Wilberforce Pendulum

Coupled spring-torsion oscillator — watching energy breathe between modes

Energy Distribution

Vertical

50%

Torsion

50%

Frequencies

ω_z (rad/s)—

ω_θ (rad/s)—

Beat Period

—

Phase

Spring const. k (N/m) 10.0

Torsion const. κ (N·m) 0.10

Coupling ε (N·m) 0.20

Mass m (kg) 0.50

Moment I (kg·m²) 0.0050

Damping γ 0.010

The Wilberforce pendulum — a mass hanging on a helical spring — has two natural modes: vertical bobbing and rotation about the spring axis. The spring geometry couples these modes: a stretched coil tends to unwind slightly, so vertical motion feeds angular momentum. When the two natural frequencies are nearly equal, energy beats slowly back and forth between pure bouncing and pure spinning in a mesmerizing exchange, a mechanical analogue of two weakly coupled quantum states.