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" 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

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

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?

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

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 •

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

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

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 = ?

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

Quiz Next Class Assignments Lesson 7.5 Page 292 Assignment A 1 – 21 odd Assignment B 23 – 37 odd Quiz Next Class