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Centroids in a planar lamina FINAL
 

Centroids in a planar lamina FINAL

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5th Period: AP Calculus

5th Period: AP Calculus

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    Centroids in a planar lamina FINAL Centroids in a planar lamina FINAL Presentation Transcript

    • Centroids in a Planar Lamina
      Young Lee
      Brian Lau
      Alvin Lau
      Jonathan Javier
    • Introduction to Centroids
      Centroid (Center of gravity)
      The center of mass of an object of uniform density
      Characteristics of a centroid
      If a region has a line of symmetry, then its centroid is on that line of symmetry
      If a region has two lines of symmetry, then the centroid is the point where they intersect
      Moment
      A measure of tendency to cause a body to rotate about a specific point
      No movement at the center of gravity
      Planar Lamina
      A thin and flat plate of material of constant density
      Density: measure of mass per unit of area
    • Illustration
      f(x)
      R
      g(x)
      a
      b
    • Illustration
      Mass of Planar Lamina (R)
    • Illustration
      f(x)
      𝜟x
      R
      f(x) — g(x)
      Subinterval, mi
      g(x)
      a
      b
    • Illustration
      Mass of subinterval
    • Illustration
      f(x)
      𝜟x
      R
      f(x) — g(x)
      g(x)
      a
      b
    • Illustration
      Moment of mi about the x-axis
      Moment of mi about the y-axis
    • Centroidal Equations
      Mass of lamina:
      Moment of planar lamina about the x-axis:
      Moment of planar lamina about the y-axis:
      Center of Mass:
    • Example
      Find the centroid of the region bounded by the graphs f(x) = x and g(x) = x4
      Centroid: (0.56, 0.37)
    • Centroids: Real Life Application
      Balance
      Balancing thin objects on a point (2-D)
      Balancing two people on a see-saw (1-D)
      Infrastructure
      Knowing the centroid of parts of buildings ensures buildings to be stable (3-D)
      Crimes
      Mapping the density of crimes in a certain area
      Locomotion
      Locating the centroid of a vehicle to increase stability