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Distinguish radial laws that share angular conservation

Any central force has zero torque about its center, so angular-momentum conservation alone cannot identify an inverse-square law. The supplied generator has inward acceleration μ/r² + q/r³, while the candidate model assumes only μfit/r². This is a synthetic model-comparison exercise, not a claim that a measured gravitational field has this correction.

Learning outcome: Fit only the first radial acceleration, predict the unused acceleration and independent potential difference, then vary the common potential offset conceptually.

Starting recap

Integrate mass moments about named axes, derive torque and angular-momentum constraints, and test orbital energy, angular and numerical consistency against independent evidence.

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Try

At q = 0, both residuals vanish for every supported μ. At q = 0.4, they become −0.05 in the respective units. Differentiating the fitted potential would reproduce the candidate force, not create an independent observation.

Open the experiment

Predict

Why does conserved central angular momentum fail to select the candidate inverse-square law?

Your prediction and reason

Experiment

Fit only the first radial acceleration, predict the unused acceleration and independent potential difference, then vary the common potential offset conceptually.

Test a central force law with withheld radial evidence

Worked teaching model · synthetic values · Synthetic inward acceleration magnitude mu/r²+q/r³ at positive radii; fit a pure inverse-square model only at r=1. Reserve acceleration at r=2 and an independently supplied potential difference1→2. Both fields are central; angular conservation alone cannot select the law.

Inverse-square coefficient: 1 m³/s² Additional inverse-cube coefficient: 0.2 m⁴/s²

Reset keeps your prediction, captures and practice history.

Calibrated and stipulated radial laws

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: stipulated generator (solid).2: fitted inverse-square law (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Calibrated and stipulated radial laws

stipulated generator

radius (m)	inward acceleration magnitude (m/s²)
1	1.2
1.025	1.137534
1.05	1.079797
1.075	1.026325
1.1	0.9767092
1.125	0.9305898
1.15	0.8876469
1.175	0.8475964
1.2	0.8101852
1.225	0.7751872
1.25	0.7424
1.275	0.7116418
1.3	0.6827492
1.325	0.6555747
1.35	0.6299853
1.375	0.6058603
1.4	0.5830904
1.425	0.5615763
1.45	0.5412276
1.475	0.5219618
1.5	0.5037037
1.525	0.4863843
1.55	0.4699406
1.575	0.4543146
1.6	0.4394531
1.625	0.4253072
1.65	0.4118318
1.675	0.3989852
1.7	0.3867291
1.725	0.3750278
1.75	0.3638484
1.775	0.3531604
1.8	0.3429355
1.825	0.3331474
1.85	0.3237715
1.875	0.3147852
1.9	0.3061671
1.925	0.2978974
1.95	0.2899577
1.975	0.2823306
2	0.275
2.025	0.2679507
2.05	0.2611686
2.075	0.2546403
2.1	0.2483533
2.125	0.2422959
2.15	0.2364572
2.175	0.2308266
2.2	0.2253944
2.225	0.2201515
2.25	0.2150892
2.275	0.2101992
2.3	0.2054738
2.325	0.2009058
2.35	0.1964883
2.375	0.1922146
2.4	0.1880787
2.425	0.1840747
2.45	0.180197
2.475	0.1764405
2.5	0.1728
2.525	0.1692709
2.55	0.1658487
2.575	0.1625292
2.6	0.1593081
2.625	0.1561818
2.65	0.1531466
2.675	0.1501988
2.7	0.1473353
2.725	0.1445527
2.75	0.1418482
2.775	0.1392188
2.8	0.1366618
2.825	0.1341745
2.85	0.1317544
2.875	0.1293992
2.9	0.1271065
2.925	0.1248741
2.95	0.1227
2.975	0.1205821
3	0.1185185

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.

fitted inverse-square law

radius (m)	inward acceleration magnitude (m/s²)
1	1.2
1.025	1.142177
1.05	1.088435
1.075	1.038399
1.1	0.9917355
1.125	0.9481481
1.15	0.9073724
1.175	0.8691716
1.2	0.8333333
1.225	0.7996668
1.25	0.768
1.275	0.7381776
1.3	0.7100592
1.325	0.6835173
1.35	0.6584362
1.375	0.6347107
1.4	0.6122449
1.425	0.5909511
1.45	0.5707491
1.475	0.5515656
1.5	0.5333333
1.525	0.5159903
1.55	0.4994797
1.575	0.4837491
1.6	0.46875
1.625	0.4544379
1.65	0.4407713
1.675	0.4277122
1.7	0.4152249
1.725	0.4032766
1.75	0.3918367
1.775	0.3808768
1.8	0.3703704
1.825	0.3602927
1.85	0.3506209
1.875	0.3413333
1.9	0.33241
1.925	0.323832
1.95	0.3155819
1.975	0.307643
2	0.3
2.025	0.2926383
2.05	0.2855443
2.075	0.2787052
2.1	0.2721088
2.125	0.2657439
2.15	0.2595998
2.175	0.2536663
2.2	0.2479339
2.225	0.2423936
2.25	0.237037
2.275	0.2318561
2.3	0.2268431
2.325	0.221991
2.35	0.2172929
2.375	0.2127424
2.4	0.2083333
2.425	0.2040599
2.45	0.1999167
2.475	0.1958984
2.5	0.192
2.525	0.1882168
2.55	0.1845444
2.575	0.1809784
2.6	0.1775148
2.625	0.1741497
2.65	0.1708793
2.675	0.1677002
2.7	0.1646091
2.725	0.1616026
2.75	0.1586777
2.775	0.1558315
2.8	0.1530612
2.825	0.1503642
2.85	0.1477378
2.875	0.1451796
2.9	0.1426873
2.925	0.1402586
2.95	0.1378914
2.975	0.1355836
3	0.1333333

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.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Inverse-square coefficient fit at radius1	1.2	m³/s²
Withheld inward acceleration at radius2	0.275	m/s²
Inverse-square prediction at radius2	0.3	m/s²
Withheld acceleration residual	-0.025	m/s²
Independent potential difference1 to2	0.575	m²/s²
Fitted potential difference1 to2	0.6	m²/s²
Independent potential residual	-0.025	m²/s²

Both candidate laws conserve central angular momentum, so that invariant cannot distinguish their radial dependence.

Calibration at r=1 gives mu_fit=mu+q. At r=2 the observed-minus-predicted inward acceleration is−q/8. Integrate the inward force to potential−mu/r−q/(2r²)+C; its independent difference1→2 also has residual−q/8 in the specified units. Constant potential offsets cancel. A differentiated fitted potential is not an independent force measurement.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Both candidate laws conserve central angular momentum, so that invariant cannot distinguish their radial dependence.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Distinguish radial laws that share angular conservation

Calibrate the candidate using only the acceleration magnitude at r = 1 m. That gives μfit = μ + q in the displayed normalized units. At r = 2, the generator gives μ/4 + q/8 while the fitted candidate predicts (μ + q)/4, leaving observed-minus-predicted residual −q/8.

An independently supplied potential difference can test the same candidate through another observable. Integrating the central force gives potential per mass −μ/r − q/(2r²) + C. From radius 1 to 2 its difference is μ/2 + 3q/8, while the fitted inverse-square prediction is (μ + q)/2. Its residual is again −q/8 in these supplied units; a common potential offset cancels.

At μ = 1 and q = 0.2, μfit = 1.2. The withheld inward acceleration is 0.275 m/s² rather than 0.3, and the independent potential difference is 0.575 m²/s² rather than 0.6. Both residuals are −0.025 in their respective units. The agreement of residual numbers does not make the physical quantities identical.

Changed-case check

Predict before reading the comparison. At q = 0, both residuals vanish for every supported μ. At q = 0.4, they become −0.05 in the respective units. Differentiating the fitted potential would reproduce the candidate force, not create an independent observation.

Retained investigation

Fit only the first radial acceleration, predict the unused acceleration and independent potential difference, then vary the common potential offset conceptually. Which evidence would cease to be independent if generated by differentiating the fitted potential?

Source and boundary

Primary conceptual reference checked 30 October 2026: OpenStax University Physics Volume 1. All models, packets and teaching prose are original. These are bounded mathematical investigations, not measured apparatus, physical procedures, practical certification or institutional credit.

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Public practice checks a specific claim. It does not establish independent mastery.

Which evidence would cease to be independent if generated by differentiating the fitted potential?

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