- #1
InvisibleMan1
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Homework Statement
To get used to finding the center of mass of an object, I have decided to start with a uniform density square. The square is centered at the origin. The center of mass should be at the center of the square, and thus at the origin.
When I tried to solve this however, my answer was offset by width/4. I've gone through the integration a couple times and I cannot find any errors in my work...
Homework Equations
Definitions:
M = Total mass of the object
A = Area of the object
M/A = The uniform density of the object.
r = Vector from the origin to a point on the object.
Original equation for the center of mass of a 2D object with uniform density.
M/A*int(int(rdy)dx)*1/M
w = The width/height of the square.
A = w^2
r = xi+yj
Inner integral:
Interval: [-w/2 w/2]
int(xi+yj)dy = xwi+(w^2)/4j
Outer integral:
Interval: [-w/2 w/2]
int(xwi+(w^2)/4j)dx = (w^3)/4i+(w^3)/4j
Substituting into the original equation:
M/(w^2)*((w^3)/4i+(w^3)/4j)*1/M = 1/(w^2)*((w^3)/4i+(w^3)/4j) = w/4i+w/4j
That answer should be 0i+0j, but it isn't.
The Attempt at a Solution
Using r=1i+ij instead of r=xi+yj looks like it would solve the problem, but that breaks the definition, since every point in the square is not located at <1, 1>.