How Do You Calculate the Center of Mass for Irregular Objects?

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To calculate the center of mass for irregular objects, one approach involves dividing the object into simpler shapes and finding the center of mass for each piece. In this case, the object was split into two sections with centers at coordinates (2,5) and (3,2). Using the grid's squares as units for measurement is acceptable for this calculation. However, there is some confusion regarding the y-axis coordinate that needs clarification. Accurate identification of coordinates is crucial for determining the overall center of mass effectively.
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Homework Statement


find center of mass

Screen Shot 2011-10-30 at 12.57.22 PM.png


Homework Equations


(r1m1+r2m2+...)/m1+m2+...

The Attempt at a Solution



cut into two pieces with one center at 2,5 and one 3,2. used each square on grid as one unit is that okay?
 
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Using the rectangles of the grid as one unit is ok. The y-axis coordinate is a little confusing.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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