Centroids in a planar lamina FINAL

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5th Period: AP Calculus

5th Period: AP Calculus

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  • 1. Centroids in a Planar Lamina
    Young Lee
    Brian Lau
    Alvin Lau
    Jonathan Javier
  • 2. 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
  • 3. Illustration
    f(x)
    R
    g(x)
    a
    b
  • 4. Illustration
    Mass of Planar Lamina (R)
  • 5. Illustration
    f(x)
    𝜟x
    R
    f(x) — g(x)
    Subinterval, mi
    g(x)
    a
    b
  • 6. Illustration
    Mass of subinterval
  • 7. Illustration
    f(x)
    𝜟x
    R
    f(x) — g(x)
    g(x)
    a
    b
  • 8. Illustration
    Moment of mi about the x-axis
    Moment of mi about the y-axis
  • 9. Centroidal Equations
    Mass of lamina:
    Moment of planar lamina about the x-axis:
    Moment of planar lamina about the y-axis:
    Center of Mass:
  • 10. Example
    Find the centroid of the region bounded by the graphs f(x) = x and g(x) = x4
    Centroid: (0.56, 0.37)
  • 11. 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