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 6kg 10kg

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 6kg 10kg

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

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) 10g 7g 12g

7 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

8 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

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

10 Try It Out! Find the centroid of the plane region bounded by y = x and the x-axis over the interval 0 < x < 4  M x = ?  M y = ?  m = ?

11 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

12 Assignments Lesson 7.5 Page 292 Assignment A 1 – 21 odd Assignment B 23 – 37 odd