Table of Contents

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Image: “Drop of water” by Tim Geers from the Netherlands – https://www.flickr.com/photos/timypenburg/4649617096/. License: CC BYSA 2.0
 Pressure
 Archimedes’ Principle & Buoyancy
 Pascal’s Law
 Ideal Hydrodynamics
Are you more of a visual learner? Check out our online video lectures and start your physics course now for free!
Table of Contents
 Are you more of a visual learner? Check out our online video lectures and start your physics course now for free! Image: “Drop of water” by Tim Geers from the Netherlands – https://www.flickr.com/photos/timypenburg/4649617096/. License: CC BYSA 2.0
 Pressure
 Archimedes’ Principle & Buoyancy
 Pascal’s Law
 Ideal Hydrodynamics
Pressure
Definition
Pressure is defined as a continuous physical force that is exerted on or against an object by something in contact with it. It can be described as the force per unit area. This concept is important in medicine especially dealing with blood flow (cardiovascular system) and with air flow (respiratory system).
Explanation
Imagine a beach ball full of air. The air inside keeps the ball inflated. Inside the ball, collisions take place against the wall of the ball that apply force. Only those collisions that are perpendicular to the wall have an effect. The air molecules moving along the side walls will not influence the balloon staying inflated.
Formula
Pressure formula is simply: P = F_{1} / A where P is the pressure, F_{1} is the force, and A is the are involved. In looking at the units, we get:
(P) = kg / ms^{2} = N / m^{2} = Pascals (Pa). Pressure is measured in the units of Pascals.
Atmospheric Pressure
When a person is standing on Earth, a weight exists on top of the person but the person doesn’t feel it. This weight is F_{air} (force of air above the person). The person doesn’t sense this due to water and air pressure inside pushing outwards. The balance of pressure doesn’t allow the person to feel the force of air above, also referred to as atmospheric pressure. Atmospheric pressure can also be measured in Pascals:
1 atm = 100,000 Pa = 100kPa
Hydrostatic Pressure
When a person is under water, there are two weights acting on the person from above: the atmospheric pressure and the water pressure. Above the person. The water pressure is referred to as hydrostatic pressure or Gauge pressure. It is denoted as: P = ρ g h where ρ is density of liquid medium, g is the gravitational force, and h is the height of water above the person. So, the total pressure above a person under water is:
P_{total} = ρ g h + P_{atm}
Archimedes’ Principle & Buoyancy
Definition
The famous Greek mathematician, Archimedes, stated that any body completely or partially submerged in a fluid at rest is acted upon by an upward, or buoyant, force. This force is equal to the weight of the fluid that the body displaced. This occurs since the pressure of the water underneath is greater than the pressure of the water above. In measuring the displaced water, the calculation is based on the volume of the object that was submerged. The pressure acting upward is F_{B} = buoyant force and the pressure acting downward is F_{g} = gravitational force.
Flotation Rules
An object will float if:
F_{B} > F_{g}
m_{water }g > m_{object} g
ρ_{water} V > ρ_{object} V
ρ_{water} > ρ_{object}
Thus buoyancy F_{B} = m_{displaced water} g
How much will emerge from the water?
If an object is floating, it will emerge from the water until:
F_{B} = F_{g}
_{ }m_{water displaced} = m_{object}
_{ }ρ_{water} V_{submerged} = ρ_{object} V_{object}
Therefore, for floating objects: V_{submerged} / V_{object} = ρ_{object} / ρ_{water}
Pascal’s Law
Definition
Pascal’s law, also called Pascal’s principle, states that pressure is equally distributed throughout a fluid. In a fluid at rest in a closed container, a pressure change in one part is transmitted without loss to every part of the fluid and to the walls of the container. This was discovered by French scientist Blaise Pascal.
Formula
In order to determine the pressures and forces associated with Pascal’s Law, the following must take place:
Since the pressure is equally distributed throughout a fluid, P_{1} = P_{2}. Since pressure is related to force and area, then the following relationship occurs:
F_{1} / A_{1 }= F_{2} / A_{2}, thus there is equal force per unit area.
Hydraulics
Pascal’s law is useful in machinery that uses hydraulics (the science involving the mechanical properties and uses of liquids). In these scenarios, one can gain a mechanical advantage by making one area greater than another such that:
F_{2} = F_{1} (A_{2} / A_{1})
Ideal Hydrodynamics
Definition
Hydrodynamics is the branch of science that is concerned with forces acting on or exerted by fluids and the forces acting on solid bodies immersed and in motion. Hydrodynamics contains six different parts: ideal flow, flow rate, continuity, Bernoulli’s principle, Venturi effect, and Pitot tube.
Ideal Flow
Ideal flow deals with an ideal fluid that is incompressible, not viscous, and laminar without turbulence. Liquids do not have the ability to be compressible, unlike gases. The flow occurs due to the lack of viscosity. The ideal flow must be laminar and orderly without turbulence or chaos. In general, laminar flow occurs at lower velocities and is smooth, not rough.
Flow Rate
The flow rate is the amount of fluid that flows in a given time. It is calculated as follows:
Flow rate = Volume per second
Q = Volume / Time = (Area x Distance) / Time
Thus, the flow Q = Area x Velocity (Q = A v) since velocity = distance / time
Continuity
When a fluid is in motion, it must move in such a way that mass is conserved. It describes the transport of some quantity. In this process, there is a conservation of mass where the flow rate must be conserved without change in mass. If the flow rate is too low, a vacuum situation will occur. If the flow rate is too high, overlap will occur. In a tube with changing diameter, the velocity is greater in places with lower cross sectional area such that at two different points:
Q_{1} = Q_{2} and A_{1} v_{1} = A_{2} v_{2}
Bernoulli’s Principle
Daniel Bernoulli explained an important principle in fluid dynamics. His principle stated that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy. At any portion along the tube, energy is conserved.
P_{1} + ½ ρ v_{1}^{2} + ρ g h_{1} = P_{2} + ½ ρ v_{2}^{2} + ρ g h_{2}
Venturi Effect
The Venturi effect is the reduction in fluid pressure that results when a fluid flows through a constricted section of a pipe. The constricted area will have a higher velocity but a lower pressure in a given system.
Pitot Tube
A pitot tube is a pressure measuring instrument used to measure fluid flow velocity. The basic setup is a tube pointing directly into the fluid flow. As this tube contains fluid, a pressure can be measured. The moving fluid is brought to rest (stagnates) as there is no outlet to allow flow to continue. This pressure is the stagnation pressure or total pressure of the fluid.
Using Bernoulli’s principle, we know stagnation pressure = static pressure + dynamic pressure.
P_{total} = P_{static} + ½ ρ v^{2}
Solving for velocity, v = √( (2 (P_{total} – P_{static})) / ρ)