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不稳定平衡的演示

不稳定平衡的演示
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2020-05-25 15:01
向量解析几何变换几何作图
①本范例通过动态演示展示了一个不稳定平衡的物理现象，通过图形变化直观呈现平衡与倾斜的关系；②涉及的核心数学概念包括向量运算、反射变换、序列生成、解析几何中的线段交点计算；③作图要点运用了GeoGebra命令：Sequence（序列生成）、Vector（向量）、Mirror（反射）、Translate（平移）、Element（元素选择）、Line（直线）、Intersect（交点）、Segment（线段）；④适合高中阶段学生学习向量、变换以及动态几何，可用于物理力学原理的数学建模演示。

GeoGebra 命令及参数

  1. Sequence(1 / n, n, 1, k) → l1
  2. Sequence(Sum[l1, j], j, 0, k) → l2
  3. Sequence((1 / 2 * Element[l2, i]), i, 1, k) → l3
  4. Vector(((-Element[l3, k]) - 1, 0, (-0.1 * k)), (0, 0, 0)) → u
  5. Sequence((Element[l3, i] + 1, -0.5, (0.1 * (i + 1))), i, 1, k - 1) → l5
  6. Sequence((Element[l3, i] + 1, 0.5, (0.1 * (i + 1))), i, 1, k - 1) → l6
  7. Sequence((Element[l3, i] + 2, -0.5, (0.1 * (i + 1))), i, 1, k - 1) → l5b
  8. Sequence((Element[l3, i] + 2, 0.5, (0.1 * (i + 1))), i, 1, k - 1) → l6b
  9. Sequence((Element[l3, i] + 2, -0.5, (0.1 * i)), i, 1, k - 1) → l7
  10. Sequence(Polygon[Element[l5, i], Element[l6, i], Element[l6b, i], Element[l5b, i]], i, 1, k - 1) → l8
  11. Sequence(Prism[Element[l8, i], 0.1], i, 1, Length[l8]) → l9
  12. Mirror(l9, (0, 0, 0)) → l9'
  13. Translate(l9', u) → l9''
  14. Element(l9'', A1) → B1
  15. Element(l9'', A2) → B2
  16. Element(l9'', A3) → B3
  17. Element(l9'', A4) → B4
  18. Element(l9'', A5) → B5
  19. Element(l9'', A6) → B6
  20. Element(l9'', A7) → B7
  21. Element(l9'', A8) → B8
  22. Element(l9'', A9) → B9
  23. Element(l9'', A10) → B10
  24. Element(l9'', A11) → B11
  25. Element(l9'', A12) → B12
  26. Element(l9'', A13) → B13
  27. Element(l9'', A14) → B14
  28. Element(l9'', A15) → B15
  29. Element(l9'', A16) → B16
  30. Element(l9'', A17) → B17
  31. Element(l9'', A18) → B18
  32. Element(l9'', A19) → B19
  33. Element(l9'', A20) → B20
  34. Element(l9'', A21) → B21
  35. Element(l9'', A22) → B22
  36. Element(l9'', A23) → B23
  37. Element(l9'', A24) → B24
  38. Element(l9'', A25) → B25
  39. Element(l9'', A26) → B26
  40. Element(l9'', A27) → B27
  41. Element(l9'', A28) → B28
  42. Element(l9'', A29) → B29
  43. Element(l9'', A30) → B30
  44. Element(l9'', A31) → B31
  45. Element(l9'', A32) → B32
  46. Element(l9'', A33) → B33
  47. Element(l9'', A34) → B34
  48. Element(l9'', A35) → B35
  49. Element(l9'', A36) → B36
  50. Element(l9'', A37) → B37
  51. Element(l9'', A38) → B38
  52. Element(l9'', A39) → B39
  53. Element(l9'', A40) → B40
  54. Element(l9'', A41) → B41
  55. Element(l9'', A42) → B42
  56. Element(l9'', A43) → B43
  57. Element(l9'', A44) → B44
  58. Element(l9'', A45) → B45
  59. Element(l9'', A46) → B46
  60. Element(l9'', A47) → B47
  61. Line(A, zAxis) → f
  62. Intersect(f, xOyPlane) → B
  63. Segment(A, B) → g
  64. Mirror(l9'', g) → l9'''
  65. Mirror(B2, g) → C2
  66. Mirror(B3, g) → C3
  67. Mirror(B4, g) → C4
  68. Mirror(B5, g) → C5
  69. Mirror(B6, g) → C6
  70. Mirror(B7, g) → C7
  71. Mirror(B8, g) → C8
  72. Mirror(B9, g) → C9
  73. Mirror(B10, g) → C10
  74. Mirror(B11, g) → C11
  75. Mirror(B12, g) → C12
  76. Mirror(B13, g) → C13
  77. Mirror(B14, g) → C14
  78. Mirror(B15, g) → C15
  79. Mirror(B16, g) → C16
  80. Mirror(B17, g) → C17
  81. Mirror(B18, g) → C18
  82. Mirror(B19, g) → C19
  83. Mirror(B20, g) → C20
  84. Mirror(B21, g) → C21
  85. Mirror(B22, g) → C22
  86. Mirror(B23, g) → C23
  87. Mirror(B24, g) → C24
  88. Mirror(B25, g) → C25
  89. Mirror(B26, g) → C26
  90. Mirror(B27, g) → C27
  91. Mirror(B28, g) → C28
  92. Mirror(B29, g) → C29
  93. Mirror(B30, g) → C30
  94. Mirror(B31, g) → C31
  95. Mirror(B32, g) → C32
  96. Mirror(B33, g) → C33
  97. Mirror(B34, g) → C34
  98. Mirror(B35, g) → C35
  99. Mirror(B36, g) → C36
  100. Mirror(B37, g) → C37
  101. Mirror(B38, g) → C38
  102. Mirror(B39, g) → C39
  103. Mirror(B40, g) → C40
  104. Mirror(B41, g) → C41
  105. Mirror(B42, g) → C42
  106. Mirror(B43, g) → C43
  107. Mirror(B44, g) → C44
  108. Mirror(B45, g) → C45
  109. Mirror(B46, g) → C46
  110. Mirror(B47, g) → C47
  111. Element(l9'', A48) → B48
  112. Mirror(B48, g) → C48
  113. Element(l9'', A49) → B49
  114. Mirror(B49, g) → C49
  115. Element(l9'', A50) → B50
  116. Mirror(B50, g) → C50
  117. Element(l9'', A51) → B51
  118. Mirror(B51, g) → C51
  119. Element(l9'', A52) → B52
  120. Mirror(B52, g) → C52
  121. Element(l9'', A53) → B53
  122. Mirror(B53, g) → C53
  123. Element(l9'', A54) → B54
  124. Mirror(B54, g) → C54
  125. Element(l9'', A55) → B55
  126. Mirror(B55, g) → C55
  127. Element(l9'', A56) → B56
  128. Mirror(B56, g) → C56
  129. Element(l9'', A57) → B57
  130. Mirror(B57, g) → C57
  131. Element(l9'', A58) → B58
  132. Mirror(B58, g) → C58
  133. Element(l9'', A59) → B59
  134. Mirror(B59, g) → C59
  135. Element(l9'', A60) → B60
  136. Mirror(B60, g) → C60
  137. Mirror(B1, g) → C1