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Flux Circle: Rigid Body Rotation

Flux Circle: Rigid Body Rotation
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2016-04-19 09:03
CircleTrigonometryCoordinate GeometryCalculus
This example demonstrates the flux circle phenomenon generated during rigid body rotation. Key math concepts include Circle, Rotation, Limits, Hyperbola, Isosceles Triangles, Normal Distribution, and Line Segment. The construction utilizes GeoGebra commands like Cartesian Curve, SelectedIndex, Element, Derivative, Sequence, Circle, Translate, Line, Sum, and Segment. Suitable for high school and above, it illustrates the connection between rigid body motion and geometric shapes.

GeoGebra Commands & Parameters

  1. CurveCartesian((a * cos(((2 * π) * t))) + x(N) - a, (a * sin(((2 * π) * t))) + y(N), t, 0, 1) → TestPath
  2. SelectedIndex(pathNames) → pathIndex
  3. Element(pathList, pathIndex) → path
  4. Derivative(path) → Dpath
  5. Sequence(Sequence[Vector[(j, k), (j + (scale * F_1(j, k)), k + (scale * F_2(j, k)))], j, xmin, xmax, xstep], k, ymin, ymax, ystep) → vField
  6. Circle(divPt, 0.5) → divGlowCircle
  7. Sequence(Segment[Rotate[curlPt - (0.5, 0), k + curlAngle, curlPt], Rotate[curlPt + (0.5, 0), k + curlAngle, curlPt]], k, 0, 2.0944, 1.0472) → spokes
  8. Translate(normalvector_1, PtPath_1) → n_1
  9. Translate(normalvector_2, PtPath_2) → n_2
  10. Translate(normalvector_3, PtPath_3) → n_3
  11. Line(I, xAxis) → c
  12. Line(K, xAxis) → e
  13. Line(L, xAxis) → g
  14. Line(M, xAxis) → h
  15. Sum(Sequence[(((F_1(x(path(i + dS / 2)), y(path(i + dS / 2))), F_2(x(path(i + dS / 2)), y(path(i + dS / 2)))) * (y(Dpath(i + dS / 2)), (-x(Dpath(i + dS / 2))))) * dS), i, 0, s - dS, dS]) → TestFlux
  16. Derivative(TestPath) → TestPath'
  17. CurveCartesian(x(path(t)), y(path(t)), t, 0, s) → partPath
  18. Segment(path(0.3), path(0.3 + 0.01)) → arrow1
  19. Segment(path(0.6), path(0.6 + 0.01)) → arrow2
  20. Sequence(Vector[path(i), path(i) + ((scale * F_1(x(path(i)), y(path(i)))), (scale * F_2(x(path(i)), y(path(i)))))], i, 0, s, 0.1) → pathVectors
  21. Sequence(CurveCartesian[(x(Element[inicialPts, i]) * cos((w * t))) - (y(Element[inicialPts, i]) * sin((w * t))), (x(Element[inicialPts, i]) * sin((w * t))) + (y(Element[inicialPts, i]) * cos((w * t))), t, 0, time], i, 1, 8) → pathlinesList
  22. Sequence(((x(Element[inicialPts, i]) * cos((w * time))) - (y(Element[inicialPts, i]) * sin((w * time))), (x(Element[inicialPts, i]) * sin((w * time))) + (y(Element[inicialPts, i]) * cos((w * time)))), i, 1, 8) → particlesList

AI Drawing Prompt Reference

Draw a diagram illustrating the flux circle phenomenon generated during rigid body rotation. The center features a rotating circle surrounded by multiple spokes and flow lines representing the force field and flux. Include various geometric shapes like circles, triangles, and curves, along with arrows indicating the vector field. The overall construction should demonstrate the connection between rigid body motion and geometric forms, with flow lines and arrows showing the direction and intensity of the force field.
The prompt above was compiled from the original drawing commands and is for reference only.