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Understanding Centroids: Center of Mass and Moments Explained

Learn about centroids, center of mass, moments, and calculating centroids for multiple points and regions between curves. Practice problems included.

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Understanding Centroids: Center of Mass and Moments Explained

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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 Quiz Next Class

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