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Centroids Lesson 7.5
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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
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Centroid Each mass in the system has a "moment" The centroid is
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
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Centroid Centroid for multiple points Centroid about x-axis
First moment of the system Also notated My, moment about y-axis Also notated Mx, moment about x-axis Total mass of the system Also notated m, the total mass
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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?
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Centroid Given 10g at (2,-1), 7g at (4, 3), and 12g at (-5,2) 7g 12g
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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, ρ and the area We sum the products of distance and mass a b •
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Centroid of Area Under a Curve
First moment with respect to the y-axis First moment with respect to the x-axis Mass of the region
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Centroid of Region Between Curves
Moments Mass f(x) g(x) Centroid
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Try It Out! Find the centroid of the plane region bounded by y = x and the x-axis over the interval 0 < x < 4 Mx = ? My = ? m = ?
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Centroid of Solid of Revolution
Mass = ρ • V Moment about x-axis = 0 Moment about y-axis f(x) a b Try evens on page 292
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Quiz Next Class Assignments Lesson 7.5 Page 292
Assignment A 1 – 21 odd Assignment B 23 – 37 odd Quiz Next Class
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