The Tilted Scale: How Far Off Level a Platform or Dial Scale Can Sit Before Its Indication Error Exceeds a Stated Fraction of the Load

A scale that is not level weighs wrong, but by how much depends on its mechanism. For a spring or load-cell scale that responds to the force along its axis or normal to its platform, the indication is W·cos θ, so the error is second order in the tilt angle θ: it reaches 0.5% only at 5.73 degrees and 2% only at 11.48 degrees, tilts that are obvious to the eye. For a mechanism with a first-order term — an internal mass whose weight shifts the zero when the case tilts, modelled here by a zero-shift coefficient s in grams per degree — the thresholds are far stricter: at s = 5 g per degree, a 1000 g load reaches 0.5% error at 1.00 degree and 2% at 4.00 degrees. The practical question is therefore not "is it level?" but "which kind of scale is it, and what is its zero shift when tilted?" A buyer can answer the second part on the spot by tilting the unloaded scale and watching its zero. Two no-tools level tests are bounded here: a water surface in a 200 mm vessel, read to 1 mm end to end, certifies level to 0.29 degrees, and a rolling coin begins to move at roughly 0.29 degrees for a rolling-resistance coefficient of 0.005. Either test is more than fine enough for a cosine mechanism, and fine enough to keep a scale with s = 5 g per degree within about 1.4 g of zero shift. AI disclosure: Dave, an AI research system, assisted in the preparation of this work.