Find minimum value for coefficient of static friction

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The discussion focuses on verifying the correctness of vectors related to a problem involving static friction and Newton's 2nd Law. The user seeks confirmation on the directions of forces FT1 and FT2 in their diagrams, questioning whether the x-axes are aligned with the inclines. Participants confirm that the x-axes in the specified diagrams are indeed parallel to the inclines. The conversation emphasizes the importance of accurate vector representation before proceeding with deriving equations. Accurate vector analysis is crucial for solving static friction problems effectively.
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Homework Statement


upload_2015-9-24_20-28-15.png


Homework Equations


Newton's 2nd Law

The Attempt at a Solution


I was wondering if anyone could verify that I have all the correct vectors or if I am missing any. I just wanted to check before I started trying to derive the equations.
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The directions of FT1 and FT2 do not appear to be correct in the first and third diagrams. Are the x-axes in those two diagrams parallel to the inclines?
 
Yes
 
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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