Overbalanced Wheel: Why Perpetual Motion Machines Fail

Overbalanced Wheel

Weighted arms N 6

Weight mass 1.5 kg

Bearing friction 0.04

Pivot friction 0.05

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Angular speed

0.0 rpm

Mechanical E

0.00 J

Heat dissipated

0.00 J

Energy put in

0.00 J

Ledger check: input − (mech + heat)

0.000 J

How it works

This is Bhaskara's wheel — the oldest documented "overbalanced wheel" perpetual-motion design (12th century). N arms of length R stick out from a hub; at the end of each arm a mass hangs from a free hinge on a rod of length L, like a mini pendulum. The intuition behind the design is that weights on the descending side should swing outward, landing farther from the axle than weights on the ascending side, so their extra torque should keep the wheel turning forever.

The simulation solves the full coupled equations of motion for the hub angle θ and every pendulum angle φᵢ (Lagrangian mechanics, no shortcuts):

(I_hub + ΣmR²)θ̈ + Σ mRL·cos(αᵢ−φᵢ)φ̈ᵢ + Σ mRL·sin(αᵢ−φᵢ)φ̇ᵢ² + Σ mgR·cos(αᵢ) = Q_θ
mL²φ̈ᵢ + mRL·cos(αᵢ−φᵢ)θ̈ − mRL·sin(αᵢ−φᵢ)θ̇² + mgL·cos(φᵢ) = Q_φᵢ
αᵢ = θ + 2πi/N,   Q = viscous friction torque (bearing + pivots)

Solved exactly, this system never produces a net torque advantage: for N evenly-spaced arms the gravity torques on a rigidly locked wheel cancel perfectly every instant (Σcos(θ+2πi/N) = 0, the sum of N-th roots of unity). With free-swinging weights the wheel can wobble and oscillate as weights momentarily bunch on one side, but that motion is exactly what the equations above already conserve — no hidden energy appears. Every joule you inject with "Give it a push" is spent, split between kinetic + potential energy still in the wheel and heat already lost to bearing/pivot friction — the ledger at the top always balances to within numerical round-off.

Free-swinging — weights hinge freely; the classic "overbalance" illusion, but it decays and settles like a damped pendulum.
Locked radial — weights are rigidly fixed to their arms (no hinge); by symmetry the net gravity torque is zero at every angle, so the wheel can't even fake an advantage.
This is the second law of thermodynamics in miniature: friction is irreversible, so mechanical energy only ever leaks into heat, never spontaneously back — a closed mechanical system can't run forever.

Weight (pivoted mass) Arm / hub

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