Kapitza Pendulum
Inverted pendulum stabilized by rapid vertical oscillation
Oscillation
Frequency ω (rad/s)
20
Amplitude a/L
0.3
Pendulum
Initial angle θ₀ (°)
175
Gravity g (m/s²)
9.81
Damping β
0.02
Reset
Pause
Kapitza pendulum:
Pivot oscillates: y_p = a·cos(ωt)
EOM: θ̈ = [g − aω²cos(ωt)]sin(θ)/L − βθ̇
Stability condition (Kapitza 1951):
aω/√(2gL) > 1
i.e., ε = a²ω²/(2gL) > 1
Effective potential:
U_eff = −(mgL)cos(θ) + (ma²ω²/4L)sin²(θ)
Inverted position stable when ε > 1.
ε = a²ω²/(2gL):
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Stability:
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θ (deg):
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θ̇ (rad/s):
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