Centers of Mass Chapter 8.3 March 15, 2006

Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find My = 6(-2) + 1(3) + 3(3) = 0 Mx = 6(5) + 1(3) + 3(-4) = 21

Find the centroid (center of mass with uniform density) of the region shown, by locating the centers of the rectangles and treating them as point masses…..

Center of Mass: 2-Dimensional Discrete Case Suppose we have a finite collection of masses in a plane. Each mass mk is located at point (xk,yk). The system mass is: The system moments are: Moment about the x-axis is given by (yk represents the VERTICAL distance from the x-axis Moment about the y-axis is given (xk represents the HORIZONTAL distance from the y-axis)

Center of Mass: 2-Dimensional Discrete Case Suppose we have a finite collection of masses in a plane. Each mass mk is located at point (xk,yk). The System’s Center of Mass is defined to be:

We can extend this idea to the more general continuous case, where we are finding the center of mass of a region of uniform density bounded by 2 functions….. We can find the area by slicing: Right: Left: Each slice has mass: (area*density) System mass: M =

We can extend this idea to the more general continuous case, where we are finding the center of mass of a region of uniform density bounded by 2 functions….. We find the Moments, by locating the centers of the rectangles and treating them as point masses….. This slice has balance point at:

We can extend this idea to the more general continuous case, where we are finding the center of mass of a region of uniform density bounded by 2 functions….. This slice has balance point at: The slice has moment about the y-axis: The system has moment about the y-axis:

We can extend this idea to the more general continuous case, where we are finding the center of mass of a region of uniform density bounded by 2 functions….. This slice has balance point at: The slice has moment about the x-axis: The system has moment about the x-axis:

Center of Mass: 2-Dimensional Case The System’s Center of Mass is defined to be:

Find the center of mass of the thin plate of constant density formed by the region y = 1/x, y = 0, x =1 and x=2. Each slice has balance point:

Find the center of mass of the thin plate of constant density formed by the region y = cos(x) and the x-axis on the interval Each slice has balance point:

Find the center of mass of the the lamina R with density 1/4 in the region in the xy plane bounded by y = 3/x and y = 7 - 4x. Bounds: Each slice has balance point:

Find the center of mass of the the lamina R with density 1/2 in the region in the xy plane bounded by y = 6x -1 and y = 5x2. Use slices perpendicular to the y-axis. Bounds: Each slice has balance point: