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# Centroids in a planar lamina FINAL

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5th Period: AP Calculus

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### Centroids in a planar lamina FINAL

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