5.1 Pressure and Buoyancy
archimedes' Principle
an object partially or wholly immersed in a fluid is acted upon by an upward buoyant force equal to the weight of the fluid it displaces -an object that has a higher density than the fluid it is in will sink -an object that has a lower density than the fluid it is in will rise
ex: if a person displaces .08 m^3 of air, what is the buoyant force he experinceS?
About 1 N. Air exerts a buyoant force on a person equal to the weight of the air he displaces. The density of air near sea level is about 1.25 kg/m^3 so .08 m^3 of air has a mass of about 0.1 kg (1.25 kg/m^3 times 0.08 m^3) and a weight of about 1N
Helium vs. Air
Replacing air particles with helium atoms: reduces the gas's density -helium atoms have less mass than air particles.
thermal motion
-Air's individual particles have such minuscule makes that even at room temperature air molecules exhibit frenetic thermal motion (a) -like tiny marbles, these air particles have sizes, masses and weights, but while marbles quickly settle to the ground when they spill them from a bag, air particles don't seem to fall at all. Why don't they pile up on earth's surface: due to the air's thermal energy, the portion of air's internal energy that's associated with temperature. -thermal energy keeps them moving, spinning and ricocheting off one another at bullet like speed (500m/s) -their frequent collisions prevent them from making much progress in any particular direction and between collisions they travel in nearly straightline paths because gravity does;t have time to make them fall very far: this vigorous thermal motion spread the air particles parts so they don't accumulate on the ground. Real marbles are too massive to exhibit noticeable thermal motion and fall to the ground.
EArth's atmosphere
-EArth is 12,700 km in diameter the atmosphere is relatively thin. -stays on earth's surface because of gravity. Every air particles has weight, ash just as a marble thrown upward eventually falls back to the ground so the particles of air keep returning toward Earth's surface. Although the particles are moving too fast for gravity to affect their motions significantly over the short term, gravity works slowly to keep them relatively near Earth's surface. -an air particles may appear to travel in a straight line at first, but it will arc over and begin to fall downward eventually. -while gravity pulls the atmosphere downward, air pressure pushes the atmosphere upward: as the air particles try to fall to Earth's surface their density increases and so does their pressure. Its this air pressure that supports the atmosphere and prevents it from collapsing into a thin pile.
Hot-Air Balloon in Air
-a rubber, hot air filled ballon: contains fewer particles than if it were cold, weighs less than the cold air it displaces, experiences an upward net force in cold air, floats in cold air, has an average density less than that of cold air.
air and air pressure
-air has no fixed shape or size. You can mold 1 kg of air into any form you like and it can occupy a wide range of volumes -air is compressible: you can squeeze a certain mass of it into almost any space. -the flexibility of size and shape originates the the microscopic nature of air, it is a gas: a substance consisting of tiny, individual particles that travel around independent. These individual particles are atoms and molecules. An atom is the smallest portion of an element that retains all the chemical characteristics of that element, a molecule is an assemly of two or more atoms, is the smallest portion of a chemical compound that retains all the characteristics of that compound. A molecule's atoms are held together by chemical bonds: linkages formed by electromagnetic forces between atoms
The lifting force on a Balloon: buoyancy
-air pressure and its variation with altitude allow air to lift hot air and helium balloons through an effect known as buoyancy. -the buoyant force originate in the forces that a fluid exerts on the surfaces of an object. We've seen that such can be quite large but tend to cancel one another out--> how do you does pressure create a nonzero force on an object -without gravity, the force would cancel each other perfectly, but gravity causes a stationary fluid's pressure to increase with distance in the downward direction. -****when nothing is moving the air pressure beneath an object is always greater than the air pressure above it. Thus air pushes upward on the object's bottom more strongly than it pushes downward on the object's top and the object' consequently experience an upward overall force from the air—buoyant force
the atmosphere
-air pressure pushes on the air itself -air near the group supports air overhead: pressure is highest near the ground, air density is highest near the ground -air pressure decreases with altitude, a balloon feels more pressure at bottom than at top, force imbalance yields an upward buoyant force. -the bottom block must support the weight of all the blocks above it and is tightly impressed, with a height of about .8 m, a density go about 1.25 kg/m^3 and a pressure of about 100,00 PA, a block father up in the stack has less weight to support and is less tightly compressed, the higher the stack: lower density and less air pressure. -since the atmosphere's density and pressure increases gradually with distance in the downward direction, the atmosphere has a downward density gradient and a downward pressure gradient: allows a balloon to float
particle density
-although the particles in hot air and cold air are similar there are fewer of them in each cubic meter of hot air than each cubic meter of cold air. -particle density: particles/volume and hot air has a smaller particle density than cold air. -since they contain similar particles, hot air also has a smaller densityy than cold air and is lifted up by the buoyant force.
helium ballons
-another way to make one gas float in another :use a as consisting of very light particles. -helium: much lighter than ai. When they have equal pressures and temperatures, helium gas and air also have early particle densities. Since each helium weights 14% as much as the average air particle 1m^3 of helium weights only 14% as much as 1 m^3 of air. Thus a helium filled ballon has only a fraction of the weight of the air it displaces, and the buoyant force carries it upward easily.
buoyant force
-equal in magnitude of the fluid the object displaces. EX: imagine replacing the object with a smirkily shaped portion of the fluid itself (a). Since the buoyant force is exerted by the surround fluid , not the object, it doesn't depend on the object's composition. A balloon filled with helium will experience the same buoyant force as a similar balloon filled with water or lead or even air. So replacing the object with a similarly shaped portion of fluid will live the buoyant force unchanged. --however, a portion of fluid suspended in more of the same fluid doesn't accelerate anywhere, it just sits there so the net force on it is zero. It has a downward weigh, but that weight must be canceled by some upward force that can come only from the surrounding fluid. This upward force is the buoyant force, and its always equal in magnitude to the weight of the object-shaped portion of fluid displaced by the object. --this buoyant principle explain how some object sink and other float. An object places in fluid experience two forces, its downward weight and an upward buoyant force. If the weight is more than the buoyant force it will accelerate downward, it is heaver than the air it displaces (b) if its weight is less than the buoyant force, it will accelerate upward., lighter than the air it displace, (c).
Speeding air molecules up by temp
-increase the rate at which air particles hit a surface by speeding them up -the hotter than air the more thermal energy it contains and the faster its particles move: thermal energy is a combination of internal KE in the particles random motion and internal PE stored as part of that random thermal motion. Since air particles are essentially independent, expect during collision, nearly all the air's thermal energy is internal KE --double the internal KE of the calm air in the tire--> double the average KE of each particle. -a particles KE is prorp to the square of its speed, doubling its KE increases its speed by a factor of √2, as a result each particle hits the surface √2 as often and exert √2 times as much average force when it hits. With each particle exerting √2 x √2, or two times as much average force the pressure doubles.
Ex. when you take an air filled plastic container out of the fridge, it warms from 2 degrees Celcius to 25 C. How much does the pressure of the air inside change>
-it increases by 8.4% -use pressure of ideal gas law equation. -2 degrees celcius is about 275 K, and 25 degress C is about 298 K: -pressure for 298 K=Boltzmann constant x particle desnity x 298 K -pressure for 275 K=Boltzmann constant x particle density x 275 K -the particle density of the air in the container can't change as it warms up because its volume is fixed. Therefore we can divide the upper equation by the lower one and cancel out the Boltzman constant and particle density on the right giving: 298K/ 275 K-=1.084--> the pressure in the container thus increases by a factor of 1.084 or about 8.4%
hot air ballon cont:
-one gas that has a lower density at atmospheric pressure is hot air. Take fewer particles than filling it will cold air, since each hot air particles is moving faster and contributes more to the overall pressure than does a cold air. —> hot air balloon has fewer particles, less mass, and weights less than if it contained cold air.—-> now a practical balloon with an average density less than the surrounding air, buoyant force is large than its weight and it floats up. -bc the air pressure inside and outside of the ballon is the same, the air has no tendency to move out, and ballon does;t need to be seal. -A large propane burner located beneath the ballon hears the air that fills the envelope. the hotter than air in the envelope, the lower its density and the less the ballon weights. The pilot controls the fame, so that the ballon's weight is very nearly equal to the buoyant force on the ballon. *** if the pilot raises the air's temperature, particles leave the envelope, the ballon's weight decreases and the balloon rise. If the pilot allow air to cool, particles enter the nevelope and the ballon's weight increases—> it descend. -as the balloon ascends, the air becomes thinner and the pressure decreases both inside and outside the envelope. Although the balloon's weight decreases as the air thing's out, the buoyant force on it decreases even more rapidly and it becomes less effective. When it has become too thin to life the balloon higher, it reaches its flight ceiling. ***for each hot air temeraure, there is a cruising altitude at which the ballon with hover. When the ballon reaches that altitude its in stable equilibrium. If it shift toward the, the net force will be upward and vice versa.
Pressure, density and temperature
-since air pressure is produced by bouncing particles, it depends on how often and how hard those particles hit a particular region of surface--> the more frequent or harder the impacts, the greater is the air pressure. -to increase the rate at which air particles hit a surface: pack more tightly. Ex: if you add another 1kg of room temp air to the truck in (a), the truck tire is sturdy and its volume doesn't change much as the number of air molecules inside it doubles. However, double the number of air parcels in the sane volume doubles the rate at which they hit each surface and therefor doubles the pressure (b). -Air's pressure is thus prop. to its density. --increase the rate at which air particles hit a surface by speeding them up (c)
EX: after you push a suction cup against a smooth wall, the elastic cup bends back and a small, empty space is created between the cup and the wall. What keeps the suction cup against the wall?
Air pressure keeps it against the wall. Because the space between the suction cup and the wall is empty, the pressure there is zero. Pressure of the surrounding air exerts large inward forces on the outsides of both the cup and the wall, squeezing them together. As long as there is no air between them to push outward, the cup and wall remain tightly attached> Once air leaks into the suction cup, its easily detached from the wall.
surface area and pressure
Ex: ignore gravity: inside a truck tire containg 1 kg of room temeprature air, the air particles whiz around inside the tire and each time a particle bounces of a wall of the tire, it exerts a force on that wall. Although the individual forces are tiny, the number of particles is not and taught they produce a large average force. The size of this total force depends on the wall's surface area: the large its surface area the more average force it experience---> instead refer to the average for the air exerts on each unit of surface area: pressure=force/surface area. -SI unit: newton per meter^2: PAscal -one pascal is a small pressure, the air around you has a pressure of about 100,000 Pa, so it exerts a force of about 100,00 N on 1-m^2 surface. -besides pushing on the walls of the tire, air also pushes on any object immersed init. Its particles bounce off the object's surface pushing them inward. AS long as the object can withstand these compressive forces, the air won't greatly affect it since the uniform are pressure ensures that the forces on al sides of the object cancel on another perfectly. -air particles also bounce off one another, so that air pressure exerts forces on air too--> a cube of air inserted into the tire experiences all the inward forces that a cube of metal would experience, the air around the cube pushes inward on it and the cube pushes outward on the air.
temperature Scales
How to convert from one temperature scale to another: -temperature measure the average internal KE per particle, the hotter the air the large is the average inthrall KE per particle and the greater the airs pressure, tire pressure increases on a hot day. -At absolute zero (-273.15 C or -459.67 F) air contains no internal KE at all and has no pressure. -room temp: 293 K. -Since air pressure is prop to both the air's density and its abolsute temp but that relationship doesn't work if you compare the pressure of two different gasses which differ in chemical composition. -real gas particles aren't completely independent of one another: if the temperature drops too low the particles begin to stick together to from a liquid and the relationship becomes invalid.
EX: can a hot air ballon lift more on a hot day or cold day?
If can lift more on a cold day, because the outside air is relatively dense and the buoyant force on a balloon with be larger than it would be on a hot day. The hot air in the balloon will cool off more quickly on a cold day, but the balloon with be able to carry a heavier load
EX: crocidiles sometimes swollen rocks to lower their heights in the water. To float with only its is able the water's surface, approx. what average density does the croc need?
It needs an average density just slightly less than water. The croc. should weight a little less than the water its displaces when fully immersed. That way, it will experience a net upward force and will rise to the water's surface. Once its eyes are above the surface and not displacing water, the net force on the croc. will drop to zero and it will hover. Since the croc. mass is slightly less than an equal volume of water, its average density is slightly less than that of water.
air and density
air has density: air particles have mass, each volume of air has a certain amount of mass, average mass per unit volume is called density -air pressure is prop. to its density. -air around is about 1.25 kg/m^3, water is about 1000 kg/m^3 density: p=mass/volume=m/V units: kg/m³
inert gases
air particles are extremely small. Most are nitrogen and oxygen molecules but others include Carbon dioxide, water methanee and hydrogen molecules, also neon argon helium etc. Those particular atoms, which don't make strong chemical bonds and rarely form molecules are inert gases: due to their chemical activity.
EX: as you drive up and down in the mountains you may feel a popping in your ears as air moves to equalize the pressures inside and outside your eardrum. What causes these pressure changes/
as you change altitude, the atmosphere pressure changes. The air inside your ear is trapped, so its temperature, density and pressure are normally constant. As your altitude changes, the pressure outside your ear changes and you eardrum experiences a net froce. It bows inward and outward, muting the sounds you hear.
pressure imbalances
balanced pressure exerts no overall force: forces on opposite sides of object cancel -unbalanced pressure exerts an overall force: forces on opposite sides of object don't cancel, forces push object toward lower pressure.
why should air and helium have the same particle densities whenever their pressures and temperatues are equal?
because a gas particles' contribution to the pressure doesn't depend on tis mass or weight. At a particular temperature each particle in a gas has the same average internal KE in its translation motion, regardless of its mass. -although a helium atom is much less massive than a typical air particle, the average helium atom moves much faster and bounce more often--> lighter but faster-moving heliumm atoms are just as effective at creating pressure as heavier slower- moving particle -if you allow the hell atoms inside a ballon to spread out until the pressure and temperature inside and out the balloon are equal, the particle densities inside and outside the balloon will also be equal. ***since the helium atoms inside the balloon are lighter than the air particles outside it, the balloon weights less than the air it displaces, and it will be lifted upward by the buoyant force. -the pressure of gas is prop. to the produce of its particle density and its absolute temperature. --> holds regardless of the gas's chemical composition.
EX: a Carbon dioxide molecule si heavier than an average air particle. if you pour carbon dioxide gas from cup which way does it flow in air?
down, it is heavier than air because its molecules are heaver than air particles. -has the same pressure and temperature as the air around it and thus the same particle density, however, each CO2 molecule weights more, so the CO2 is the denser gas and floats downward in air.
ideal gas law
pressure=Boltzmann constant x particle density x absolute temperature -only applies perfectly to independent particles, real particles are not completely independent -boltzmann constant k=1.381 x 10^-23 Pa x m³.(particle x K) -temeprature in Kelvins -pressure proportinal to particle density times abolstue temp, and with an associated constant of proportionality is call the ideal gas law.: relates pressure, particle density and absoltue temperature for a gas in which the particles are perfectly indecent.
hot air balloons
since air is light, few objects float in it. One of these rare objects is a ballon with a vacuum inside it. Assuming that the ballon has a very thin out shell or envelope, it will weight almost nothing and have a density near zero. Bc its neglibgle weight is less than the upward buoyant force it experience, the empty balloon with float upward nicely. -this empty ballon wont last long, because't its surrounded by atmospheric pressure air, each square meter of its envelope will experience an inward force of 100,000 N. With nothing inside the balloon to support its envelop against this force, it will smash. So an empty balloon won't work. -What will work is a balloon filled with something that exerts an outward pressure on the envelope equal to the inward pressure of the surrounding air. Then each portion with experience a zero net force. You could fill the balloon with outside air, but that would make its average density too high. Instead need a gas that has the same pressure as the surrounding air but a lower density.
EX: if you remove a partially filled container of food from the fridge and allow it to warm to room temperature, the lid will often bow outward and may even pop off. what has happened
the pressure of the air trapped in the container increases as the its temperature increases, causing the to bulge outward. Whenever a trapped quantity of gas changes temperature it also changes volume or pressure o both. In this case ,warming the air trapped in the container causes its pressure to increase. The unbalance pressures inside and outside the container cause the lid to bow outward or even to pop off.