Question

Belinda needs tofind the center of mass of a sculpture she has
made so that it will hang in a gallery correctly. The sculpture is
all in one plane and consists of various shaped uniform objects
with masses and sizes as shown. The length of the rectangle is L
= 1.6 m and the mass of the octagon is M = 2 kg. Where is the
center of mass of this sculpture? Assume the thin rods connecting
the larger pieces have no mass and place the reference frame
origin at the top left corner of the sculpture.

Answer 1

To calculate the CM, we treat all the masses as if they were point masses at their centers. We can write the coordinates of the centers of each piece. Thenwe can average the x coordinates (weighted average... weighted by the mass of each piece), and average the y coordinates. The CM is just the weightedaverage of the coordinates of the individual pieces.

So... top block, mass is 2.0, center is (0.8,-0.25)
left sphere, mass is 5.0, center is (0, -2.5)
hanging block, mass is 2.0, center is (1.0, -1.9)
right sphere, mass is 2.0, center is (1.6, -3.0)

Now... to find the average x position, we take mass times x-coord for each one, add them together, and divide by the total mass (note: total mass is2+2+5+2 = 11)

so now... x coord of cm = (2*.8 + 5*0 + 2*1 + 2*1.6)/11 = 0.618

and y coord of cm = (2*-0.25 + 5*-2.5 + 2*-1.9 + 2*-3) / 11 = -2.07

so the center of mass is at (0.618, -2.07)