## Force and Pressure in Fluids

The three common states, or phases, of matter, are solid, liquid, and gaseous. The **solid **maintains a fixed shape and size; even when a large force is applied, a solid will not change easily in form or volume. A **liquid** does not maintain a fixed form, but takes that of its container; Like solids, it does not compress easily, but its volume can change appreciably if a very large force is applied. A **gas** has no fixed form or volume; it expands and fills its container.

For example, when air is pumped into a car tire, the air is not concentrated at the bottom, as would a liquid, but fills the entire volume of the tire. Since liquids and gases do not maintain a fixed form, they have the ability to flow; that is why they are generically called **fluids**.

## Density and specific gravity

Density is a characteristic property of each substance and gives an idea of how heavy the atoms are and how close they are: the same mass of different substances occupies different volumes.

d = M/V

The specific gravity of a substance is defined as the ratio between density and water density at 4.0 ° C. The specific gravity (GE) is a pure number, without dimensions or units. Since the density of water is 1.00 g / cm^{3} = 1.00 x 10^{3} kg / m^{3}, the specific gravity of any substance is exactly equal to its density in g / cm^{3} or 10^{-3} multiplied by its density expressed in kg / m^{3}.

## Pressure

A solid on contact with another, exerts a force on its surface trying to penetrate it. The deforming effect of that force or the penetration capacity depends on the intensity of the force and the contact area. Pressure is the magnitude that measures that capacity.

P = F/S

Its unit in the International System is the Pascal (Pa=1N/m^{2})

In the case of solids and fluids, by applying an external force to a moving wall of a container that contains a fluid, it creates a pressure that compresses it. The force distributed over the surface of the movable wall gives the value of the pressure. The volume occupied by the fluid decreases with increasing pressure. The comprehensibility is almost zero in liquids.

Even without external force, the weight of the liquid will exert a hydrostatic pressure on its lower layers. This pressure generates a force acting from the inside of the liquid outwards and perpendicular to all the walls of the container.

The pressure is a scalar, has no direction or sense, but the force that creates against the walls is a vector, has direction perpendicular to the surface and outward direction.

The concept of pressure is especially useful in the study of fluids. It is an experimental fact that a fluid exerts a pressure in all directions. This is well known to swimmers and divers who feel the pressure of water in all parts of their body. At a certain point of a fluid at rest, the pressure is the same in all directions. If it were not so, the net force would not be zero and would move until the pressure was equal. If the fluid does not flow, then the pressures must be equal.

Another important property of a fluid at rest is that the force due to its pressure always acts perpendicular to any surface that is in contact with it. If there were a component of force parallel to the surface, according to Newton’s third law, the surface would exert a force opposite to that of the fluid, which, in turn, would also have a component parallel to the surface. This component would cause the fluid to flow, which contradicts our hypothesis that the fluid is at rest. So, the pressure is perpendicular to the surface.

## Atmospheric pressure, hydrostatic pressure and gauge pressure

The pressure of the Earth’s atmosphere, **atmospheric pressure,** as in any fluid, decreases as depth decreases (or height increases). But the terrestrial atmosphere is somewhat more complicated, because not only varies the density of the air with altitude, but there is no defined external surface, from which you can measure the height “h” for the following equation:

P = dgh

However, we can calculate the approximate pressure difference between two altitudes with the following equation:

The pressure of the air in a certain place varies slightly according to the climate. At sea level, the atmosphere pressure, on average, is 1.013×10^{5}N / m^{2}. This value is used to define another pressure unit of much use: the atmosphere (abbreviated atm).

Another pressure unit that is sometimes used is the bar, which is defined as 1.00×10^{5}N / m^{2} = 100kPa. Thus, the normal atmospheric pressure is slightly greater than 1 bar.

Hydrostatics treats liquids at rest. A liquid enclosed in a container creates a pressure in its breast and exerts a force on the walls that contain it.

The **hydrostatic pressure** at a point inside a liquid is directly proportional to the density of the fluid, d, to the depth, h, and to the gravity of the place, g.

P = dgh

Fluids also exert a pressure on any body submerged in them. The pressure will be greater the more dense the fluid and the greater the depth. All points located at the same depth have the same pressure.

We can verify that the hydrostatic pressure increases when descending inside a liquid seeing that the speed with which the liquid comes out is greater the further down the hole is made in the side wall of the container, since:

The pressure on the walls increases downwards and therefore also the force on them. If we drill holes at different depths, the exit velocity becomes greater as the depth increases.

The pressure due to the weight of the atmosphere is exerted on all objects submerged in this great ocean of air, including our bodies. How is it that a human organism can withstand the enormous pressure? The answer is that living cells maintain an internal pressure that exactly balances external pressure. The pressure inside a balloon also balances the pressure outside it, due to the atmosphere. Because of its rigidity, a car tire can maintain pressures much greater than external pressure.

However, care must be taken when determining tire pressure because all pressure gauges, including tire gauges, measure the pressure that exceeds atmospheric pressure. This pressure is called gauge pressure. Thus, to obtain the absolute pressure P, the atmospheric pressure P_{A} must be added to the gauge pressure P_{M}:

## The Principle of Pascal

The atmosphere puts pressure on all objects with which it is in contact, including the other fluids. The external pressure acting on a fluid is transmitted through it. The principle of Pascal states that *“**the pressure applied to a point of a static and incompressible fluid enclosed in a container is transmitted entirely to all points of the fluid**”.*

** ** If an external force F is exerted on a piston of section S, a pressure originates in the entire liquid mass.

P = F/S

The pressure, as we have said before, is a scalar quantity, it has no definite direction, but the inner force that it produces is a vector perpendicular to the surface on which it acts. Therefore within a wait it is perpendicular, at each point, to the inner surface.

The jet of liquid does not come out with more force through the inner hole, as one might think when the external force pushes the plunger in that direction, but it comes out through all the holes with equal velocity.

There are several devices that use the principle of Pascal, for example, the hydraulic brakes of a car and the hydraulic ramp or the well-known **“hydraulic jack”.**

The hydraulic jack is used in workshops to raise cars. It is a tank with two pistons of different sections S_{1} and S_{2} connected to it. The pressure exerted by the plunger when pressing on the surface of the liquid is transmitted completely to the entire liquid. The pressure is the same in the points next to the two pistons. P_{1} = P_{2}.

The force F_{1} applied to the small piston is applied in an amplified factor **k** such that: F_{2} in the large piston is k · F1. In addition to amplifying the value of F_{1 }changes its direction of use, because F_{2} will be where we connect to the tank of the second piston.

## Buoyancy and Principle of Archimedes

Objects submerged in a fluid appear to weigh less than when they are outside the fluid. For example, a rock that could be lifted from the ground only with difficulty will be able to rise more easily from the bottom of a stream. When the rock exceeds the surface of the water, it seems much heavier. Many objects, such as wood, float on the surface of the water. These are two examples of **buoyancy** or **flotation**. In each of them, the force of gravity acts downwards, but in addition, the liquid exerts a floating force upwards.

Flotation force occurs because the pressure of a fluid increases with depth. Thus, the upward pressure exerted on the lower surface of a submerged object is greater than the downward pressure on its upper surface. After various studies Archimedes reached his principle in which: *“Every body submerged in a fluid suffers a vertical force and upwards equal to the weight of fluid that dislodges the submerged part of the body.”*

If the fluid is water:

How the evicted mass is equals to the submerged volume of the body by the density (m = V·d):

In our day to day we use this principle with balloons or boats. The ascension of a balloon occurs because the interior density is less than that of the air and the weight of the dislodged air is greater than the sum of the weight of the interior gas, the basket, the ballast and the ropes.

Boats float because they displace a weight of water that is equal to the weight of the boat itself. For there to be balance and not oscillate, in addition to the equality between the weight of the body and the thrust, it is required that the center of gravity of the body and the submerged part remain on the same vertical. If the weight and the push are not in the vertical direction, a pair of forces originates.

In the diagram where holes are poked in a bottle, the fluid streams exit the bottle with some positive vertical velocity. Why is that?