# Bewegung des Massenschwerpunkts

Interaktive Visualisierung von Mehrteilchensystemen und Massenschwerpunktdynamik

> Kanonische Seite: https://elysiatools.com/de/visualizations/center-of-mass-motion

- **Kategorie:** Physics

## Überblick

Interactive visualization of center of mass motion in multi-particle systems with projectile motion, explosion dynamics, and momentum conservation. Features the fundamental equations: Center of mass position r_cm = (Σmᵢrᵢ)/(Σmᵢ) (weighted average position of all mass), COM velocity v_cm = (Σmᵢvᵢ)/(Σmᵢ) (mass-weighted average velocity), Total momentum P = M·v_cm = Σmᵢvᵢ (system momentum equals total mass times COM velocity), External force F_ext = M·a_cm (only external forces affect COM motion), Internal forces ΣF_int = 0 (internal forces cancel out, no effect on COM), Motion decomposition rᵢ = r_cm + r'_i (any particle's motion equals COM motion plus motion relative to COM). Real-time visualization includes: (1) System Display canvas showing 3 particles with different masses (m₁, m₂, m₃) and sizes proportional to mass, color-coded particles (red=m₁, blue=m₂, green=m₃), real-time COM marker (× symbol) showing weighted center position, velocity vectors on each particle showing direction and magnitude, particle trails showing individual trajectories, ground plane with gravitational field, time indicator and scenario label; (2) COM Trajectory canvas plotting the parabolic path of the center of mass regardless of internal motions, coordinate axes with distance (x) and height (y) scales, trajectory trace showing historical COM positions, current position marker, comparison with individual particle paths; (3) Momentum Analysis displaying individual particle momentum vectors (P₁, P₂, P₃) with magnitude proportional to mass×velocity, total momentum vector P (COM motion) showing conservation principle, vector addition demonstration, real-time momentum values (Total Momentum P, COM Velocity v_cm, Total Mass M), momentum conservation verification during explosions; (4) Position Coordinates showing bar charts of individual particle x-positions (x₁, x₂, x₃), COM x-position (x_cm) for comparison, real-time position updates, relative positions demonstrating motion decomposition. Interactive scenarios: (a) Projectile Motion - standard projectile launch where COM follows parabolic trajectory while particles may rotate or move internally; (b) System Explosion - explosive forces push particles apart symmetrically while COM continues on original trajectory, demonstrating internal forces don't affect COM motion, explosion parameters (force magnitude, timing, duration); (c) Sand Pendulum - leaking pendulum showing mass redistribution effects, sand stream visualization, COM motion with changing mass; (d) Motion Decomposition - visualization separating COM motion from relative motion, showing how complex motion breaks into simple translation plus rotation/vibration. Interactive parameters: Particle masses m₁, m₂, m₃ (0.5-10 kg each), Initial velocity v₀ (5-30 m/s), Launch angle θ (10-80°), Initial height h₀ (0-20 m), Explosion force (10-200 N), Explosion time (0.2-3 s), Explosion duration (0.05-0.5 s), Gravity g (1.6-20 m/s²), Air resistance k (0-0.5), Animation speed (0.1-3x). Display options: toggle particle trails, COM marker, motion decomposition view, velocity vectors. Quick presets: Cannonball (m=[5,3,2]kg, v₀=20m/s, standard projectile), Firework Explosion (m=[0.5,0.3,0.2]kg, symmetric explosion at peak), Shrapnel (high explosion force, air resistance), Space Debris (low gravity g=1.6m/s²). Educational content covers COM definition and calculation (weighted average position, balance point concept, for discrete particles and continuous objects), COM velocity and momentum (mass-weighted velocity, total momentum equals M×v_cm, conservation principles), Projectile motion of systems (COM always parabolic regardless of internal complexity, tumbling objects, rotating systems, why athletes can rotate while COM follows predictable path), Explosions and internal forces (momentum conservation during explosion, symmetric fragment distribution, COM trajectory unchanged, firework physics examples, space applications), Sand pendulum demonstration (leaking pendulum with changing mass, period independence from mass for slow leaks, COM of pendulum plus fallen sand), Motion decomposition (separating COM translation from internal motion, kinetic energy K = K_cm + K_internal, applications in molecular physics and biomechanics), Real-world applications (Sports: gymnastics, diving, throwing events; Ballistics: bullet trajectories, fragmenting projectiles; Robotics: spacecraft attitude control, robot stability; Vehicle dynamics: COM height affects stability, rollover prevention; Structural engineering: building oscillations, earthquake response), Historical context (Archimedes' center of gravity concept 3rd century BCE, Lagrange's 1788 analytical mechanics formalization, development in 18th-19th century mechanics, modern applications in quantum mechanics and astrophysics). Multi-language support (zh, en, de, fr, es, pt, ru).

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