Moments, Center of Mass, Centroids Lesson 7.6

Mass Definition: mass is a measure of a body's resistance to changes in motion  It is independent of a particular gravitational system  However, mass is sometimes equated with weight (which is not technically correct)  Weight is a type of force … dependent on gravity

Mass The relationship is Contrast of measures of mass and force SystemMeasure of Mass Measure of Force U.S.SlugPound InternationalKilogramNewton C-G-SGramDyne

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

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

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

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) 10g 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 = x and the x-axis over the interval 0 < x < 4  M x = ?  M y = ?  m = ?

Theorem of Pappus Given a region, R, in the plane and L a line in the same plane and not intersecting R. Let c be the centroid and r be the distance from L to the centroid L R c r

Theorem of Pappus Now revolve the region about the line L Theorem states that the volume of the solid of revolution is where A is the area of R L R c r

Assignment Lesson 7.6 Page 504 Exercises 1 – 41 EOO also 49