Name: fluid pressure in an accelerating elevator
Uploaded: 2023-05-16T08:18:41.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

Alabama Course of Study Science

9-12

Physics

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

9-12.P.1.4.d

Accelerating Fluids

Considérons un bécher placé dans un élévateur qui accélère vers le haut. Dans le bécher, supposons qu’il y ait un cylindre mince de hauteur h avec une section transversale infinitésimale.
Sur le liquide contenu dans ce cylindre infinitésimal, trois forces verticales agissent.
Il s’agit de la force vers le haut due au liquide présent sous la surface inférieure, de la force vers le bas due au liquide au-dessus de la surface supérieure et de la force vers le bas due à son poids.
Comme le liquide accélère, une relation est obtenue à partir de la deuxième loi de Newton.
La représentation de la masse de l'élément fluide en termes de densité simplifie l'équation et l'expression de la différence de pression pour le fluide accélérateur est obtenue.
Supposons qu’un corps soit plongé dans le même liquide d’accélération. Il subit la force de flottabilité et la force due à son poids.
Pour simplifier, le corps est remplacé par un volume égal du même liquide. Cela fait de l’ensemble du liquide à l’intérieur du bécher une masse homogène subissant la même accélération.
D'après la deuxième loi de Newton, la force de flottabilité est exprimée en termes d'accélération.

<ol>
	<li>H.C. Verma Vol 1. (2012). Concepts de physique, Section 13.6 pp.262</li>
</ol>

fluides accélérés

Lorsqu'un fluide est en accélération constante, les équations de pression et de force de flottabilité sont modifiées. Supposons qu'un bécher soit placé dans un ascenseur accélérant vers le haut avec une accélération constante, a. Dans le bécher, on suppose qu'il y a un cylindre mince de hauteur h avec une aire transversale infinitésimale, ΔS.

Le mouvement du liquide à l'intérieur de ce cylindre infinitésimal est considéré pour obtenir la différence de pression. Trois forces verticales agissent sur ce liquide :

<ol>
	<li>Une force vers le haut due au liquide présent en dessous de la surface inférieure du cylindre.</li>
	<li>Une force vers le bas due au liquide au-dessus de la surface supérieure du cylindre.</li>
	<li>Une force vers le bas due au poids de l'élément cylindrique.</li>
</ol>

Sous l'effet de ces trois forces, le liquide accélère vers le haut. En utilisant la deuxième loi de Newton, on obtient l'expression suivante :

<img alt="Equation1" src="/files/ftp_upload/14713/14713_Equation_1.svg" style="text-align: center; height: 16px;" />

La représentation de la masse de l'élément fluide en termes de densité (ρ) simplifie l'équation, et l'expression de la différence de pression pour un fluide accéléré est obtenue :

<img alt="Equation2" src="/files/ftp_upload/14713/14713_Equation_2.svg" style="text-align: center; height: 18px;" />

Pour obtenir la force de flottabilité, supposons qu'un corps soit plongé dans le même liquide accéléré. Il subit une force de flottabilité et une force due à son poids. Pour simplifier, le corps est remplacé par un volume équivalent du même liquide. À partir de la deuxième loi de Newton, la force de flottabilité est exprimée en fonction de l'accélération, et l'expression suivante est obtenue :

<img alt="Equation3" src="/files/ftp_upload/14713/14713_Equation_3.svg" style="text-align: center; height: 18px;" />

Fluides accélérés

Lorsqu’un fluide est en accélération constante, les équations de pression et de force de flottabilité sont modifiées. Supposons qu’un bécher soit placé dans un ascenseur en accélérant ...

/files/media/science-education/science-education-thumbs/14713.jpg

Education

Enseignement

JoVE Core

Physique

Unit 1

Unité 1

Fluid Mechanics

Mécanique des fluides

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

fluid pressure in an accelerating elevator

Fluid pressure changes when a liquid accelerates inside a moving container. A beaker placed in an elevator that moves upward with constant acceleration is one example. In this setup, the pressure and buoyant force equations must be modified.

To find the pressure difference, the liquid is modeled as a thin vertical cylinder with height h and an infinitesimal cross-sectional area, ΔS. Three vertical forces act on this small fluid element. The liquid below the bottom surface pushes upward, the liquid above the top surface pushes downward, and the element’s weight also acts downward. Because the cylinder accelerates upward, Newton’s second law is used to relate these forces.

The mass of the fluid element is written in terms of density, ρ, to simplify the result. This gives the pressure difference for an accelerating fluid. The same idea is then used to determine buoyant force.

For buoyancy, a body is placed in the same accelerating liquid. It experiences buoyant force and its own weight. To make the analysis easier, the body is replaced by an equal volume of the same liquid. Newton’s second law then gives the buoyant force in terms of the acceleration, leading to the final expression.

Fluid Pressure in an Accelerating Elevator

fluid-pressure-in-an-accelerating-elevator

characteristics of fluids

Fluid Behavior Under Force

Fluid behavior under force differs from the behavior of solids. When a force acts parallel to the top surface of a solid, the solid resists it because of internal friction between its layers. This is called shearing resistance. If the force is removed, the shearing forces help the solid return to its original shape. Other deformation forces can also change the shape of a solid for a short time, as long as the force stays below a threshold magnitude. Solids usually keep their shape, which makes...

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density

Density, Floating, and Specific Gravity

Density helps explain why objects sink or float in a fluid. It is also useful for identifying a substance and for learning about its phase and substructure. The SI unit of density is kg/m3, and the cgs unit is g/cm3. Liquids and solids usually have densities that are roughly comparable. That matches the fact that their atoms are in close contact. Gases have much lower densities than liquids and solids because their atoms are separated by large amounts of empty space. Density does not always...

pressure of fluids

Fluid Pressure in Everyday Life

Fluid pressure appears in many everyday situations, including blood pressure and weather systems. The same force can have a very different effect depending on the area it acts on. A force spread over 1 mm² creates 100 times more pressure than the same force spread over 1 cm². That is why a sharp needle can pierce skin with a small force, while the same force from a finger usually does not. Pressure is especially important when studying fluids. A fluid does not offer much resistance to a force...

variation of atmospheric pressure

Atmospheric Pressure and Altitude

Atmospheric pressure and altitude are closely linked. As height increases, atmospheric pressure decreases. This happens because the gravitational force per unit area becomes smaller as we move away from Earth’s surface. To describe this change, the atmosphere is treated with a constant air temperature at a given altitude. The ideal gas law of thermodynamics is also used as a good approximation for the air. Let p(y) be the atmospheric pressure at height y. The density ρ at y, the temperature T...

/files/media/science-education/science-education-thumbs/14711.jpg

pascal's law

Pascal’s Principle in Static Fluids

Pascal’s principle explains how pressure changes move through a fluid that is not flowing. A fluid at rest is in static equilibrium. If the fluid is water, it is in hydrostatic equilibrium. In this state, the net force on any part of the fluid must be zero, or the fluid will begin to flow. Blaise Pascal described this idea in 1653 in Treatise on the Equilibrium of Liquids. He showed that when pressure changes in an enclosed fluid, that change is transmitted without loss through the fluid and...

application of pascal's law

Hydraulic Force Transfer with Pascal's Law

Hydraulic force transfer uses Pascal's law to move pressure through an enclosed fluid. Pascal showed experimentally that a change in pressure applied to a confined fluid is transmitted undiminished throughout the fluid and to the walls of its container. That idea is the basis of hydraulics, a major part of modern mechanical technology. Hydraulic systems appear in automotive brakes, hydraulic jacks, and many other machines. A simple hydraulic system uses two pistons at the same height. Because...

pressure gauges

Gauge vs Absolute Pressure

Pressure gauges often show pressure relative to atmospheric pressure. Many gauges, including those on scuba tanks, are set to read zero at atmospheric pressure. That reading is called gauge pressure. When the pressure inside the tank is higher than atmospheric pressure, the gauge shows a positive value. Some gauges can also measure negative pressure. This is useful in a vacuum chamber, which is a rigid chamber where some air has been pumped out. Because the pressure inside the chamber is below...

/files/media/science-education/science-education-thumbs/14712.jpg

archimedes' principle

Buoyant Force from Fluid Pressure

Archimedes' principle explains buoyant force in a fluid. An upward buoyant force acts on an object that is partially or fully immersed in a fluid. This force is equal to the weight of the fluid displaced by the object. To see why an object floats, imagine removing a submerged object from the fluid. The space it occupied would be filled by fluid with weight w_fl. That weight is supported by the surrounding fluid, so the buoyant force must match the weight of the displaced fluid. The principle...

density and archimedes' principle

How Buoyancy Depends on Density

Buoyancy and density explain why some objects float and others sink. A lump of clay sinks in water, but the same clay can float when it is shaped like a boat. The change in shape lets it displace more water, which increases the buoyant force even though its mass stays the same. The same idea applies to steel ships. An object with an average density lower than the surrounding fluid will float. The fluid has more mass in the same volume, so it has more weight. The buoyant force equals the weight...

accelerating fluids

Fluid pressure changes when a liquid accelerates inside a moving container. A beaker placed in an elevator that moves upward with constant acceleration is one example. In this setup, the pressure and buoyant force equations must be modified. To find the pressure difference, the liquid is modeled as a thin vertical cylinder with height h and an infinitesimal cross-sectional area, ΔS. Three vertical forces act on this small fluid element. The liquid below the bottom surface pushes upward, the...

surface tension and surface energy

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/files/media/science-education/science-education-thumbs/14714.jpg

excess pressure inside a drop and a bubble

excess-pressure-inside-a-drop-and-a-bubble

Surface Tension in Drops and Bubbles

Surface tension sets the pressure difference in small drops and bubbles. For a tiny liquid drop, gravity can be neglected, so the drop is treated as spherical. It can also be viewed as two equal hemispherical drops joined together by surface tension. The forces on the drop come from three sources. The liquid inside pushes outward, the air outside pushes inward, and surface tension acts along the diameter between the two hemispheres. Since surface tension is force per unit length, it creates a...

/files/media/science-education/science-education-thumbs/14715.jpg

buoyancy

Buoyant Force in Sinking and Floating

Buoyant force explains why objects float, sink, or stay suspended in a fluid. A fluid can be a liquid or a gas, and every object placed in it experiences this upward force. A metal ball sinks, while a rubber ball floats. A submarine also changes its buoyancy to move up or down. The force comes from pressure changes in the fluid. Fluid pressure increases with depth because the weight of the fluid above also increases. That means the bottom of an object feels more pressure than the top. This...

contact angle

Wetting and Surface Curvature

Wetting and surface curvature describe how a liquid behaves where it meets a solid. When a solid is dipped into a liquid, the liquid surface can bend near the contact line. In some cases, the liquid rises along the solid. In others, the surface becomes convex or depressed near the solid. This behavior comes from cohesive and adhesive forces. Adhesive force is the attraction between molecules of different materials, such as a solid and a liquid. Cohesive force is the attraction between...

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rise of liquid in a capillary tube

Capillary Action in Thin Tubes

Capillary action explains why liquid can rise or fall in a very thin tube compared with the liquid around it. A capillary is a narrow cylindrical tube, and the effect becomes noticeable when it is dipped into a liquid. The movement happens because two forces act in opposite directions. Cohesive forces make liquid molecules stick to each other, so the liquid tends to hold together and form a rounded shape. Adhesive forces pull the liquid toward the walls of the container. In a narrow glass...

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laminar and turbulent flow

Fluid Motion Patterns in Air and Water

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equation of continuity

Fluid Flow in Narrowing Pipes

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bernoulli's equation

Fluid Pressure and Speed in Motion

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bernoulli's principle

Fluid Pressure and Speed Changes

Bernoulli's equation links fluid pressure and fluid speed in a static, incompressible fluid. For horizontal flow, where gravitational energy density stays constant, the equation is especially useful. In that setting, the result is known as Bernoulli's principle. Bernoulli's principle says that fluid pressure drops when speed increases, and pressure rises when speed decreases. This pressure-speed relationship helps explain how moving fluids behave in simple flow situations. It also provides a...

bernoulli's principle: applications

Bernoulli's Principle in Fluid Devices

Bernoulli's principle helps explain how many fluid devices work when the fluid moves at a constant height. It is useful for devices that use fast flow to create low pressure and for devices that measure fluid speed. Some devices use this pressure change to pull one fluid into another. These are called entrainment devices. A Bunsen burner uses an adjustable gas nozzle to entrain air for proper combustion. An atomizer uses a squeeze bulb to make a jet of air that pulls perfume drops into the...

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energy conservation and bernoulli's equation

energy-conservation-and-bernoulli-s-equation

Bernoulli’s Equation in Streamlined Flow

Bernoulli’s equation describes energy in a moving fluid. It comes from the conservation of energy principle, or the work-energy theorem, applied to an incompressible, inviscid fluid in laminar, steady, irrotational flow. The equation says that pressure energy, potential energy, and kinetic energy per unit volume stay constant along a streamline. Each term has the units of energy per unit volume. The kinetic energy per unit volume is called kinetic energy density, and the potential energy per...

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viscosity

Viscosity and How Fluids Flow

Viscosity describes how easily a fluid flows. Water moves quickly, while honey and maple syrup flow slowly and stick to a surface. That difference comes from fluid friction between liquid layers and between a fluid and the material around it. In this example, water has a lower viscosity than honey and maple syrup. Viscosity is measured in newton-second per meter squared (N·s/m2) or pascal-second (Pa·s). It also changes with temperature and can differ from one fluid to another by several orders...

poiseuille's law and reynolds number

Fluid Resistance, Viscosity, and Flow Regime

Fluid resistance, viscosity, and flow regime shape how liquid moves through a horizontal tube. Fluid flows from high pressure to low pressure. The flow rate, Q, is the ratio of the pressure difference to the resistance in the tube. A larger pressure difference gives a higher flow rate. Poiseuille's law describes laminar flow of an incompressible fluid in a tube. In this relation, the factors that affect flow rate except pressure are grouped into resistance. Resistance depends on the tube size...

stokes' law

Terminal Velocity in a Viscous Fluid

Viscous forces act like friction in a liquid. They resist the relative motion of molecules and pull back on a solid body moving through the fluid. The size of this force depends on the body’s shape and size, its speed, and the liquid’s viscosity, density, and temperature. For a spherical solid object, the force in a fluid is described by Stokes’ law. This law applies only at low Reynolds number, when the flow is laminar and not turbulent. Experiments show that the force is proportional to the...

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key terms and concepts

See how constant acceleration changes fluid pressure and buoyant force in an elevator example. Learn the force balance and Newton’s second law result.

Fluid Pressure in an Accelerating Elevator | JoVE K12

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Fluid Pressure in an Accelerating Elevator

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