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Pendulum with Unrolled Dial

Pendulum with Unrolled Dial
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2019-12-27 01:16
FunctionTrigonometrySequenceCalculusSlider Animation
This GeoGebra example demonstrates a pendulum motion while simultaneously displaying the sine and cosine values of the swing angle, along with the pendulum bob's trajectory, on an unrolled dial. The core mathematical concepts involve trigonometric functions (sine and cosine), arithmetic mean, median, and indefinite integrals (used for arc length or area calculations). The construction uses GeoGebra commands such as Circle, Segment, Intersect, Dilate, Sequence, Insert, Element, Rotate, Vector, and Textfield to build the pendulum geometrycars, with multiple lists (list1 to list9) storing values at different times, and a slider θ controlling the swing angle for dynamic updates. It is suitable for high school or early college students to intuitively understand trigonometric functions and periodic motion, and can serve as a visual aid in physics courses for simple harmonic motion.

GeoGebra Commands & Parameters

  1. Circle(C, c) → d
  2. Segment(C, D) → d_1
  3. Intersect(d, d_1) → E,
  4. Dilate(E, 1 / n, C) → D'
  5. Sequence(((a * cos((((i * 2) * π / n) * k))) + (b * sin((((i * 2) * π / n) * k))), (-((a * sin((((i * 2) * π / n) * k))) - (b * cos((((i * 2) * π / n) * k)))))), i, 1, n) → list1
  6. Sequence(A + Sum[First[list1, i]], i, 1, n, 20) → list2
  7. Insert(A, list2, 1) → list3
  8. Sequence(Segment[Element[list3, i], Element[list3, i + 1]], i, 1, n) → list4
  9. Segment(C, E) → e
  10. Sequence(Rotate[F, (-i), O], i, 0, 6.283185307179586, 0.5235987755982988) → liste1
  11. Sequence(Rotate[F - (0, 0.2), (-i), O], i, 0, 6.28319, 0.5236) → liste2
  12. Sequence(Segment[Element[liste1, i], Element[liste2, i]], i, 1, Length[liste1]) → liste3
  13. Circle(O, r) → f
  14. Segment(C, B) → g
  15. Sequence(Rotate[F - (0, 0.3), (-i), O], i, 0, 6.28319, 0.5236) → liste2b
  16. Sequence(Text[i - 1, Element[list3, i + 1] - (0.1, 0.25)], i, 1, Length[list3] - 1) → liste5
  17. Sequence(G + Sum[First[list1, i]], i, 1, n, 4) → listemin
  18. Sequence(Rotate[F - (0, 0.075), (-i), O], i, 0, 6.28319, 0.10472) → liste60
  19. Sequence(Rotate[F, (-i), O], i, 0, 6.28319, 0.10472) → liste60a
  20. Sequence(Segment[Element[liste60a, i], Element[liste60, i]], i, 1, Length[liste60]) → liste6
  21. Sequence(Segment[Element[listemin, i], Element[listemin, i + 1]], i, 1, n) → liste8
  22. Sequence(G + Sum[First[list1, i]], i, 1, n, 20) → liste9
  23. Sequence(Text[i, Element[listemin, i + 1] - (0.071, 0.25)], i, 0, Length[listemin] + 5, 5) → liste7
  24. Circle(O, r + 0.1) → h
  25. Circle(O, r + 0.035) → p
  26. Circle(O, r + 0.175) → q
  27. Circle(O, r + 0.225) → s
  28. Sequence(Text[i, Element[liste2b, i + 1] - (0.065, 0.025)], i, 1, Length[liste2b] - 1) → liste4
  29. Element(list2, Length[list2]) → H
  30. Rotate(F - (0, 0.5), (((-h1) * 2) * π / 12), O) → F'
  31. Rotate(F - (0, 0.2), (((-m1) * 2) * π / 60), O) → I
  32. Vector(O, F') → u
  33. Vector(O, I) → v
  34. Element(list3, Mod[h1, 12] + 2) → J
  35. Vector(K, J) → w
  36. Element(listemin, Mod[m1, 60] + 1) → L
  37. Vector(M, L) → z
  38. Element(listemin, Length[listemin]) → N
  39. Textfield(h1) → ChampTexte1
  40. Textfield(m1) → ChampTexte2

AI Drawing Prompt Reference

Draw a pendulum with a swinging angle controlled by a slider. Simultaneously display the sine and cosine values of the swing angle on an unrolled dial, and draw the pendulum's trajectory on the dial. The dial should also have hour and minute indicators, as well as areas for numerical displays.
The prompt above was compiled from the original drawing commands and is for reference only.