PhysicsLAB Resource Lesson
Fluids At Rest

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On earth, there are three dominate phrases of matter: solids, liquids and gases.* The predominate distinction between these types lies in the forces between and the energies within their molecules. In gases, the molecules are extremely energetic and are essentially independent of each other; while in liquids, less-energetic molecules are loosely attracted to each other and easily wander past each other. Both of these phases take on the shapes of their containers, with gases also taking on the container's volume. They are often grouped together as fluids - that is, they are materials that flow readily under the action of an applied force. In solids, the molecules vibrate about fixed positions and are so strongly attracted that the material appears to be rigid.
Several properties of fluids need to be established before we concentrate our discussion on hydrostatic pressure, Archimedes' Principle and Pascal's Principle. The first of these is density.
Density is defined as the ratio of mass per unit volume. It is generally represented by the Greek letter rho, ρ, and measured in terms of kilograms/cubic meter, or kg/m3.
Since the volume of a fluid expands and contracts, the density of fluids vary with temperature. The most common fluid, water, has maximum density of 1000 kg/m3 at 4ºC. Air, a mixture composed principally of the gases nitrogen (78%) and oxygen (21%), has a density of 1.29 kg/m3 at 0ºC and 1.20 kg/m3 at 20ºC.
How a liquid's density compares to that of water at 4ºC is called its specific gravity. If a liquid has a specific gravity of 0.9, then its density is 0.9 times that of water, or 0.9 x 1000 = 900 kg/m3.
The second property is a fluid's compressibility. This quantity represents how easily the material can be compressed or reduced in volume by a change in pressure. It is generally represented by the letter  and measured in terms of pascals-1.
Air has a compressibility value of 1.0 x 10-5 Pa-1 while water has a compressibility of 4.5 x 10-10 Pa-1. Because of this extremely small value, liquids are often referred to as being incompressible and we do not need to worry about volume changes in our calculations.
Hydrostatic pressure
Pressure is defined as the ratio of force per unit area
where the force is perpendicular to the cross-sectional area.
Pressure is a scalar quantity measured in pascals, Pa, where 1 Pa = 1 N/m2.
 Suppose in the above diagram that the applied force equals 1 N, and the area of the defined rectangle equals 20 cm2. What is the pressure on the specified rectangle?

The force exerted by a fluid at rest acting on any rigid surface is always perpendicular to the surface. For if there were to be a non-perpendicular component, then the fluid would no longer be at rest, but would respond to the reaction force supplied by the surface and motion would result.
Suppose a tank is filled with water and at a certain depth within the water a small piece of aluminum foil, having a cross-sectional area, A, is floating with neutral buoyancy.
What would be the hydrostatic pressure on the foil's top surface resulting from the water column located above the foil's surface?

This additional hydrostatic pressure exerted on the submerged foil is often referred to as the gauge pressure. The total pressure on the surface of the foil would be the combination of atmospheric pressure plus this hydrostatic pressure.
In each of the three containers shown below, the total pressure at depth h would be the same - it is independent of the shape of the container, volume of water above the surface, or the exposed surface area.
Refer to the following information for the next question.

This water pressure is exerted equally in all directions at each depth. For example, a scuba diver would feel the same pressure on his ears, as on his hands and feet.
 At what depth below the surface of a fresh-water diving tank would the total pressure equal two atmospheres?

Consider for a moment of 2-liter bottle of soda securely sealed so that no atmospheric pressure is exerted on the surface of the liquid within the bottle. Now punch three small holes, aligned vertically, in the side of the bottle. While the top remains securely closed, no fluid flows from the holes because the atmospheric pressure on the outside of the bottle is greater than the hydrostatic pressure within the bottle. But all of that changes when the cap is loosened or removed. At that time, the fluid column within the bottle will also be exposed to the external atmospheric pressure and the combination of atmospheric and hydrostatic pressure will result in fluid escaping from the holes in the bottle. If all three holes have the same cross-sectional area then the lowest hole will release liquid with the greatest instantaneous horizontal acceleration.
Buoyancy and Archimedes' Principle
When a non-porous object is completely submerged in water, it displaces a volume of water equal to its own volume. In the diagram shown above, notice that the upward forces against the bottom surface of the object are greater than the downward forces against its top surface. This net force is called the buoyant force. Also note that the horizontal forces on its vertical sides cancel and are removed from consideration.

Archimedes' Principle states that a buoyant force equal to the weight of the volume of this displaced water will be exerted upward on the object.
Refer to the following information for the next question.

Consider a 10 cm x 20 cm x 50 cm (V = 0.01 m3) lead brick that weighs 1100 N in the air.
 How much will it weigh while submerged in water?

When an non-porous object is placed in a fluid, there are three possible outcomes:
  • the object floats when the density of the object is less than the density of the displaced fluid
  • the object sinks when the density of the object is greater than the density of the displaced fluid
  • the object has neutral buoyancy (neither rising nor falling) when the density of the object equals the density of the displaced fluid
So, how can objects made of aluminum or iron float? The secret lies in increasing the volume of the displaced water. Although a small cube of iron will immediately sink when placed in water, a large boat can float by adjusting the amount of water it displaces. A sheet of aluminum foil can float when formed into a "barge" with a large surface area whereas the same size sheet will immediately sink when crushed. In all cases, you are reducing the density by increasing the volume. Remember, an object floats by displacing a volume of fluid which has a weight equal to the object's weight.
Hydraulic lifts and Pascal's Principle
When a confined fluid is completed enclosed a change in pressure in one location is transmitted through the fluid. Consider a water balloon with negligible air. Squeezing one side of the balloon transmits the pressure to all other regions usually resulting in the opposite of the balloon being pushed outward, stretching the balloon.
Pascal's Principle states that a change in the pressure at any point in an enclosed fluid that is at rest is transmitted undiminished to all points in the fluid and in all directions.
Let's look at a simplified version of a hydraulic lift. Applying Pascal's Principle allows us to see that the pressure applied to the small piston will be disseminated through the confined fluid (usually oil) and transmitted to the larger piston.
At first glance it might look as if we are getting "free" energy since an initial force F applied to an area A
P1 = F/A
produces a force of 6F applied to the area 6A
P2 = 6F / 6A = P1
However, energy must be conserved and the work done in lifting must equal the work done during compression. That is, in order to lift the large piston 10 cm, we must push the small piston down through a distance of 60 cm so that comparable amounts of work = force x distance will be done on each side of the lift:
F(0.60) = 6F(0.10)
Because of this trade off of force and distance, service stations often place heavy cars on lifts having small pistons and use compressed air to push down the larger area piston (buried underground) a shorter number of centimeters.

*In our universe there are four different phases of matter: solids, liquids, gases, and plasmas. Plasmas represent a state of matter that conducts electricity and consists of atoms, positive ions and free electrons. Even though they only exist at extremely high temperatures, they represent the predominant phase of matter in the stars and intergalactic regions within our universe. Although this phase is rare on the earth, glowing plasmas can be produced in fluorescent lamps and energized gas discharge tubes, for example, neon advertising signs. The most spectacular natural views of plasmas on earth are the auroras: northern and southern lights.

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Catharine H. Colwell
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    Mark Acton