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Rainbow Pendulum: Multi-Ball Swing Simulation

Rainbow Pendulum: Multi-Ball Swing Simulation
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2021-09-24 14:48
SequenceTransformationParametric EquationSlider AnimationLocus
In collections Pendulum simulations
① This GeoGebra example demonstrates a pendulum system with multiple colored balls swinging under gravity and damping, creating a rainbow-like visual effect. ② Core mathematical concepts include simple harmonic motion, numerical integration of differential equations, vector decomposition, iterative updates of angle and angular velocity, and the effect of damping on amplitude. ③ Construction highlights: using commands such as Rotate, Translate, Segment to create pendulum arms and spheres, Sequence and Zip to generate multiple pendulums in batch, Slider to control initial angle, damping, and time step, and objects like Length, Sphere, angleV, force, speed, damping, along with lists of angles (angle1 to angle16) and angular velocities (angleV1, etc.). ④ Suitable for high school physics or math classes teaching vibration, periodic motion, and introductory numerical simulation, as well as university physics demonstrations to help students intuitively understand discrete solutions to differential equations.

GeoGebra Commands & Parameters

  1. Rotate(Segment[O, (posx, posy)], π / 2, xAxis) → s
  2. Rotate((posx, posy), π / 2, xAxis) → P
  3. Sphere(P, 1) → sph
  4. Zip("angle" + (Text[k]), k, Ln) → LnamAng
  5. Zip("angleV" + (Text[k]), k, Ln) → LnamAngV
  6. Zip("angleA" + (Text[k]), k, Ln) → LnamAngA
  7. Sequence(k, k, 0, 0.15, 0.01) → Ldiff, {0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.060000000000000005, 0.07, 0.08, 0.09, 0.09999999999999999, 0.10999999999999999, 0.11999999999999998, 0.12999999999999998, 0.13999999999999999, 0.15}
  8. Length(Ldiff) → size
  9. Translate(Rotate[((L * sin(angle1)), (L * cos(angle1))), π / 2, xAxis], Vector[(0, 1, 0)]) → P1
  10. Translate(Rotate[((L * sin(angle2)), (L * cos(angle2))), π / 2, xAxis], Vector[(0, 2, 0)]) → P2
  11. Translate(Rotate[((L * sin(angle3)), (L * cos(angle3))), π / 2, xAxis], Vector[(0, 3, 0)]) → P3
  12. Translate(Rotate[((L * sin(angle4)), (L * cos(angle4))), π / 2, xAxis], Vector[(0, 4, 0)]) → P4
  13. Translate(Rotate[((L * sin(angle5)), (L * cos(angle5))), π / 2, xAxis], Vector[(0, 5, 0)]) → P5
  14. Translate(Rotate[((L * sin(angle6)), (L * cos(angle6))), π / 2, xAxis], Vector[(0, 6, 0)]) → P6
  15. Translate(Rotate[((L * sin(angle7)), (L * cos(angle7))), π / 2, xAxis], Vector[(0, 7, 0)]) → P7
  16. Translate(Rotate[((L * sin(angle8)), (L * cos(angle8))), π / 2, xAxis], Vector[(0, 8, 0)]) → P8
  17. Translate(Rotate[((L * sin(angle9)), (L * cos(angle9))), π / 2, xAxis], Vector[(0, 9, 0)]) → P9
  18. Translate(Rotate[((L * sin(angle10)), (L * cos(angle10))), π / 2, xAxis], Vector[(0, 10, 0)]) → P10
  19. Translate(Rotate[((L * sin(angle11)), (L * cos(angle11))), π / 2, xAxis], Vector[(0, 11, 0)]) → P11
  20. Translate(Rotate[((L * sin(angle12)), (L * cos(angle12))), π / 2, xAxis], Vector[(0, 12, 0)]) → P12
  21. Translate(Rotate[((L * sin(angle13)), (L * cos(angle13))), π / 2, xAxis], Vector[(0, 13, 0)]) → P13
  22. Translate(Rotate[((L * sin(angle14)), (L * cos(angle14))), π / 2, xAxis], Vector[(0, 14, 0)]) → P14
  23. Translate(Rotate[((L * sin(angle15)), (L * cos(angle15))), π / 2, xAxis], Vector[(0, 15, 0)]) → P15
  24. Translate(Rotate[((L * sin(angle16)), (L * cos(angle16))), π / 2, xAxis], Vector[(0, 16, 0)]) → P16
  25. Zip(Vector[(0, 0, 0), (0, k, 0)], k, Ln) → Lv
  26. Zip(Translate[O, Vector[v]], Vector[v], Lv) → TO
  27. Segment((0, 1, 0), P1) → segment1
  28. Segment((0, 2, 0), P2) → segment2
  29. Segment((0, 3, 0), P3) → segment3
  30. Segment((0, 4, 0), P4) → segment4
  31. Segment((0, 5, 0), P5) → segment5
  32. Segment((0, 6, 0), P6) → segment6
  33. Segment((0, 7, 0), P7) → segment7
  34. Segment((0, 8, 0), P8) → segment8
  35. Segment((0, 9, 0), P9) → segment9
  36. Segment((0, 10, 0), P10) → segment10
  37. Segment((0, 11, 0), P11) → segment11
  38. Segment((0, 12, 0), P12) → segment12
  39. Segment((0, 13, 0), P13) → segment13
  40. Segment((0, 14, 0), P14) → segment14
  41. Segment((0, 15, 0), P15) → segment15
  42. Segment((0, 16, 0), P16) → segment16

AI Drawing Prompt Reference

Describe a multi-ball pendulum simulation with multiple colored balls swinging under gravity and damping, creating a rainbow effect. Balls are connected by pendulum arms, with arms of possibly equal length and balls at different heights. The simulation uses numerical integration to update angles and angular velocities iteratively. Construction order: first create pendulum arms and ball centers, then create arms, then simulate the swing.
The prompt above was compiled from the original drawing commands and is for reference only.