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Introduction to Finding the Center of Mass of a Lamina

Introduction to Finding the Center of Mass of a Lamina
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2021-08-19 00:53
PolygonVectorCoordinate GeometryCalculusIntegral
This example demonstrates how to calculate the center of mass of a plane lamina using integration. Key math concepts include integral calculus, vector operations, and polygon manipulation. Construction highlights involve using FitPoly to fit polygonal boundaries, Integral to compute area and centroid coordinates, and Sequence/Polygon for graphical representation. Suitable for high school and above students learning centroid calculation methods.

GeoGebra Commands & Parameters

  1. FitPoly(l1, 3) → f
  2. Integral(f, 1, 9) → a
  3. Sequence((1 + ((k - 1) * d), 0), k, 1, n + 1) → l3, {(1, 0), (1.8, 0), (2.6, 0), (3.4000000000000004, 0), (4.2, 0), (5, 0), (5.800000000000001, 0), (6.6000000000000005, 0), (7.4, 0), (8.2, 0), (9, 0)}
  4. Sequence((b + ((k - 1) * d), f(b + ((k - 1) * d))), k, 1, n + 1) → l2, {(1.4, 3.4118779378900292), (2.2, 3.445108344369352), (3, 3.4004365133494336), (3.8000000000000003, 3.324094732171628), (4.6, 3.2623152881772897), (5.4, 3.261330468707774), (6.200000000000001, 3.367372561104435), (7, 3.6266738527086266), (7.800000000000001, 4.085466630861706), (8.6, 4.78998318290502), (9.4, 5.7864557961799346)}
  5. Sequence((x(l3(2)) + ((k - 1) * d), f(b + ((k - 1) * d))), k, 1, n) → l5, {(1.8, 3.4118779378900292), (2.6, 3.445108344369352), (3.4000000000000004, 3.4004365133494336), (4.2, 3.324094732171628), (5, 3.2623152881772897), (5.8, 3.261330468707774), (6.6000000000000005, 3.367372561104435), (7.4, 3.6266738527086266), (8.200000000000001, 4.085466630861706), (9, 4.78998318290502)}
  6. Sequence((1 + ((k - 1) * d), f(b + ((k - 1) * d))), k, 1, n) → l4, {(1, 3.4118779378900292), (1.8, 3.445108344369352), (2.6, 3.4004365133494336), (3.4000000000000004, 3.324094732171628), (4.2, 3.2623152881772897), (5, 3.261330468707774), (5.800000000000001, 3.367372561104435), (6.6000000000000005, 3.6266738527086266), (7.4, 4.085466630861706), (8.2, 4.78998318290502)}
  7. Sequence(Polygon[(1 + ((k - 1) * d), 0), (x(l5(1)) + ((k - 1) * d), 0), (x(l5(1)) + ((k - 1) * d), f(b + ((k - 1) * d))), (1 + ((k - 1) * d), f(b + ((k - 1) * d)))], k, 1, n) → l6, {2.729502350312024, 2.7560866754954816, 2.7203492106795473, 2.659275785737301, 2.609852230541832, 2.6090643749662163, 2.693898048883547, 2.9013390821669027, 3.2683733046893693, 3.8319865463240177}
  8. Polygon(N, O, Q, P) → q1, n_1, o, q, p
  9. Point(xAxis) → R
  10. Segment(R, S) → h
  11. Point(xAxis) → T
  12. Segment(T, U) → i
  13. Vector(W, V) → u
  14. Vector(A_1, Z) → v
  15. Midpoint(V, Z) → B_1
  16. Vector(C_1, D_1) → w
  17. Vector(D_1, C_1) → j
  18. Midpoint(C_1, D_1) → E_1
  19. Vector(F_1, H_1) → k
  20. Vector(H_1, F_1) → l
  21. Midpoint(F_1, H_1) → I_1

AI Drawing Prompt Reference

Describe a schematic diagram illustrating the calculation of the center of mass of a plane lamina. The diagram includes a polygonal region representing the lamina's shape, points indicating the center of mass coordinates, and line segments and vectors representing integration and vector operations. The overall purpose is to demonstrate the center of mass calculation method, suitable for students learning calculus and vector operations.
The prompt above was compiled from the original drawing commands and is for reference only.