Why Are There Only 3 Pi Groups in Dimensional Analysis?

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The discussion centers on the confusion regarding the number of Pi groups in dimensional analysis, specifically questioning why there are only three instead of four. One participant calculates six dimensions and concludes there should be four Pi groups based on their interpretation. However, another participant clarifies that with six quantities involving three fundamental dimensions (mass, length, time), the correct calculation yields three dimensionless expressions. This highlights a common misunderstanding in applying dimensional analysis principles. The conversation emphasizes the importance of correctly identifying the number of fundamental dimensions involved in the analysis.
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Homework Statement



Is the question wrong ?
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Homework Equations

The Attempt at a Solution


I found that the number of dimension = 6 , primary dimensionof velocity is L / T , which is 2 , number of repeating variable = 2 , so , there are 6-2 = 4 pi group , why the question only gt 3 pi group ?
 
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How to do this question?
 
foo9008 said:
y is L / T , which is 2 , number of repeating variable = 2 , so , there are 6-2 = 4 pi group , why the question only gt 3 pi group
Not sure where you get that formula from. As I understand it, you have 6 quantities that, between them, involve 3 dimensions (MLT), so the equation can be written relating 6-3=3 dimensionless expressions.
 
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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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