Why do I keep getting a slightly different result from the following two ways of determining the center of mass of a rigid, geometrically simple object?
The object is a rectangular 5(x) by 7(y) sheet of uniform rigid material, with a 2(x) by 3(y) rectangle missing from the upper right corner.
The official formula in my textbook:
x center of mass = ( m1x1 + m2x2 ) / ( m1 + m2 )
y center of mass = ( m1y1 + m2y2 ) / ( m1 + m2 )
This gives results of x = 2.18 and y = 3.09.
A torque calculation that would seem more intuitive to me:
The xy coordinate plane is parallel to floor and I watch it from above. Object is in the first quadrant (top right), touching the x and y axes.
I imagine balancing the object on a razor blade parallel to the y-axis. I solve for x by assuming that the torques on the left and right side must cancel out.
I rotate the razor blade so that it's parallel to the x-axis. I balance the object again. I solve for y by assuming again that the torques on each side must cancel out.
This gives results of x = 2.07 and y = 2.9.
The difference between the results given by the two methods is small, but significant. What's going on?
I checked my math several times and even tried a different problem with a simpler geometry. Again the results differed by a small but significant value. I'm fine with having to learn the textbook method, but would like to know why the torque approach gives results that are 3-5 % off.