I Derivation of Fluid Pressure In A Gravitational Field

AI Thread Summary
The derivation of fluid pressure under gravity presented in the discussion contains a critical flaw regarding the assumption of uniform pressure over the volume. The equation used, P = 2ρgh, incorrectly implies that pressure is constant, which contradicts the principles of fluid mechanics. The value of h is misapplied, as it represents both the height for work done and a factor in volume calculations. This misunderstanding leads to an incorrect conclusion about fluid pressure. Accurate derivation must consider the variation of pressure with depth in a fluid.
bmarc92
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Given that ##P = ρgh##, there's obviously a problem with the following derivation of fluid pressure under gravity. Can someone spot the flaw?

$$W = mgh$$
$$W = ρVgh$$
$$F \cdot dh = ρVgh$$
$$F \cdot dh = ρ(Ah)gh$$
$$F \cdot dh = ρgAh^{2}$$
$$\frac{d(F \cdot dh)}{dh} = \frac{d(ρgAh^{2})}{dh}$$
$$F = 2ρgAh$$

$$\frac{dF}{dA} = \frac{2ρgAh}{dA}$$
$$P = 2ρgh$$
 
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Your derivation assumes that the pressure is uniform over the volume V. This is not correct.
 
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