Centers of Mass

Particles of masses 6, 1, 3 are located at (-2,5), (3,3) & (3,-4) respectively. Find x , y( )

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…..

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

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( )

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⎛⎝⎜⎞⎠⎟

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:

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:

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:

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: