at 40 mm, adjust only the sliding rider. Nudge each end down. The pointer must return from both directions. If it stays where released, rubs, or changes zero after swapping the empty cradles, stop.
The weighing that cancels the beam
A folded beam has an unknown fixed torque. One reading inherits that offset. Reversal changes which load sees which side while the beam’s own torque stays fixed. Averaging the two positions cancels that fixed term to first order.
x̄ = (xR + xL)/2
mu = mc rc/x̄ = 452.0/x̄ grams
The physical protocol
- Put one 20c coin at 40.0 mm and the unknown on the other arm. Approach level from above and below at least three times. Record the midpoint as xR.
- Swap both loads. Keep the 20c coin at 40.0 mm. Repeat for xL.
- Reject if either station falls outside 35–110 mm, repeats span more than 2.0 mm, reversal differs by more than 5%, or the pointer sticks.
- If it passes, report a coarse coin-referenced apparent mass. Compare against another published coin or a kitchen scale if available, remembering a 1 g display step is already coarse here.
What cancels, exactly
For vertical loads on a nearly horizontal beam, torque about the pivot is τ = rxmg. At balance, mugx = mcgrc. The same local g multiplies both sides, so the ratio does not depend on its magnitude.
A force spring scale is different: gravity competes with a spring’s restoring force, so a mass-number inferred from extension depends on local gravity unless adjusted or compensated. This balance is not simply “immune to gravity”. Gravity gradients, buoyancy, magnetic and electrostatic forces, deformation, levelling and zero still exist.
If the printer gives both arms the same scale factor s, that factor cancels too. If the reference and unknown distances acquire different factors, the reported ratio changes by sr/sx. That is why two axes are measured and why one correction number is refused. Longer arms reduce ideal station and friction-equivalent errors, but they are not an unlimited improvement: paper sag, twist and off-page size grow too.
Read the nonlinear scale
With a nominal 20c coin at 40.0 mm, station and nominal mass are reciprocals. Equal gram steps are not equal millimetre steps.
| x (mm) | nominal g | coin reference contribution | 1 mm station contribution |
|---|
The visible check
The coin table comes from the Australian statutory instrument linked below. The distinction between mass comparison and force indication follows NIST Handbook 44 Appendix B (2014 edition). The air-buoyancy qualification follows BIPM Monographie 1.
Sources and limits
Currency (Australian Coins) Determination 2019, Schedule 2019, Part 1, Division 1. NIST Handbook 44, Appendix B, 2014 edition. 3rd CGPM Resolution 2. BIPM Monographie 1.
No prototype result is claimed here. The reader’s free-return, repeat and reversal gates decide whether their individual build earns a number. Coin wear, soil and actual household-printer behaviour were not measured by this page.