![Page 1: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/1.jpg)
Centers of Mass
![Page 2: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/2.jpg)
Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find x , y( )
![Page 3: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/3.jpg)
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…..
![Page 4: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/4.jpg)
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….. x + y2 =4 and x−y=2
We can find the area by slicing:
ρ
![Page 5: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/5.jpg)
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…..
ρx =4 −y2 and x =2 + y
We find the Moments, by locating the centers of the rectangles and treating them as point masses…..
This slice has balance point at:
horizontal,vertical( )
![Page 6: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/6.jpg)
Center of Mass: 2-Dimensional Case The System’s Center of Mass is defined to be:
x =My
My =
Mx
M
x , y( )
M x = m ki=1
n
∑ yk
M y = m ki=1
n
∑ xk
M = m ki=1
n
∑ M = ρ⋅Aρeaxy⎛⎝⎜⎞⎠⎟a
b
∫ dxy⎛⎝⎜⎞⎠⎟
M y = ρ⋅Aρeaxy⎛⎝⎜⎞⎠⎟a
b
∫ xk
xy⎛⎝⎜⎞⎠⎟
⎛⎝⎜
⎞⎠⎟d
xy⎛⎝⎜⎞⎠⎟
M x = ρ ⋅Aρeaxy⎛⎝⎜⎞⎠⎟a
b
∫ yk
xy⎛⎝⎜⎞⎠⎟
⎛⎝⎜
⎞⎠⎟d
xy⎛⎝⎜⎞⎠⎟
![Page 7: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/7.jpg)
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.
ρ
Each slice has balance point:
![Page 8: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/8.jpg)
Find the center of mass of the thin plate of constant density formed by the region y = cos(x) and the x-axis
ρ
Each slice has balance point:
![Page 9: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/9.jpg)
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.
Each slice has balance point:
Bounds:
![Page 10: Centers of Mass. Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find](https://reader036.vdocuments.us/reader036/viewer/2022082621/5a4d1b477f8b9ab0599a3c01/html5/thumbnails/10.jpg)
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.
Each slice has balance point:
Bounds: