B Validity of hydrostatic pressure equation

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The hydrostatic pressure equation P=hρg is valid despite the downward force of weight because pressure in fluids acts in all directions due to the nature of fluid equilibrium. When a force is applied to a fluid, such as pressing down on a water balloon, the pressure is transmitted equally in all directions, demonstrating that pressure is not limited to the direction of the applied force. In a sealed system, like connected pipes, the pressure remains constant throughout, illustrating that the forces exerted on different sides of a fluid parcel balance out, resulting in no net force. This behavior aligns with Pascal's law, which states that pressure changes in an enclosed fluid are transmitted undiminished throughout the fluid. Therefore, the hydrostatic pressure equation remains applicable in understanding fluid behavior under pressure.
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The hydrostatic pressure equation, P=hρg is derived from P=F/A where F=mg and A=cross sectional area of the water column. As weight only acts downwards and it's force is only exerted in downward direction. Then why does pressure acts in all directions? How is the P=hρg still valid?
 
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pgirl1729 said:
The hydrostatic pressure equation, P=hρg is derived from P=F/A where F=mg and A=cross sectional area of the water column. As weight only acts downwards and it's force is only exerted in downward direction. Then why does pressure acts in all directions? How is the P=hρg still valid?
A short and not very revealing answer is that fluids are fluid. At equilibrium, they cannot sustain a shear stress. [Stress is a generalization of pressure that one may eventually learn about]

Perhaps more appealing is to think about a water balloon. If you press down on its top, it expands out the sides. Your top pressure has become a side pressure.

Or consider a vertical pipe connected by an elbow to a horizontal pipe. Seal the horizontal pipe at its free end. Fill the pipes with water and apply pressure at the top. Would you agree that the longitudinal pressure in both pipe segments should be the same? Remove the pipes and replace them with a bath of water. Maintain the pressure on the top of the bath. Would you expect the pressure anywhere to change?
 
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pgirl1729 said:
As weight only acts downwards and it's force is only exerted in downward direction. Then why does pressure acts in all directions?

The weight of the balls in the below image is also only exerted in downward direction, and yet they will push sideways against the walls. You don't need complex fluid dynamics to understand the basic principle qualitatively.

hex_packing.png
 
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pgirl1729 said:
Then why does pressure acts in all directions? How is the P=hρg still valid?
The pressure on which the water column fron the left is "pressed" is equal to the pressure with which it is fron the right and they cancel each other out.
 
Pisica said:
The pressure on which the water column fron the left is "pressed" is equal to the pressure with which it is fron the right and they cancel each other out.
Yes. This is a correct observation. If we imagine a small parcel of water (perhaps a cube), the force from the left will match the force from the right. So there is no net force in that direction. The parcel remains at rest, not accelerating.

This must be the case if the fluid is in an equilibrium state.

This is the sense in which the left and right pressures "cancel out". It is not that there is no pressure. It is that there is no acceleration of the fluid parcel.
 

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