Coupled Pendulum

pendulum ℓ, m pinned to slider M, spring k, rail μ

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(M + m) ẍ + mℓ sin(φ) φ̈ + mℓ φ̇² cos(φ) + kx + (M + m) g = 0
ℓ φ̈ + ẍ sin(φ) + g sin(φ) = 0

trace path force vectors

t = 0.00 s physics dt = 1/240 s (fixed) sim steps/frame ≈ –

Settings & Telemetry

Pendulum

ℓ — length (m)

m — bob mass (kg)

φ₀ — initial angle (deg)

φ̇₀ — initial rate (deg/s)

Block & Spring

M — block mass (kg)

k — stiffness (N/m)

x₀ — initial position (m)

ẋ₀ — initial velocity (m/s)

Environment & Dissipation

g — gravity (m/s²)

c_x — block damping (N·s/m)

c_φ — pendulum damping

μ — rail friction

Live State

x block position0

ẋ block velocity0

φ pendulum angle0

φ̇ angular velocity0

Forces (Newton check)

spring force F=−kx0

rail normal force N0

friction force0

block accel ẍ0

pendulum accel φ̈0

Energy

kinetic T0

potential V0

total E = T+V0

dissipated (∫ damping+friction)0

conservation drift0

total kinetic potential