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We were introduced fluid mechanics in class today I am thoroughly confused on some basic points

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  • What is the proof of Pascal's law? And what does it exactly mean? I know that it states that the pressure is transmitted equally to every point in fluid. Let's take a point in the left beaker at depth h from the piston. So does it have a pressure of $P(atm)$ + $P$ + $\rho*g*h$? ($P(atm)$= atmospheric pressure). If so, then that pressure must be same throughout the horizontal line at h. So if the piston in right beaker is at height h' from that horizontal line, does that imply $P(atm) + P + rho*g*h= P(atm) + P+ \rho*g*h'? h=h'$? So what does it really mean to be transmitted to all points? Does it mean that pressure at every point will increase by P than original value?

  • Why is pressure even $\rho*g*h$ at depth $h$? My understanding is that pressure in fluid is really just the normal force the layer applies on upper layers as in the table applies on a book and the book applies the pressure back on table. So if it is a uniform cylinder then it makes sense that the weight above the layer at h must be $\rho*g*h*A$ and the pressure $\rho*g*h$. What about when shape changes? Why is pressure same at all points at a depth? Then the weight above layer isn't equal to what mentioned above. It just doesnt make sense no matter how many answers I read here or on other sites.

  • Proof for archimedes principle is simple enough if it is a cylinder or cuboid by taking pressure difference at top and bottom. What about irregular shapes?

  • When a container accelerates on an incline with acceleration other than g*sin(theta) where theta is the angle of inclined plane, why is the water surface slanted? Can you explain with an FBD?

Please don't close this question. I know it's a bit too basic but I have tried to understand these answers for hours and I can't find a satisfying explanation elsewhere.

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  • $\begingroup$ you can easily prove pascals law by energy conservation and volume conservation (because this law assumes that the liquid is incompressible) ill give the link to the easy proof here $\endgroup$
    – Amit Verma
    Commented Sep 19 at 12:24

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When working with a hydraulic system, it is often assumed that the pressure is high enough that you can ignore the differences in height. Notice in your equation that the atmospheric pressure drops out. (That may not be true in a different situation.) Then you are left with a pressure difference of ρg(h-h'). Archimedes principle is based on the fact that if you remove an object which is subject to a buoyant force, then the volume of fluid that had been displaced (but now returns), must now be supported by the same buoyant force. That is independent of the shape of the volume displaced.

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  • $\begingroup$ But what's the proof of archimedes principle? And in the Pascal's law diagram does the pressure increase by P at every point? $\endgroup$
    – Qwerty
    Commented Apr 8, 2021 at 15:28
  • $\begingroup$ What would consider more proof than simple logic? In a fluid, the pressure is the same at every point that is at the same height. $\endgroup$
    – R.W. Bird
    Commented Apr 8, 2021 at 19:01
  • $\begingroup$ I agree that it's somewhat obvious that (neglecting atmosphere) pressure is due to the weight of the water column above, and thus varies with height. However, I don't think it's obvious that pressure has to be equal in all directions. See this recent post physics.stackexchange.com/questions/824179/… $\endgroup$
    – Mariano G
    Commented Aug 13 at 17:22

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