11 Gases
(air pressure, barometer, Bernoulli's equation in \cp1\liquids)
demo:
ball in reverse vacuum cleaner stream
put a little water in a soda can, heat it,
then plunge it inverted into cold water
heat a larger can until it collapses
soap bubbles as large as a person
extend a board from under a newspaper and then smack the board
place two empty soda cans slightly apart and blow between them
through a straw
total mass of air in a frigerator is same as a grapefruit
room is ...
atmosphere is barely larger than Earth's diameter
place a board on a chair and try to lift the board from its center
by raising a suction cup
whap a toilet plunger onto the wall so pressure keeps it there
ears are pressure meters, they squeeze in a meter of water,
or when going over hills, up in airplanes...
try to drink from two straws, one in and one outside the soda
the outside straw lets air in so none is removed from your mouth
dunk an inverted glass in water, see trapped air which compresses
as the glass is lowered. Scuba divers have to deal with this.
estimate the weight of a car from its tire pressure and area
trucks spread the weight over 18 wheels
Boyle's law, compressing without changing the temperature as shown in fig 13.13 page 241: P1V1=P2V2
A solid cares how far you strech it, F=kx
A liquid cares how fast you try to move it, F=kv.
This means we swim by moving our hand quickly backwards and slowly forwards.
Pressure = force / area
Pressure of my car = estimate weight / area
foot = estimate weight / area
high heel =
ice-skate =
auto on tires =
A sharp knife has less surface area and hence greater cutting pressure for the same force
Pressure is the same in all directions within a fluid, poke a hole in a can and the water flows outward.
Pressure increases with increasing depth, as our ears know when diving
see this visible hand vein when arm is lowered becomes invisible when my hand is raised because pressure is lowered. Our blood pressure is measured at the level of our heart.
Density: float or sink: http://www.vanderbilt.edu/blogs/leandek/ken/archives/penny2.mov
City water pressure from elevated buckets, sometimes placed on a hill and sometimes on a tower (in the plains states)
P = force / area = weight of water above / area
water mass density = 1 g/cc = 1000 kg / m3
its weight density = 9800 N /m3
= 64 lbs /ft2,
salt water density is a bit more dense so we float more easily = Salt Lake in Utah. Or a barge moving from the last Panama freshwater canal lock into the salty ocean. Or the homework project suggest making an egg that sank in fresh water float by adding salt to the water.
density of ice = 0.92 * density of fresh water
density of salt water = 1.03 * density of fresh water
Weight of a column of fluid = mass density * g * volume = DgV
The resulting pressure P = F / A = D V / A = DAh/A = D g height
Let d = weight density = g * mass density then P = dh
Pressure at the bottom of a dam is dgh.
Champagne less bubbly in deep tunnels having higher air pressure
Pressure at any level in a fluid depends only on the height of the fluid above it and not on shape of the container, see page 217.
Weight of a column of air above each m2 causes the air pressure = 1.013 x 105 N/m2.
This matches the weight of 32 feet of water or 30 inches of mercury.
One N/m2 = Pa. Pa = 1 N/m2
So air pressure is ≈ 105 N/m2 = 14.7 lbs/in2 = 2116 lb/ft2
This air pressure will hold fluid in an inverted, covered tube.
Why Pascal?
Pascal made a barometer out of 46 feet of red wine.
Pressure decreases on a rising bubble, so the bubble expands as it moves up, same thing for a rising weather baloon.
Bouyant force from difference in pressures at top and bottom of the object--sides cancel.
At the top: P = d y (y = depth under water surface)
At the bottom: P = d (y+object height=l)
difference in pressures = P-bottom - P-top = d(y+l - y) = dl
delta P = dl = fluid's weight density * objects length
delta Force = Area of object * delta P = Adl = d * volume of object
The bouyant force = weight of fluid that would fill the volume of the object. This is archimedes principle (Syracuse, 287? - 212 b.c.). King's crown was not all gold and he refused to pay due attention to a Roman soldier wanting to question him. "Wait till I finish this math problem" and was killed. Submarine takes in external water or expels water to change its mass. The changed net force of weight - bouyant force raises or lowers it.
Iron floats in mercury,
as does the 100 inch telescope on Mt. Palomar.
What is the bouyant force of air on you?
The floating ship hull displaces a larger volume of less-dense water. The floating ship displaces a greater volume of less-dense water while it is floating. After it has sunk it will displace only its own volume of water. When the ship sinks it displaces only its own volume of water, which is less than the volume of water it had been displacing. The level of the ocean decreases.
But for a melting ice cube, the water level stays the same. Think of a glass full of ice and water. The level doesn't rise or fall after melting because a 5 gram piece of ice will dsiplace 5 grams of water and after melting will become 5 grams of water. The ship doesn't melt and get denser, plus, melting usually expands a material. When an ice cube containing an air bubble melts, the water level still stays the same.
An object sinks if its density is more than water, or whatever fluid its in. A hot air balloon rises.
An icecube is floating. What portion of its volume is submerged? The bouyant force B (= dwater * Volume under water = Vsubmerged) equals its weight W =diceVice
So dwater * Vsubmerged = diceVice or
Vsubmerged / Vice = dice / dwater 0.92 / 1.03 = 0.89
So 89% of an an ice cube is below the water surface.
Mountains of less-dense material float on more dense mantle--density of mountain=.85 density of mantle so is 85% submerged. And move upward as they are eroded. Takes 50 or 100 million years to build and erode a mountain range. Build as tectonic plates collide, one flowing under an the other over. New mountains are steep and tall: Himylaya and Sierra. Old are smoothed and low like Apalachians or Carpathians in East Europe. Earth's lifetime of 4 billion years means there has been enough time to build and erode 40 generations of mountains.
Bouyant force makes less dense materials float and more dense things sink. This viscosity of the fluid is also involved in the sink-time. Within a few tens of millions of years the molten Earth separated into most dense at the center = iron and less dense at the surface.
Air pressure decreases exponentially with altitude. (Fisher went North.)
How do you use a barometer to measure the height of a building. 1) dh 2) as a ruler, giving height in barometer units 3) bribe the building manager with the barometer so he'll tell you the building's height 4) throw it off the roof and time the fall 5) compare shadow lengths 6) place it such that when you view passed it you see the top of the building at the top of the barometer. The similar triangles tan(Θ) tell you the height of the building. 7) Swing it as a pendulum to measure g at the top and bottom of the building. 8) Height of meniscus such that the surface tension force is balanced by the weight of the column of height h. This involves g. As g changes from bottom to the top of the building, the height of the msniscus will also change. Measuring this height measures g and change in g gives building height. 9) Tie a string to the barometer and lower it to the ground. Then measure the length of the string. 10) Any equation containing 'g' or pressure or density will do.
A barometer is an upside down tube of fluid. Makes a vacuum at the top. Same as holding a glass upside down over a piece of cardboard. The barometer was actually the first vacuum created by people. Before then, they thought a vacuum was an imposibility.
Pressure drag on an object in a flow is integral of pressure throughout surface. For a sphere its Fnet in direction of flow. For an airplane wing--a bent sphere--its up and back.
Drag on a ball. Golf ball drag decreased by roughness resulting in less diference of pressure on front and back because the turbulence creeps forward into the direction of motion and beyond the top and bottom of ball.
Pascal's (1623-1662) principle takes advantage of the fact that fluid pressure only depends on depth.
A1 A2
┌─┐ ┌────────────────┐ ┌─┐
│ └─┘ └──────────┘ │
│ │
└─────────────────────────────────┘
Force = Pressure * area. Connect a large and a small piston to a volume of fluid. Press with force F1 on the smaller piston having area A1 get pressure P= F1/A1 = F2/A2. The pressure is transmitted to the larger piston having area A2 and pushed with a force
F2=(A2/A1)F1.
A small area supports a small weight wlittle=PAlittle but a large area supports a large weight wbig = PAbig
Hewitt page 225 shows how you lift a car on a hydraulic jack at the repair shop.
Huge forces to squeeze carbon into diamond under earth's surface
Cohesion is the force between the constituent atoms of a drop of water. Adhesion is the attractive force between differing materials.
Adhesive force allows you to pick up a scrap of paper by touching it with your finger. It also allows bugs to walk up the wall and along the ceiling. Does the fly twist or rotate to land on the ceiling?
Surface tension. page 227 picture shows a net pull downward.
Surface tension decreases with temperature (because atoms move faster and get more away from neighbors) so heating a film of liquid from below will make a dry spot above the flame.
And hot water drops splash more than do cold water drops, and hot water passes through smaller cracks in its container.
Oil drops get larger on cold soup because surface tension has increased compared to dispersed oil when the soup was hot.
Surface tension
holds soap bubbles together taking smallest surface area because
a sphere has the smallest surface to volume ration of any shape
a water drop in space forms a sphere
makes spilled milk splash into bubble-topped crown
pulls two cheerios together
lets bugs walk on water (ratio of surface tension force / weight=Je)
put oil on water and the surface has less tension and quits waving
moves air from smaller balloon into larger when joined
also holds the water in an inverted glass covered with cardboard
allows a drop to stay intact over a smooth water surface
decreased water tension when under pressure makes earthquakes begin at depth under surface
pulls water into a sphere when falling over a plate held under the faucet
soap decreases tension so dirt can be pulled out of a pit between cloth fibers
makes capillary action that will:
move melted wax up a candle wick to the flame
will pull water out of a bucket by lowering a rag below the bucket's level = capillary action, same for 1988 mustangs because grandma didnt like to lift the car and turn it upside down.
a Stenus beetle walks on water. It drops a fluid that changes the surface tension of the water behind it, resulting in a net forward force giving a speed of 0.7 m/s. Get a free ride.
moves fluid up or down a tube depending on contact angle
Capillarity transfers water from you to the fibers of the towell, whose surface area is like "fractal" bumps.
A drop of water cares which material it rests on. The force between adjacent water molecules is cohesion. There is also a force between the water molecules at the boundary and the material they lie on. This adhesive force depends on the material. The ratio of cohesive to adhesive forces determines whether the liquid flattens out of balls up. Place a fluid on a material. The fluid will either spread out or curl up depending on the two materials. This is the contact angle. Waterproof by coating a surface with an agent giving an increased contact angle.
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. .
. . . .
. Θ . . .
. . . .
. . . Θ .
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Force of adhesion and cohesion make the surface curve up or down.
Capillarity: Height of water in the tube is determined by the raised weight equalling the force of adhesion between liquid and tube. The weight = density times volume, where volume is area times height πr2h so smaller area gets greater height.
Hewitt problems:
How can you cut a pie into eight pieces by slicing it three times?
problem 7, 9 & 10 page 233.
We had, Vsubmerged / Vobject = dobject-floating / dwater
If x% of an object is submerged then its density is x% that of the supporting fluid: problem 7 and 9.
7) 90% of the total voume is submerged. Shaving it to 9 cm then .9 cm would be above water.
If an object is totally submerged then the bouyant force of the fluid supports (fluid density / object density )% of the object's weight: problem 10.
10) Steel is four times as dense as water. This means the bouyant force will support only 1/4th of the weight of steel. The ship's crane has to lift 3/4 of the weight of steel. The crane can not lift one times the weight but two boats can lift 1.5 times the weight of steel.
9) If 90% of the vacationer is submerged then his density is 90% that of water.
Hewitt problems 1 and 2 give the same pressure of 50 N/cm2
Water expands when freezing, this is contrary to most other chemicals. Decreasing the external pressure allows water to freeze at a higher temperature. Increasing the pressure on ice results in lowering its freezing temp. Ice skates work this way. This effect is combined with the heating due to the friction of the blade against the ice. Estimate the pressure under an ice skate, shoe, high heel, car tire, snow shoe.
skip:
When something freezes it gives off energy. This slightly warms its surroundings. The atmosphere is warmed a bit when cloud water freezes.
Refrigerator works by
How much does the height of a 300-meter steel TV tower increase when its temperature changes from 5 to 30 °C?
dL=αL(Tf-Ti) = (11x10-6/C°)(300 m)(30C°-5C°)=0.0825 m = 8 cm.
The design must account for this. Guy-wires also expand. The height of the tower is 1/4 or 1/2 the emitted wavelength=cf, say 20 MHz.
The 1400-foot tall Sears tower in Chicago grows by one foot as the temperature changes from 0 to 100 °F.
Temperature compensated pendulum:
The length of a pendulum increases when the temperature increases. Sit a second material on the end of the pendulum such that it expands upward as the rest of the temperature expands downward, keeping the center of mass in the same spot.
│ │Iron Iron material expands downward by dLFe=αL(Tf-Ti)
│ Hg │ and the Mercury expands upward by dLHg=αL(Tf-Ti)
│ ^ │ Since the ratio
│ | │ mass of mercury / mass of Iron =13.6/7.87 x103 kg/m3
├─────┤ we see Hg must expand 7.87/13.67 as much as dLFe
│ │ to keep the center of mass in the same place
│ │ The ratio of expansion coefficients is
│ │ αFe/αHg = 12x10-6/60x10-5/C° = 0.02
└─────┘ Both materials have the same area.
We need
dLHg=αHgLHg(Tf-Ti) = 7.87/13.67 dLFe=αFeLFe(Tf-Ti) cancel (Tf-Ti)
or LHg = 7.87/13.67 αFeLFe/αHg = ( 7.87 / 13.67 )(0.02) LFe =
0.0115 LFe
If LFe=1 meter then LHg=1.15 cm.
The Fe-Ni material Invar has α = 0.7 x 10-6/C°, which is small hence good for clock pendulum material.
When water freezes it expands and bursts pipes, streets, crumbles buildings, windshields or rocks as it gets into cracks and then expands.
Compressiblity = 1 / Bulk modulus
A pressure of 1 N/m2 compresses a steel bar by one part in 1011 of its original volume, or keeps one from expanding thermally. A pressure of 109 N/m2 compresses a steel bar by one part in 100 of its original volume, or keeps one from expanding thermally by one part in 100.
The volume of a cube of water decreases by 0.464 of its original volume for each 100-thousand added atmospheres of pressure (or to keep one from expanding thermally). 1 atm=1.013 x 105 N/m2.
Hg expands through a constriction in a max-temp meter and must be shaken (not stirred) to get it back through that constriction.
To measure how many kg/s od water flow in a river, its (rho)(Area)(Velocity) = (kg/m3)(m2)(m/s) = kg/s.
Since rho is constant unless subject to thousands of atmospheres of pressure, AV=constant is a statement of the conservation of mass.
As pipes narrow the flow must speed up.
The speed of the wind increases as it squeezes through a constriction between adjacent buildings.
This happens when you watch the narrowing of water falling out of the kitchen faucet.
If pipeway makes a right-angled turn, the water flow begins spinning. We see this resulting spinning exhaust from car pipes.
Why doesn't the dust blow off your car when driving at the posted speed limit, or off a room fan? Why do pipes plug up with goop and not wear away from the inside out? There is no motion of fluid relative to the car, fan, or pipe in the last millimeter adjacent to the object. This is the boundary layer. The speed of water flowing within a pipe has a parabolic profile in the pipe.
Bernouli's equation is the conservation of energy per unit volume of fluid: W/V = K.E./V + P.E./V
We saw that W=PV or that W/V = P
Kinetic energy per unit volume is K/V=1/2(rho)v2
Gravitational potential energy per unit volume = mgh/V=(rho)gh
Bernoulli's equation is P + 1/2(rho)v2 + (rho)gh = constant
This describes how the pressure, speed, and height of a piece of fluid varies as that piece moves about. The changes in height are usually small. More often P + 1/2(rho)v2 = constant (skip h).
P is called static pressure and 1/2(rho)v2 the dynamic pressure (skip).
Some Bernoulli examples (to wet your appetite for a sesond physics course containing more math):
A ball will remain within a vacuum exhaust flow. Higher speed within stream also has less pressure. When the ball tries to move out of the stream flow it feels higher pressure on outer edge than on inner edge and moves inward.
Same thing to place an egg in a glass of water and then let the aucet run over the egg. The egg will move upward.
Same thing to hold a spoon halfway in the faucet flow. The spoon feels an inward force.
speed of fow at outer edge of tornado
->
-->
--->
---->
----->
------>
------->
speed of fow at inner edge of tornado (or poured bottle contents)
The speed profile across a tornado shows less speed on the outside and so more pressure on the outside than the inside. The difference in pressure across a streamline supplies the centripetal force to make a piece of air spin. This is stable.
When pouring fluid out of a bottle, the flow is curved. The outer edge (farthest from the bottle) of the flow has less speed (like the tornado) and more pressure so the outer flow is pressed inward. The fluid drips down the bottle.
is more curved, has a shorter arc to sweep, and has more speed and less pressure. The outer edge of the flow is more curved, has a longer arc to sweep, and has more speed and less pressure.
Speed of air decreases and pressure increases as a bug nears a windshield. The difference in pressure on front and back of insect pops it before hitting the windshield. Not many legs hit the window.
Blow over the top of a piece of paper and it rises
Place a straw in a fluid and blow over the straw. The fluid will rise in the straw.
Lift of airplane wing and propeller, and maple-seed helicopter
Curve of a pitched, spinning baseball.
Flow over a prairie dog home:
---------> highest v . . and lowest pressure
-----> higher v . │ │ .
--> low v . │ │ .
. │ │ . low speed high P
────────────── │ │ ──────────────────┐ ┌────
│ │ │ │
│ └──────────────────────┘ │
└──────────────────────────┘
There is a resulting flow of air from ground-level hole on right toward the hole in the high peak. The dogs use Bernouli's equation to avoid suffocation and keep breathing.
Same for a tall chimney in our home. The height of the chimney reaches up to higher wind speeds and lower pressures. This pulls the smoke out of the house. An ventilator exhaust pipe on the roof does the same thing.
A fish profile is streamlined and its eyes are placed where the pressure is lowest.
. .
.█ .
. .
. .
. .
. .
█ . .
Re = rho length speed / viscosity
more examples:
Pressure: Serway 9.15
Each of four car tires have an area A1 of 0.024 m2 and a pressure of 2.0 x 105 N/m2 due to the weight of the car? What is its weight?
P = F / A = mg / 4A1 so mg = 4A1 P = (4)(0.024 m2)(2.0 x 105 N/m2) = 19200 N = 4250 lbs (divide Newtons by 4.5 to get pounds). Trucks have 18 wheels to support more weight.
9.18
A large pressure at ground level is need to have water in a pipe reaching the top of the 1200-foot (366 m) Empire State Building. What is the water pressure at this depth?
P=rho gh = (1000 kg/m3)(9.8 m/s2)(366 m) = 3.3 x 106 N/m2.
9.19
What is the water pressure at a depth of 7 mile (11200 m)?
P=rho gh = (1000 kg/m3)(9.8 m/s2)(11200 m) = 1 x 108 N/m2. What is the net force on a 0.15 meter diameter window?
Fnet = Fin - Fout ≈ Fin = PA = Pπ(d/2)2 = 1.8 x 106 Newton.
9.21
The pressure in an elevated bag pushes glucose into an arm artery having pressure 1.33x104 N/m2. The density of glucose is 1.02 times the density of water. What height of glucose is needed to have as much pressure as in the artery? (Lowering the bag below your arm will instead fill the bag with your blood.)
P=rho gh so
h = P/(rho g)
= (1.33 x 104 N/m2) / [(1.02)(1000 kg/m3)(9.8 m/s2)] = 1.3 m
Archimedes:
The buoyant force is that force needed to push a beach ball under water = 64 lbs for a 1 ft3 ball because water weighs 64 lbs/ft3.
Dervive buoyant force = weight of fluid filling the volume.
Bouyant force from difference in pressures at top and bottom of the object--sides cancel.
At the top: P = rho g y (y = depth under water surface)
At the bottom: P = rho g (y+l, which is the object's height)
difference in pressures=P-bottom - P-top = rho g (y+l - y) = rho g l
delta P = rho g l = fluid's weight density * objects height
delta Force = Area of object * delta P = rho g Al = rho g * volume of object
The bouyant force = weight of fluid that would fill the volume of the object. This is archimedes principle (Syracuse, 287? - 212 b.c.). King's crown was not all gold and he refused to pay due attention to a Roman soldier wanting to question him. "Wait till I finish this math problem" and was killed. Submarine takes in external water or expels water to change its mass. The changed net force of weight - bouyant force raises or lowers it.
What is the bouyant force of air on you? A few ounces. If I am a rectangle.
An icecube is floating. What portion of its volume is submerged? The bouyant force B (= g rhowater Vsubmerged) equals its weight
W = g rhoice Vtotal
So g rhowater * Vsubmerged = g rhoiceVtotal or
Vsubmerged / Vice = rhoice / rhowater 0.92 / 1.03 = 0.89
So 89% of an an ice cube is below the water surface.
A 0.227 m3 (8 cubic foot) piece of steel is suspended under water by a cable. The iron has mass 1800 kg (4000 lb). What is the buoyant force on the block of steel?
Fb=rho gV = (1000 kg/m3)(9.8 m/s2)(0.227 m3) = 2043 N (460 lbs)
rho-steel = 8000 kg/m3
rho-water = 1000 kg/m3
1800 kg of steel weighs (1800 kg)(9.8 m/s2) = 17640 N
What is the apparent weight of the steel?
Fapp = Fnormal - Fbuoyant = 17640 N - 2043 N = 15600 N
Notice that water supports rho-water/rho-steel times weight of steel = 1000 kg/m3 / 8000 kg/m3 times weight of steel = 1/8 weight so the apparent weight = 7/8 actual weight = 7/8 (1800 kg)(9.8 m/s2) = 15600 N.
What is the bouyant force of water on a fully submerged balloon having a radius of 0.5 m? Its the weight of water that would fill the balloon.
Fb=rho gV = (1000 kg/m3)(9.8 m/s2)(4/3 π r3) = 5129 N = 1140 lbs.
Serway 9.34
A 4x6 meter boat floats on water. It sinks 0.04 m when a truck is placed on the boat. What is the weight of the truck? It equals the weight of the water filling the additional submerged volume:
W=rho gV = (1000 kg/m3)(9.8 m/s2)(4x6x0.04 m3) = 9400 N.
Serway 9.30
What is the bouyant force of the air on a baloon having a radius of 0.5 m? Its the weight of air that would fill the balloon.
Fb=rho gV = (1.29 kg/m3)(9.8 m/s2)(4/3 π r3) = 6.6 N. The balloon weighs 0.1176 N when empty. If the balloon is tied to a holder, what is the tension in the string?
Fb =
^
|
█
|
\/
T + W
where W = weight of balloon plus weight of helium
weight of baloon = 0.1176 Newton
weight of helium = rho g Vol
= (0.181 kg/m3)(9.8 m/s2)(4/3 π r3) =0.85 Newton
T + mg = Fb or T = Fb - mg = 6.6 N - 0.1176 N - 0.85 N = 5.6 N
Replace T with some cargo, which is another weight. We see the balloon could support a cargo of 5.6 N.
Try to calculate this with Pete's balloon. Estimate volume, weight of balloon, weight of helium within the balloon, then max cargo.
A1 A2
┌─┐ ┌────────────────┐ ┌─┐
│ └─┘ └──────────┘ │
│ │
└─────────────────────────────────┘
Force = Pressure * area. Connect a large and a small piston to a volume of fluid. Press with force F1 on the smaller piston having area A1 get pressure P= F1/A1 = F2/A2. The pressure is transmitted to the larger piston having area A2 and pushed with a force F2=(A2/A1)F1.
for A1 = 2 cm2
for A2 = 4 cm2
for F1 = 10 N
we get F2=(A2/A1)F1 = (4/2) 10 = 20 N
see \cp1\fractal for a lab.
An animals surface area grows as l2 (l=length)
volume l3
mass l3 = volume
animal density always about equal to water, so it is equal for most every animal.
how can you cut a pie into eight pieces by slicing it three times?
Strength of a support grows as area = l2
The maximum force of bones, tendons, and muscles goes with cross sectional area, or l2
Doubling the height of an animal will double a bone length and the animal's mass and volume cube = 8 times but the support ability goes up as a square = 4 times. We can't have 100 foot tall ants having the same shape as a 1 mm high ant. Making the ant twice as high requires it also become 4 times as thick and 8 times as voluminous and heavy. Elephant legs are thicker than our legs, which are thicker than mice, which are thicker than ants--thicker means a larger proportion of height. Class estimates some leg thickness and heights for various species.
Consider an animal falling from a standing height, which is proportional to l. The speed of the collision is 1/2mv2=mgl so v goes as sqrt(l) but kinetic energy, which causes breakage, goes as 1/mv2. With m going as l3 and v2 going as l we get breakage going as l4 and momentum p=mv going as l3 times l = l4. Infants fall just a few inches and aren't hurt but adults do get hurt. Dogs less likely than horse, cow, or elephant.
Bouyancy goes with volume. We can always think of mass when thinking of volume.
frequency of sound produced goes as 1/l, think of the buzz of smaller to larger mosquitoes.
Female-attracting male toads cheat with an expanding air pouch but also by sitting in the cooler locations where the speed of sound speed = wavelength times frequency = sqrt( gamma RT ), when speed goes down then wavelength of transmitted sound goes up because frequency of generator can't change.
We retain heat with our volume and lose it with our surface. We have enough volume to retain heat when the outside temperature is 60 degrees less than our body temperature of 98.6 (average of many persons, your temp might be 97 or 98 or 99). We have too little cooling surface area to bear being surrounded by air just two degrees above our body temperature. We wrap many blankets around babies because they are all surface and no volume.
We get warm when walking. Did our primate ancestors loose hair to not retain heat when walking between dwindling forest areas? As we got taller we also increased our volume/surface ratio and lost heat-retention hair?
Surface to volume ratio changes from cooling to retaining at 1 square meter size, which is a large dog. Elepants cool with ear area, dogs with tongue sweating. We sweat.
Plants absorb sunlight and exchange atmospheric gas through their surfaces. The 200 leaves of a tree result in an area of 200 m2 and a gas exchange area of 6000 m2 = 1.5 acres. The surface area of the ten billion root hairs of a square meter of Kentucky Bluegrass lawn add up to 350 m2, which is double the floor space area of a large home.
The volume of our lungs is about 6 liters but its 30 million alveoli have a surface area of 50 to 100 m2, which is about the floor area of this class room.
Our muscles exchange oxygen and CO2 through 6000 m2 surface area, as does that tree.
We chew food to increase its surface area, making it more readily dissolved by digestion acids and chemicals than if it were voluminous blocks. We process food through the surface area of our intenstines. They are 7 meters long but the surface has villi projections which themselves have yet smaller microvilli projections givnig a total intestinal area of 200 m2 = 0.5 acres. See that a hamster tonque has bumps on bumps in Cameron page 334.
Surround a car's gas tank or a lantern with a screen to reduce flare up or explosions. The large surface area radiates lots of heat and lessens, or removes until much greater temperatures, the chance of an explosion. The holey lantern also keeps wind out when walking on windy nights.
Fractal coastline is same from far or closer or closer or even closer. Same for the wiggles in the price of cotton over a century, dacde, year, month, week, day, or hour. If you see a plot of the stock market going up or down, you can not know whether it is showing a time span of a century or a day.
The air resistance or drag force on flying birds, swimming fish, and standing plants and animals goes with area. Trees reduce their area in the wind to reduce drag force.
Homework
page 212 problems
12) strength of rope determined by its cross sectional area. To double its strength one must multiply its radius (hence area) by 22=4.
17) Each piece of a rope suspended from its two ends will naturally fall parallel to the net force on it. The shape of the rope is a record of how the direction of the net force changes along its length. The upside down rope is an arch that is everywhere tangent to the net force. There is no sideways force on a piece of the arch. No piece is being pushed outward, that is, perpendicular to the arch at any point. It has no tendency to fall apart unless you remove any piece.
15-16) Dams, barrels, and spherical cars are stronger because the forces result in compression. Deeply diving subs are spheres to better withstand the force of the water.
21) More total surface area of lots of little apples or potatoes or ... They result in more peel and less stuff.
22) Suntan lotion or paint proportional to surface area. If a person is a sphere then volume = 4/3 π r3 and surface = 4 π r2. Mass goes as volume, or having a volume then radius goes as volume to the 1/3 power. If the mass doubles, how does the radius change--and hence the surface? S2/S1 = (r2/r1)2 = (m2/m1)2/3=22/3=1.587
23) Heating, cooling, dissolving sugar, exploding dust-sized particles, burning (problems 22-31) determined by surface area not length or volume. Our fingers, ears, and nose cool faster than the rest of us because of the increased surface area.
33) Small animals radiate more heat per kg than large do animals. The blood flow has to be larger to get more blood there. The heartbeats have to be frequent. A life time is about one billion heart beats. Slower heart beat rates for larger animals means it takes a longer time for those billion beats to occur. The lifetimes of animals is proportional to mass.
36) A cell receives life-supporting chemicals through its surface. The upper limit of surface to volume is reached and not exceeded. Cells are the same size no matter the size of the animal. An exception is an egg, which is a single cell.
38) People sweat through their surface. Children have more surface per volume or kg and will be thirstier--drink more often.
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