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🌐 3D SPHERES
📈 F vs r
⚖️ EQUAL FORCE
🕯️ CANDLE
⚙ Parameters
📊 Data
g = GM/r²
Patch A = Ω·r²
Same patch on all spheres
g·A = const ✓
👆 drag · pinch zoom · tap sphere
📡 Selected
🔴 PATCH AREA
—
Same size on every sphere
g×Area = const ✓
📐 Patch Compare
Key: The red patch is the same physical size on every sphere. g differs, but Force = g×Area stays constant — geometry explains 1/r².
⚙ Graph
Overlays
F=GMm/r²
dF/dr=−2GMm/r³
d²F/dr²=+6GMm/r⁴
📍 At r=—
🧮 Derivation
1ST DERIV (power rule)
dF/dr=GMm·(-2)·r⁻³
2ND DERIV
d²F/dr²=+6GMm/r⁴
⚖️ Equal Force
Overlays
A=F₀·r²/GM
A∝r² for same F
g∝1/r² → g·A=const
🎯 Target F₀
REQUIRED FORCE
—
Newtons per patch
Per-Sphere Areas
FORCE ON PATCH
F=g·A=GM/r²·A
SOLVE FOR AREA
A=F·r²/GM ∝ r²
Key: To feel the same force at each sphere, the patch must grow as r². This is the inverse-square law in reverse.
🕯️ Settings
Overlays
E=I₀/r²
vs E=I₀/r
Sq: 2×→¼, 3×→1/9
Lin: 2×→½, 3×→1/3
Real light obeys 1/r²
LINEAR LAW E=I₀/r (hypothetical)
📊 Comparison
INVERSE SQUARE E
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I₀/r² lux
LINEAR / INV.SQ. RATIO
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Linear stays brighter at distance
Distance Table
Key: With 1/r², brightness collapses fast — at 4m already 1/16. With 1/r only 1/4. This is why stars look so dim!