Derive a radial barrier from angular momentum
In a central force, angular momentum per mass h = r²θdot is conserved. Tangential speed is h/r, so its kinetic energy per mass is h²/(2r²). Combining this with gravitational potential −μ/r gives the radial effective potential Ueff = h²/(2r²) − μ/r.
Learning outcome: Derive Ueff, its stationary radius and curvature, then compare an actual circular energy with a higher-energy passage and a forbidden radius.
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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At h = 1 and r = 0.25 m, Ueff = 4 m²/s², above E = −0.2. That radius is rejected for the specified energy; clipping its negative radial kinetic energy to zero would falsely turn a forbidden state into a turning point.
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At h = 1, r = 1 and E = −0.2, the radial speed magnitude is:
Experiment
Derive Ueff, its stationary radius and curvature, then compare an actual circular energy with a higher-energy passage and a forbidden radius.
Separate radial accessibility from angular motion
Worked teaching model · synthetic values · Unit test mass in a fixed Newtonian central field with mu=1 m³/s². Effective potential per mass h²/(2r²)−mu/r, fixed specific energy−.2 m²/s². No central-body surface, encounters or relativistic corrections.
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Angular barrier plus attractive potential
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: h²/(2r²)−1/r (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Angular barrier plus attractive potential
h²/(2r²)−1/r
| radius (m) | effective potential per mass (m²/s²) |
|---|---|
| 0.25 | 4 |
| 0.296875 | 2.304709 |
| 0.34375 | 1.322314 |
| 0.390625 | 0.7168 |
| 0.4375 | 0.3265306 |
| 0.484375 | 0.06659729 |
| 0.53125 | -0.1107266 |
| 0.578125 | -0.2337473 |
| 0.625 | -0.32 |
| 0.671875 | -0.3807463 |
| 0.71875 | -0.4234405 |
| 0.765625 | -0.4531445 |
| 0.8125 | -0.4733728 |
| 0.859375 | -0.4866116 |
| 0.90625 | -0.4946492 |
| 0.953125 | -0.4987906 |
| 1 | -0.5 |
| 1.046875 | -0.4989975 |
| 1.09375 | -0.4963265 |
| 1.140625 | -0.4924001 |
| 1.1875 | -0.4875346 |
| 1.234375 | -0.481974 |
| 1.28125 | -0.4759072 |
| 1.328125 | -0.469481 |
| 1.375 | -0.4628099 |
| 1.421875 | -0.4559836 |
| 1.46875 | -0.449072 |
| 1.515625 | -0.4421299 |
| 1.5625 | -0.4352 |
| 1.609375 | -0.4283156 |
| 1.65625 | -0.4215023 |
| 1.703125 | -0.4147799 |
| 1.75 | -0.4081633 |
| 1.796875 | -0.4016635 |
| 1.84375 | -0.3952887 |
| 1.890625 | -0.3890445 |
| 1.9375 | -0.3829344 |
| 1.984375 | -0.3769608 |
| 2.03125 | -0.3711243 |
| 2.078125 | -0.3654248 |
| 2.125 | -0.3598616 |
| 2.171875 | -0.354433 |
| 2.21875 | -0.3491371 |
| 2.265625 | -0.3439715 |
| 2.3125 | -0.3389335 |
| 2.359375 | -0.3340204 |
| 2.40625 | -0.3292292 |
| 2.453125 | -0.3245568 |
| 2.5 | -0.32 |
| 2.546875 | -0.3155557 |
| 2.59375 | -0.3112208 |
| 2.640625 | -0.3069921 |
| 2.6875 | -0.3028664 |
| 2.734375 | -0.2988408 |
| 2.78125 | -0.2949123 |
| 2.828125 | -0.2910778 |
| 2.875 | -0.2873346 |
| 2.921875 | -0.2836798 |
| 2.96875 | -0.2801108 |
| 3.015625 | -0.2766249 |
| 3.0625 | -0.2732195 |
| 3.109375 | -0.2698922 |
| 3.15625 | -0.2666405 |
| 3.203125 | -0.2634622 |
| 3.25 | -0.260355 |
| 3.296875 | -0.2573168 |
| 3.34375 | -0.2543454 |
| 3.390625 | -0.2514388 |
| 3.4375 | -0.248595 |
| 3.484375 | -0.2458123 |
| 3.53125 | -0.2430887 |
| 3.578125 | -0.2404226 |
| 3.625 | -0.2378121 |
| 3.671875 | -0.2352558 |
| 3.71875 | -0.2327519 |
| 3.765625 | -0.2302991 |
| 3.8125 | -0.2278957 |
| 3.859375 | -0.2255405 |
| 3.90625 | -0.223232 |
| 3.953125 | -0.2209689 |
| 4 | -0.21875 |
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 |
|---|---|---|
| Effective potential per mass | -0.5 | m²/s² |
| Specified total energy per mass | -0.2 | m²/s² |
| Selected radius energetically accessible | 1 | flag |
| Stable circular radius for selected angular momentum | 1 | m |
| Effective potential curvature at circular radius | 1 | 1/s² |
| Radial equation acceleration | 0 | m/s² |
| Compatible radial speed magnitude | 0.7745967 | m/s |
A radial effective potential contains tangential kinetic energy; it is not solely gravitational potential.
Conserved h=r² θdot gives tangential speed h/r. Substitution into total energy yields radial kinetic energy plus Ueff. Its derivative vanishes at r=h²/mu; for mu=1 the second derivative there is1/h^6>0. A negative available radial kinetic energy rejects the radius at this energy; no imaginary or clipped speed is reported.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A radial effective potential contains tangential kinetic energy; it is not solely gravitational potential.
Conserved h=r² θdot gives tangential speed h/r. Substitution into total energy yields radial kinetic energy plus Ueff. Its derivative vanishes at r=h²/mu; for mu=1 the second derivative there is1/h^6>0. A negative available radial kinetic energy rejects the radius at this energy; no imaginary or clipped speed is reported.
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Why is the angular barrier not an additional physical repulsive force?
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