Pascal's Principle

Fluids: Pascal's Principle

        Fluids are commonly found in almost every corner of our planet. The first thing that comes to mind with the word fluid is water. This is a common misconception of fluids as there are various kinds of fluids, all of which can be classified as either liquid or gas.
        When we refer to an object as a fluid, we expect that the object to have no definite shape and can be effortlessly subjected to external pressure. Moreover, fluids must possess density, surface tension, and viscosity.

Pascal's Principle states that:

If any external pressure is applied to a fluid, that pressure is evenly distributed throughout the fluid.

What does this mean? 

        This means that if we were to apply pressure P to a system, pressure P shall be evenly applied throughout all directions of the system. 
        To illustrate this, if pressure A is applied on one side of a hydraulic system, the other side of the system should have an output of pressure B that is equal to the input of pressure A. This is because there are two openings, and since one opening is used as an input, the other should act as an output.

Hence,

Pressure A = Pressure B

        On the other hand, if we were to block one opening of the hydraulic system and apply pressure on side A, that pressure shall be evenly distributed throughout the system and will try to find an opening.

Hence,

Pressure A =  Net Pressure of the System

 

Moving on,  you may be wondering, 

PRESSURE

    Well, for starters, it simply means, force(F) exerted over an area(A).

Quite simple right? Well, yes and no.

     Say we were to get the pressure experienced by an object under the ocean. We can't just say Force over Area, it simply doesn't make sense. Even if we have the area, how can we even begin to calculate the Force of the ocean on the object? 

    Hence, physicists eventually came up with this equation. 

    Or simply, 

This equation is derived from the first equation, P = F/A.

The derivation is as follows (refer to the diagram),

We know that Force is defined as F = mass * acceleration.

To further simplify this equation, we know that Density

is defined as ρ = mass / Volume. Hence, by isolating the mass, we get m = ρV.
Plugging it into the equation, it becomes

For the finishing touch, we define Volume as V = Area * height. Plug this into our equation, cancel out the Area, and we got P = ρha

Since we are dealing with hydrostatic pressure, meaning fluid at rest, then our only acceleration is that of the planet's gravity, thus

a = g

Do note that ρ refers to the density of the fluid rather than that of the object.

This equation, however, does not account for atmospheric pressure.

There are two types of pressure, these are named as gauge pressure and absolute pressure. The difference between these two is the presence of atmospheric pressure. Gauge pressure only calculates pressure exerted by a fluid to an object, while absolute pressure is gauge pressure with the addition of atmospheric pressure. 

Gauge Pressure

Absolute Pressure

~kaku