[Physics] Force on the bottom of a tank full of liquid – Hydrostatic Pressure or Gravity

fluid-staticsforcespressure

Imagine a tank filled with water that has some height $h$ and at the bottom area $A$ but as it goes up, for example at height $h/2$, it's area is now $A/2 $. What's the correct way to calculate the force at the bottom of the tank? (Let's ignore atmospheric pressure for now)

  1. If I use $W=mg$, we get $F=W=ρVg=ρ(\frac{Ah}{2}+\frac{Ah}{4})g=\frac{3}{4}ρghA$
  2. If I calculate the hydrostatic pressure at the bottom, it's $p=ρgh$, and then $F=pA=ρghA.$

Which one is the correct one and why?

Best Answer

As @Berend mentioned you are calculating two different things. The first calculation gives you the weight of all water in the tank which is what an scale would read.(internal forces explained in the second part cancel out.) In the second case though you are calculating hydrostatic pressure of water. Theses answers are different because water in the tank applies force $f$ to the tank upwards as shown in the figure and according to Newton's laws the tank applies force $f$ downwards so hydrostatic pressure gets bigger than water weight and their difference is as much as the weight of water in the stripped region. enter image description here

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