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Penguin on Slippery Ice: Friction, Forces, and Center of Mass

Penguin on Slippery Ice: Friction, Forces, and Center of Mass
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2021-01-21 20:25
VectorTrigonometryOptimizationGeometric Construction
This example uses a “penguin on ice” scenario to demonstrate how friction and the center-of-mass location determine whether the motion remains stable and which directions are physically feasible. It is built around 2D vector geometry to represent forces, trigonometric relationships to encode direction/angle dependencies, and ratio-style comparisons related to friction limits (e.g., effective force versus maximum available friction). Construction is done in GeoGebra by creating key points and geometric references (Point; Line; OrthogonalLine for perpendicular structure; Segment for bounded connections) and then drawing force vectors (Vector) such as effective force and maximum friction capacity. Named objects like the friction coefficient μ, force vectors (F, F_eq, F_max), and center-of-mass-related elements (e.g., cofmoverfoot) connect the physical constraint to a clear geometric visualization. Suitable for high school advanced classes and early undergraduate modeling/physics-math lessons on vectors and trigonometry.

GeoGebra Commands & Parameters

  1. Point(yAxis) → G
  2. OrthogonalLine(G, yAxis) → f
  3. Point(f) → I
  4. Point(f) → H
  5. Segment(H, I) → g
  6. Point(g) → C
  7. Vector(C, D) → u
  8. Vector(D, D + (u * scale)) → W
  9. Vector(D, D + (((0, 1) * W) * Vector[(0, -1)])) → R
  10. Vector(D, D - (Vector[(1, 0)] * (W * (1, 0)))) → F_{eq}
  11. Vector(D, D + ((UnitVector[F_{eq}] * μ) * abs(R))) → F_{max}
  12. Point(xAxis) → F
  13. Point(xAxis) → E
  14. Point(xAxis) → A
  15. Point(xAxis) → B
  16. Point(xAxis) → J
  17. Point(xAxis) → K
  18. Point(yAxis) → L
  19. Point(xAxis) → M
  20. Line(L, M) → h
  21. Point(h) → N
  22. Point(h) → O
  23. Segment(C, D) → i

AI Drawing Prompt Reference

Draw a penguin on slippery ice with arrows indicating the friction force and maximum friction force. Start by creating the penguin's center of mass point C on the y-axis, then create points I and H on the perpendicular line f, and connect H and I to form the perpendicular line f. Next, create the point D where the friction force acts on f, draw the vector u representing the friction coefficient from D, and then draw the vectors F_eq and F_max for the effective and maximum friction forces. Finally, create boundary points F, E, A, B, J, K, L, and M on the x-axis, draw the ice surface line h, and create points N and O on h. Draw a line connecting the center of mass point C and the point D where the friction force acts.
The prompt above was compiled from the original drawing commands and is for reference only.