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Centroids

Centroids. Lesson 7.5. Centroid. Center of mass for a system The point where all the mass seems to be concentrated If the mass is of constant density this point is called the centroid. 4kg. 10kg. 6kg. •. Centroid. Each mass in the system has a "moment"

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Centroids

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  1. Centroids Lesson 7.5

  2. Centroid • Center of mass for a system • The point where all the mass seems to be concentrated • If the mass is of constant density this point is called the centroid 4kg 10kg 6kg •

  3. Centroid • Each mass in the system has a "moment" • The product of the mass and the distance from the origin • "First moment" is the sum of all the moments • The centroid is 4kg 10kg 6kg

  4. Also notated Mx,moment about x-axis First moment of the system Also notated My, moment about y-axis Also notated m, the total mass Total mass of the system Centroid • Centroid for multiple points • Centroid about x-axis

  5. Centroid • The location of the centroid is the ordered pair • Consider a system with 10g at (2,-1), 7g at (4, 3), and 12g at (-5,2) • What is the center of mass?

  6. Centroid • Given 10g at (2,-1), 7g at (4, 3), and 12g at (-5,2) 7g 12g 10g

  7. a b Centroid • Consider a region under a curve of a material of uniform density • We divide the region into rectangles • Mass of each considered to be centered at geometric center • Mass of each is the product of the density, ρand the area • We sum the products of distance and mass •

  8. Centroid of Area Under a Curve • First moment with respectto the y-axis • First moment with respectto the x-axis • Mass of the region

  9. Centroid of Region Between Curves f(x) • Moments • Mass g(x) Centroid

  10. Try It Out! • Find the centroid of the plane region bounded by y = x2 + 16 and the x-axis over the interval 0 < x < 4 • Mx = ? • My = ? • m = ?

  11. Centroid of Solid of Revolution f(x) • Mass = ρ • V • Moment about x-axis = 0 • Moment about y-axis b a Try evens on page 292

  12. Assignments • Lesson 7.5 • Page 292 • Assignment A1 – 21 odd • Assignment B23 – 37 odd

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