⚙️ Maxwell's Wheel

Maxwell's wheel (a yo-yo on a string) converts gravitational potential energy into translational and rotational kinetic energy as it descends, then back again as it rises. Adjust the wheel's mass distribution and observe how the period and maximum speed change.

Wheel Parameters

Mass m 0.5 kg Axle radius r 0.02 m Wheel radius R 0.10 m Damping β 0.02

Energy

KE trans—

KE rot—

PE grav—

Total E—

α = r²/(R²+r²)—

I = ½mR² (solid disk)
a = g/(1 + I/mr²)
= g·r²/(R²+r²) [disk]
T = 2√(2h/a)
ω_max = v_max/r
v_max = √(2gh·r²/(R²+r²))

Rotational Dynamics

As Maxwell's wheel descends with string unrolling from its axle (radius r), both translational velocity v and angular velocity ω increase with the constraint v = ω·r. For a solid disk, moment of inertia I = ½mR², so the equation of motion gives acceleration a = g·r²/(R² + r²). The effective "gyration radius" k² = I/m = R²/2 determines how much of the gravitational work appears as rotation vs. translation. When r≪R, almost all energy goes into rotation and the descent is very slow. The wheel bounces back perfectly (no damping) or loses amplitude due to string imperfection (modeled by damping β).