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Rigid Body Rotation: Circulation

Rigid Body Rotation: Circulation
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2016-04-05 06:49
VectorTrigonometryCoordinate GeometryCalculusOptimization
This example demonstrates the flow field of rigid body rotation. Key math concepts include vector fields, velocity distribution, and integral calculations. It is constructed using GeoGebra commands like Textfield, SolveODE, Sequence, SlopeField, Midpoint, SelectedIndex, Element, CurveCartesian, Sum, Segment, Line, etc. Suitable for high school students and math enthusiasts studying rigid body motion and fluid dynamics.

GeoGebra Commands & Parameters

  1. Textfield(x_{min}) → inputBox2
  2. Textfield(x_{max}) → inputBox3
  3. Textfield(y_{min}) → inputBox4
  4. Textfield(y_{max}) → inputBox5
  5. SolveODE(f, x(A), y(A), x_{max}, 0.01) → SLVF
  6. SolveODE(f, x(A), y(A), x_{min}, 0.01) → SLVB
  7. Sequence(SolveODE[f, 0, t1, x_{max}, 0.01], t1, -2, 2, 0.25) → SLVFL
  8. Sequence(SolveODE[f, 0, t1, x_{min}, 0.01], t1, -2, 2, 0.25) → SLVBL
  9. SlopeField(f, n, a, x_{min}, y_{min}, x_{max}, y_{max}) → SF
  10. Sequence(Sequence[Vector[(j, k), (j + (a * v_x(j, k)), k + (a * v_y(j, k)))], j, x_{min}, x_{max}, xstep], k, y_{min}, y_{max}, ystep) → vField
  11. SolveODE(f, x(B), y(B), x_{max}, 0.01) → numericalIntegral1
  12. SolveODE(f, x(C), y(C), x_{max}, 0.01) → numericalIntegral2
  13. SolveODE(f, x(D), y(D), x_{max}, 0.01) → numericalIntegral3
  14. Midpoint(K, L) → M
  15. SelectedIndex(pathNames) → pathIndex
  16. Element(pathList, pathIndex) → path
  17. CurveCartesian(x(path(t)), y(path(t)), t, 0, s) → partPath
  18. Sequence(Vector[path(s), path(s) + ((a * v_x(x(path(s)), y(path(s)))), (a * v_y(x(path(s)), y(path(s)))))], s, 0, 1, 0.1) → pathVectors
  19. Sum(Sequence[((path(i + dS) - path(i)) * (v_x(x(path(i + dS / 2)), y(path(i + dS / 2))), v_y(x(path(i + dS / 2)), y(path(i + dS / 2))))), i, 0, s - dS, dS]) → lineIntegral
  20. Segment(path(0.2), path(0.21)) → arrow1
  21. Segment(path(0.5), path(0.51)) → arrow2
  22. Segment(path(0.8), path(0.81)) → arrow3
  23. Textfield(time) → InputBox2
  24. SolveODE(f, x(F), y(F), x_{max}, 0.01) → numericalIntegral4
  25. SolveODE((-y), x, x(E), y(E), x_{max}, 0.01) → numericalIntegral5
  26. CurveCartesian((x(E) * cos((w * t))) - (y(E) * sin((w * t))), (x(E) * sin((w * t))) + (y(E) * cos((w * t))), t, 0, time) → stream_1
  27. CurveCartesian((x(A) * cos((w * t))) - (y(A) * sin((w * t))), (x(A) * sin((w * t))) + (y(A) * cos((w * t))), t, 0, time) → stream_2
  28. CurveCartesian((x(B) * cos((w * t))) - (y(B) * sin((w * t))), (x(B) * sin((w * t))) + (y(B) * cos((w * t))), t, 0, time) → stream_3
  29. CurveCartesian((x(C) * cos((w * t))) - (y(C) * sin((w * t))), (x(C) * sin((w * t))) + (y(C) * cos((w * t))), t, 0, time) → stream_4
  30. CurveCartesian((x(D) * cos((w * t))) - (y(D) * sin((w * t))), (x(D) * sin((w * t))) + (y(D) * cos((w * t))), t, 0, time) → stream_5
  31. CurveCartesian((x(F) * cos((w * t))) - (y(F) * sin((w * t))), (x(F) * sin((w * t))) + (y(F) * cos((w * t))), t, 0, time) → stream_6
  32. CurveCartesian((x(Q) * cos((w * t))) - (y(Q) * sin((w * t))), (x(Q) * sin((w * t))) + (y(Q) * cos((w * t))), t, 0, time) → stream_7
  33. Line(S, xAxis) → c
  34. Line(T, xAxis) → d

AI Drawing Prompt Reference

Draw a flow field diagram for rigid body rotation, showing velocity distribution and integral calculations. Start by plotting direction vectors to represent the velocity field at different points. Then, draw two symmetric curves representing velocity trajectories starting from the minimum and maximum x-values. Add several more curves to show trajectories from different initial points. Finally, include streamlines to depict the motion trajectories over time, and label key areas of the velocity field and paths.
The prompt above was compiled from the original drawing commands and is for reference only.