Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to talk about Pascal's Principle and Hydraulics.
So, without further ado, let's get straight into it!
Static Equilibrium
When fluids are not flowing (not in motion) they are considered to be in static equilibrium and known as static fluids. In the case of liquids, like water, it's also possible to say that the liquid is in hydrostatic equilibrium.
For a fluid to be in such a state, the total net force on it must be zero, as otherwise the fluid would start to flow.
Pascal's Principle
One very important principle of static fluids is Pascal's principle (or law), which states that when there is an increase (or decrease) in pressure at any point of confined (enclosed) fluid, this change is transmitted to every other point of the fluid. The change in pressure is transmitted undiminished.
It's important to note that the pressure will not be the same at all points of the fluid, because as we already known the pressure varies with height / depth. What is the same is the change in pressure.
So, let's consider a cylindrical liquid container of some height h and area A. Adding a movable mass m on top of it (like a piston) the increase in pressure at the top would be:
According to Pascal's principle the pressure at all points of the liquid will change by that same amount. So, the pressure at the bottom would also be:
In other words:
Hydraulic Systems
Applications of Pascal's principle include hydraulic systems (liquid) and pneumatic systems (air). In such systems fluid is used in order to transmit forces from one location to another. Incompressible fluid in the case of hydraulic systems and compressible air in the case of pneumatic systems.
Consider the following hydraulic system.
Pascal's law allows force to be multiplied. As such, applying a force F1 to the smaller area A1, will yield a force F2 to the bigger area A2. This increase is known as a mechanical advantage and is commonly used in hydraulic car jacks.
Let's get into the mathematics now.
Because the pressure is the same throughout the whole container, the forces and areas are related as follows:
So, if F2 equals the weight of some car, the force F1 that would be required to lift it is much less than the weight of the car and given by:
Additionally, the volume decrease on the left side has to be equal to the volume increase on the right side, which gives us:
where d1 and d2 are the decrease and increase in height respectively.
