Derivation of Fluid Pressure In A Gravitational Field

In summary, the derivation of fluid pressure in a gravitational field explains how pressure increases with depth in a fluid due to the weight of the fluid above it. It begins with the fundamental principles of hydrostatics, illustrating that pressure at a given depth is the result of the gravitational force acting on the fluid column above. The relationship is mathematically expressed by the hydrostatic pressure equation, P = P₀ + ρgh, where P is the pressure at depth, P₀ is the surface pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the depth. This concept is essential for understanding fluid behavior in various applications, including engineering and meteorology.
  • #1
bmarc92
8
0
Given that ##P = ρgh##, there's obviously a problem with the following derivation of fluid pressure under gravity. Can someone spot the flaw?

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

$$\frac{dF}{dA} = \frac{2ρgAh}{dA}$$
$$P = 2ρgh$$
 
Last edited:
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  • #3
Your derivation assumes that the pressure is uniform over the volume V. This is not correct.
 

FAQ: Derivation of Fluid Pressure In A Gravitational Field

What is fluid pressure?

Fluid pressure is the force exerted by a fluid per unit area on the surfaces in contact with it. It is a measure of the intensity of the force acting on an area and can vary with depth in a fluid due to the weight of the fluid above.

How is fluid pressure derived in a gravitational field?

The derivation of fluid pressure in a gravitational field begins with the concept of hydrostatic pressure. According to Pascal's principle, pressure at a point in a fluid at rest is transmitted equally in all directions. The pressure at a depth 'h' in a fluid is given by the equation P = P0 + ρgh, where P0 is the pressure at the surface, ρ is the fluid density, g is the acceleration due to gravity, and h is the depth below the surface.

What factors affect fluid pressure in a gravitational field?

The factors that affect fluid pressure in a gravitational field include the density of the fluid, the depth of the fluid column above the point of interest, and the gravitational acceleration. As depth increases, the pressure increases due to the weight of the fluid above.

Why does fluid pressure increase with depth?

Fluid pressure increases with depth because the weight of the fluid above a given point exerts a downward force. This force is distributed over the area of the point, resulting in an increase in pressure. The deeper you go, the more fluid there is above, leading to greater pressure.

What is the significance of hydrostatic pressure in engineering?

Hydrostatic pressure is significant in engineering as it influences the design of structures such as dams, tanks, and pipelines. Understanding how pressure varies with depth is crucial for ensuring that these structures can withstand the forces exerted by the fluids they contain or are in contact with, thus preventing failures and ensuring safety.

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