Break a density-gradient and origin-offset ambiguity
A reported center coordinate can combine a true mass-gradient effect with an additive origin offset. One number cannot normally identify both. Reuse the 2 m normalized linear-density family, but now treat its positive gradient α and coordinate offset δ as unknown: the reading is xcm(α) + δ.
Learning outcome: Invert the independent fraction, propagate its deterministic endpoints in the correct order and then infer offset bounds. Preserve the original assumed density family.
Starting recap
Integrate signed force and normalized mass moments. Specify the system before conserving momentum, and independently test energy and observation constraints.
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At α = 1 and δ = 0, the mass fraction is 5/12 and the true center is 10/9 m. The inverse fraction recovers α = 1 without using the centroid reading; only then does the reading constrain δ.
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Why does a separately supplied mass fraction help distinguish α and δ?
Experiment
Invert the independent fraction, propagate its deterministic endpoints in the correct order and then infer offset bounds. Preserve the original assumed density family.
Break a centroid-origin ambiguity with a separately weighed segment
Worked teaching model · synthetic values · Length2 m, normalized total mass3 kg rod with lambda proportional to1+alpha x/L. Reported centroid has an unknown additive origin offset. Independent left-half mass fraction has feasible bound±.005; it is separately supplied, not inferred from the same position readout. Bound is deterministic, not a confidence interval.
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Compatible origin offsets before independent mass evidence
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: reported center minus model center (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Compatible origin offsets before independent mass evidence
reported center minus model center
| hypothesized density gradient alpha | compatible origin offset (m) |
|---|---|
| 0 | 0.2666667 |
| 0.05 | 0.2585366 |
| 0.1 | 0.2507937 |
| 0.15 | 0.2434109 |
| 0.2 | 0.2363636 |
| 0.25 | 0.2296296 |
| 0.3 | 0.2231884 |
| 0.35 | 0.2170213 |
| 0.4 | 0.2111111 |
| 0.45 | 0.2054422 |
| 0.5 | 0.2 |
| 0.55 | 0.1947712 |
| 0.6 | 0.1897436 |
| 0.65 | 0.1849057 |
| 0.7 | 0.1802469 |
| 0.75 | 0.1757576 |
| 0.8 | 0.1714286 |
| 0.85 | 0.1672515 |
| 0.9 | 0.1632184 |
| 0.95 | 0.159322 |
| 1 | 0.1555556 |
| 1.05 | 0.1519126 |
| 1.1 | 0.1483871 |
| 1.15 | 0.1449735 |
| 1.2 | 0.1416667 |
| 1.25 | 0.1384615 |
| 1.3 | 0.1353535 |
| 1.35 | 0.1323383 |
| 1.4 | 0.1294118 |
| 1.45 | 0.12657 |
| 1.5 | 0.1238095 |
| 1.55 | 0.1211268 |
| 1.6 | 0.1185185 |
| 1.65 | 0.1159817 |
| 1.7 | 0.1135135 |
| 1.75 | 0.1111111 |
| 1.8 | 0.1087719 |
| 1.85 | 0.1064935 |
| 1.9 | 0.1042735 |
| 1.95 | 0.1021097 |
| 2 | 0.1 |
| 2.05 | 0.09794239 |
| 2.1 | 0.09593496 |
| 2.15 | 0.0939759 |
| 2.2 | 0.09206349 |
| 2.25 | 0.09019608 |
| 2.3 | 0.08837209 |
| 2.35 | 0.08659004 |
| 2.4 | 0.08484848 |
| 2.45 | 0.08314607 |
| 2.5 | 0.08148148 |
| 2.55 | 0.07985348 |
| 2.6 | 0.07826087 |
| 2.65 | 0.07670251 |
| 2.7 | 0.0751773 |
| 2.75 | 0.07368421 |
| 2.8 | 0.07222222 |
| 2.85 | 0.07079038 |
| 2.9 | 0.06938776 |
| 2.95 | 0.06801347 |
| 3 | 0.06666667 |
| 3.05 | 0.06534653 |
| 3.1 | 0.06405229 |
| 3.15 | 0.06278317 |
| 3.2 | 0.06153846 |
| 3.25 | 0.06031746 |
| 3.3 | 0.0591195 |
| 3.35 | 0.05794393 |
| 3.4 | 0.05679012 |
| 3.45 | 0.05565749 |
| 3.5 | 0.05454545 |
| 3.55 | 0.05345345 |
| 3.6 | 0.05238095 |
| 3.65 | 0.05132743 |
| 3.7 | 0.0502924 |
| 3.75 | 0.04927536 |
| 3.8 | 0.04827586 |
| 3.85 | 0.04729345 |
| 3.9 | 0.04632768 |
| 3.95 | 0.04537815 |
| 4 | 0.04444444 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Reported center coordinate | 1.266667 | m |
| Independent left-half mass fraction | 0.375 | ratio |
| Recovered nominal gradient | 2 | ratio |
| Gradient lower bound | 1.846154 | ratio |
| Gradient upper bound | 2.166667 | ratio |
| Recovered nominal origin offset | 0.1 | m |
| Offset lower bound conditional on exact centroid | 0.09333333 | m |
| Offset upper bound conditional on exact centroid | 0.1066667 | m |
A centroid reading alone leaves density gradient and origin shift confounded; an independent mass fraction constrains the gradient.
Left-half mass fraction f=(1/2+alpha/8)/(1+alpha/2). Solve alpha=(1/2−f)/(f/2−1/8); the inverse decreases over this positive-density domain. Bound f by±.005 and evaluate opposite endpoints, then propagate through the monotone center function. The offset bounds treat the centroid readout as exact except for its unknown additive offset; additional measurement errors would enlarge them. Separately weighed segment fractions can cancel a common multiplicative mass gain, not arbitrary additive tare errors.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A centroid reading alone leaves density gradient and origin shift confounded; an independent mass fraction constrains the gradient.
Left-half mass fraction f=(1/2+alpha/8)/(1+alpha/2). Solve alpha=(1/2−f)/(f/2−1/8); the inverse decreases over this positive-density domain. Bound f by±.005 and evaluate opposite endpoints, then propagate through the monotone center function. The offset bounds treat the centroid readout as exact except for its unknown additive offset; additional measurement errors would enlarge them. Separately weighed segment fractions can cancel a common multiplicative mass gain, not arbitrary additive tare errors.
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Which additional error source would widen the displayed offset interval?
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