Torricelli's theorem Derivation help

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In the discussion about Torricelli's theorem, participants question why atmospheric pressure is considered equal at two points (P1 = P2). The connection to Bernoulli's Principle is emphasized, highlighting that total pressure, which includes both fluid and atmospheric pressure, must be accounted for. The role of atmospheric pressure in fluid dynamics is clarified, indicating that it influences the pressure calculations in the theorem. Understanding these pressure dynamics is crucial for accurately applying Torricelli's theorem. The conversation ultimately underscores the importance of recognizing the contributions of atmospheric pressure in fluid mechanics.
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Homework Statement



In torricellis theorem why we take P1=p2=atmospheric pressure?

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The Attempt at a Solution

 
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Torricelli is a special case of Bernoulli's Principle. You must look at the total pressure (fluid + atmospheric) which acts.
 
so what must be the pressure of the fluid?
 

Thread 'Magnitude of buoyant force in fluids of different densities'

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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