Manometer -- determine height from relative pressure

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The discussion centers on calculating height using a manometer and relative pressure. A participant asserts that the correct height is 5.6 meters but is uncertain about their calculations. There is confusion regarding the equations used, specifically the relationship between gauge pressure and absolute pressure. One participant questions the accuracy of the initial equation presented, suggesting an alternative formulation. The conversation highlights the importance of correctly applying pressure equations in fluid mechanics.
daphnelee-mh
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Homework Statement
A Burdon gauge is mounted on the top of the vessel at point M . The relative pressure at M is -980Pa, determine h1 if H=1.5m, h2=0.2m
Relevant Equations
Pm=Pa- Pmabs
The correct answer is 5.6m, not sure where I made the mistake.
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##P_{M gauge}=P_{atm}-P_{M abs}##?
Doesn't look right to me.
 
I get ##P_{Mabs}=p_{atm}+P_M##
 
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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