1. Centers of Mass
Chapter 8.3
March 15, 2006

2. 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

3. 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…..

4. 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)

5. 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:

6. 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 =

7. 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:

8. 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:

9. 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:

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

11. 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:

12. 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:

13. 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:

14. 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: