A thin rectangular plate of uniform area density 1 = 1.13 kg/m2 has a length a =

Question

A thin rectangular plate of uniform area density 1 = 1.13 kg/m2 has a length a = 0.420 m and a width b = 0.180 m. The lower left corner is placed at the origin, (x, y) = (0, 0). A circular hole of radius r = 0.034 m with center at (x, y) = (0.048 m, 0.048 m) is cut in the plate. The hole is plugged with a disk of the same radius that is composed of another material of uniform area density 2 = 5.22 kg/m2. What is the distance from the origin of the resulting plate's center of mass?

A thin rectangular plate of uniform area density sigma 1 = 1.13 kg/m2 has a length a = 0.420 m and a width b = 0.180 m. The lower left corner is placed at the origin, (x, y) = (0, 0). A circular hole of radius r = 0.034 m with center at (x, y) = (0.048 m, 0.048 m) is cut in the plate. The hole is plugged with a disk of the same radius that is composed of another material of uniform area density sigma 2 = 5.22 kg/m2. What is the distance from the origin of the resulting plate's center of mass?

Explanation / Answer

 

This is the same as taking the original plate, full rectangle, and adding to it a circle with density   5.22 – 1.13  =   4.09

 

First, we need to know the mass of the additional circle, the mass of the full rectangle and the total mass.

mass of circle = pi r^2 density = pi * 34^2 * 4.09 = 14854 (note: I'm putting distances in mm to make the numbers easier... dont worry about this)
mass of full rectangle = length * width * density = 420 * 180 * 1.13 = 85428
mass of total = 14854 + 85428  =  100282


Now... we can write the coordinates of the center of mass of each piece:

circle: (48, 48) remember, I’m using mm

full rectangle: (210, 90)

both together: (x, y) both unknown... we’re trying to find these

Finally, using the center of mass equation:

(mass * xcoord) of total = (mass * xcoord) of circle + (mass * x coord of full rectangle)

 

Or...

100282 * x  = 14854 * 48 + 85428 * 210

Solve for x...

x = 186

Do the same thing for y coord:

100282 * y = 14854 * 48 + 85428 * 90

y = 83.78

Distance from origin? square root (x^2 + y^2) =

= sqrt (186^2 + 83.78^2) = 204 millimeters = 0.204 m 

This is the same as taking the original plate, full rectangle, and adding to it a circle with density   5.22 – 1.13  =   4.09

 

First, we need to know the mass of the additional circle, the mass of the full rectangle and the total mass.

mass of circle = pi r^2 density = pi * 34^2 * 4.09 = 14854 (note: I'm putting distances in mm to make the numbers easier... dont worry about this)
mass of full rectangle = length * width * density = 420 * 180 * 1.13 = 85428
mass of total = 14854 + 85428  =  100282


Now... we can write the coordinates of the center of mass of each piece:

circle: (48, 48) remember, I’m using mm

full rectangle: (210, 90)

both together: (x, y) both unknown... we’re trying to find these

Finally, using the center of mass equation:

(mass * xcoord) of total = (mass * xcoord) of circle + (mass * x coord of full rectangle)

 

Or...

100282 * x  = 14854 * 48 + 85428 * 210

Solve for x...

x = 186

Do the same thing for y coord:

100282 * y = 14854 * 48 + 85428 * 90

y = 83.78

Distance from origin? square root (x^2 + y^2) =

= sqrt (186^2 + 83.78^2) = 204 millimeters = 0.204 m

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