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Simple Harmonic Motion Derivation

Simple Harmonic Motion Derivation
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2013-11-01 23:03
VectorFunctionTrigonometryCalculusGeometric Construction
This example demonstrates how to derive the equation of motion for a particle attached to one end of a horizontal spring with a fixed other end. It involves core mathematical concepts like differential equations, functions, trigonometry, and vectors. The construction uses GeoGebra commands/objects such as Point, Polygon, Line, Intersect, Mirror, Segment, Circle, Vector, Ray, Midpoint, and CurveCartesian. Suitable for high school and above students to understand the mathematical principles of simple harmonic motion.

GeoGebra Commands & Parameters

  1. Point(xAxis) → O
  2. Point(Circle[O, 2]) → E
  3. Polygon(O, E, E', E'', E''', E'''', E''''') → poly1, h'', h''', h'''', h''''', h'''''', h, h'
  4. Point(Circle[O, abs(l_0)]) → x_{eq}
  5. Line(x_{eq}, yAxis) → b
  6. Point(h'') → Z
  7. Line(Z, xAxis) → a''
  8. Intersect(f, b, 1) → P
  9. Mirror(P, Line[x_{eq}, y = x]) → P'
  10. Line(P', yAxis) → a
  11. Intersect(a, a'') → Z'
  12. Segment(O, P') → Spring
  13. Circle(P', 1) → c
  14. Intersect(c, a, 2) → N'
  15. Intersect(c, a, 1) → W'
  16. Vector(P', N') → N
  17. Vector(P', W') → W
  18. Ray(P', x_{eq}) → l
  19. Intersect(c, l) → H',
  20. Vector(P', H') → H
  21. Midpoint(O, P') → S
  22. Vector(S, S') → s
  23. Vector(I, I') → i
  24. Point(b) → U
  25. Point(a) → V
  26. Line(U, xAxis) → b'
  27. Line(V, xAxis) → a'
  28. Intersect(b', h'') → U'
  29. Intersect(a', h'') → V'
  30. Midpoint(U, U') → l'
  31. Vector(l', U) → u
  32. Vector(l', U') → u'
  33. Midpoint(V, V') → l''
  34. Vector(l'', V') → v'
  35. Vector(l'', V) → v
  36. Intersect(b, a') → V''
  37. Midpoint(Z, Z') → X'
  38. Vector(X', Z) → z'
  39. Vector(X', Z') → z
  40. CurveCartesian(Spring * t / (2 * π * Loops) + Coiling * cos(t) - Coiling + x(O), Width * sin(t), t, 0, 2 * π * Loops) → Coil

AI Drawing Prompt Reference

Draw a diagram illustrating simple harmonic motion with a particle attached to one end of a horizontal spring fixed at the other end. Show the particle oscillating around an equilibrium point, with key points like the equilibrium, particle position, vertical and horizontal segments, and spring deformation. Use circles and vectors to represent the motion and plot the particle's trajectory.
The prompt above was compiled from the original drawing commands and is for reference only.