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How Do Airplanes Fly?
How do airplanes fly? And for that matter, how do sharks swim through water? Both are massive objects with interesting shapes moving through a fluid (both air and water are fluids.)
After all this time you’d think that we know all the details of how an airplane flies. There must be some specific and agreed-upon explanation. Air hits a plane, air and plane then follow laws of physics, and voilà, the plane flies, right?
Although flight indeed is in accord with the laws of physics, the specific ideas about how this happens are incomplete and controversial.
What’s the controversy about? What new ideas are being proposed?
Bernoulli theorem idea
(Here the class will look into the general idea.)
image from http://www.thaitechnics.com
This explanation is from the Scientific American article.
Newton’s laws of motion
(Here the class will look into the general idea.)
This explanation is from the Scientific American article.
New Theories of lift
These ideas are also from the Scientific American article.
How do fish fly through water?
Just as aerodynamics explains how airplanes generate lift and fly through the air. hydrodynamics explains how fish generate lift and fly through water.
And yes, fish do fly through water. If they stop moving, then they literally fall down to the bottom of the ocean.
We all know how interesting hammerhead sharks are. Why do their heads have this peculiar shape?
Part of it has to do with the fact that their head is a giant electromagnetic sensor; it can detect the EM fields of nearby prey. But evolution optimizes body design in more than one way. Sharks need to do more than sense prey, they need to move efficiently.
This recent paper, A hydrodynamics assessment of the hammerhead shark cephalofoil, shows that the shape of their head may increase maneuverability as well as produce dynamic lift similar to a cambered airplane wing.
See Gaylord, M.K., Blades, E.L. & Parsons, G.R. A hydrodynamics assessment of the hammerhead shark cephalofoil. Scientific Reports 10, 14495 (2020). https://doi.org/10.1038/s41598-020-71472-2
Also see : Pioneers of flight, How do airplanes fly?, the graveyard spiral, and breaking the sound barrier – Flight
References
No One Can Explain Why Planes Stay in the Air: Do recent explanations solve the mysteries of aerodynamic lift? By Ed Regis
Scientific American, February 2020, Volume 322, Issue 2
Aerodynamic Lift, Part 1: The Science, Doug McLean, The Physics Teacher Vol. 56, issue 8, 516 (2018)
https://doi.org/10.1119/1.5064558
Aerodynamic Lift, Part 2: A Comprehensive Physical Explanation, Doug McLean, The Physics Teacher Vol. 56, 521 (2018)
https://doi.org/10.1119/1.5064559
Understanding Aerodynamics: Arguing from the Real Physics, Doug McLean. Wiley, 2012
You Will Never Understand Lift. Peter Garrison, Flying; June 4, 2012.
Flight Vehicle Aerodynamics. Mark Drela, MIT Press, 2014.
#Flight #aerodynamics #Bernoulli #Lift
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion is a mathematical model describing change in motion (the acceleration) of objects when acted on by a net force.
A FRAMEWORK FOR K-12 SCIENCE EDUCATION: Practices, Crosscutting Concepts, and Core Ideas
PS2.A: FORCES AND MOTION
How can one predict an object’s continued motion, changes in motion, or stability?
Interactions of an object with another object can be explained and predicted using the concept of forces, which can cause a change in motion of one or both of the interacting objects… At the macroscale, the motion of an object subject to forces is governed by Newton’s second law of motion… An understanding of the forces between objects is important for describing how their motions change, as well as for predicting stability or instability in systems at any scale.
NGSS
2016 High School Technology/Engineering
HS-ETS1-2. Break a complex real-world problem into smaller, more manageable problems that each can be solved using scientific and engineering principles.
HS-ETS1-4. Use a computer simulation to model the impact of a proposed solution to a complex real-world problem that has numerous criteria and constraints on the interactions within and between systems relevant to the problem.
College Board Standards for College Success: Science
PS-PE.1.2.2 Analyze force diagrams to determine if they accurately represent different real-world situations.
PS-PE.1.2.4 Given real-world situations involving contact, gravitational, magnetic or electric charge forces and an identified object of interest:
PS-PE.1.2.4a Identify the objects involved in the interaction, and identify the pattern of motion (no motion, moving with a constant speed, speeding up, slowing down or changing [reversing] direction of motion) for each object.
PS-PE.1.2.4b Make a claim about the types of interactions involved in the various situations. Justification is based on the defining characteristics of each type of interaction.PS-PE.1.2.4c Represent the forces acting on the object of interest by drawing a force diagram.
PS-PE.1.2.4d Explain the observed motion of the object. Justification is based on the forces acting on the object.
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Iridescence and thin film interference
Iridescence is a spectacular optical trick – it is the creation of color without pigment!
Consider surfaces that gradually change color as the angle changes. Soap bubbles, feathers, butterfly wings, some seashells, and certain minerals. Let’s dig in to what causes this phenomenon.
The word iridescence comes from Iris (Ἶρις) the Greek goddess of the rainbow.
There are three ways to get color
Additive color – mixing together light of two or more different colors. Red, green, and blue are the additive primary colors normally used in additive color systems such as smartphones, TVs, projectors and computer displays.
Subtractive color – uses dyes, inks, or pigments to absorb some wavelengths of light and not others. The color that we see comes from the wavelengths of light that are not absorbed by these chemicals.
But iridescence is nature’s special, third way of producing color. In this method, color is created by wave interference with tiny physical structures on the scale of the color’s wavelength.
Iridescence in animals
Iridescence: a functional perspective
Iridescence in minerals
Bismuth is a great example of thin-film interference.
The colors come from a thin film of bismuth(III) oxide that forms on the surface if the crystals are formed in air.
Chemistry.stackexchange What causes the iridescent colour in bismuth?
The physics of thin film interference
Thin-film interference is a natural phenomenon.
In it, light waves reflected by the upper and lower boundaries of a thin film interfere with one another. The result either enhances or reduces the reflected light.
When the thickness of the film is an odd multiple of one quarter-wavelength of the light on it, the reflected waves from both surfaces interfere to cancel each other.
Since the wave cannot be reflected, it is completely transmitted instead.
When the thickness is a multiple of a half-wavelength of the light, the two reflected waves reinforce each other, thus increasing the reflection and reducing the transmission.
Thus when white light, which consists of a range of wavelengths, is incident on the film, certain wavelengths (colors) are intensified while others are attenuated.
Thin-film interference explains the multiple colors seen in light reflected from soap bubbles and oil films on water.
It is also the mechanism behind the action of antireflection coatings used on glasses and camera lenses.
https://en.wikipedia.org/wiki/Thin-film_interference
http://physics.highpoint.edu/~jregester/potl/Waves/InterferenceColors/interfcolors.html
Videos for thin film interference
Apps
Molecular Expressions Interference Phenomena in Soap Bubbles
https://micro.magnet.fsu.edu/primer/java/interference/soapbubbles/
Optical Interference – Java Tutorial
https://www.olympus-lifescience.com/en/microscope-resource/primer/java/interference/
Molecular Expressions – Interference Between Parallel Light Waves
https://micro.magnet.fsu.edu/primer/java/interference/waveinteractions2/index.html
Physics Hanukkah Fun
A goal of Social Studies is to expose students to the diversity of ethnic, religious, and cultural observances in our world. The College, Career, and Civic Life (C3) Framework for Social Studies State Standards notes that students should be able to describe how religions are embedded in culture and cannot only be isolated to the “private” sphere, and identify which religious communities are represented or obscured in public discourse.
A goal of science education is to see how basic laws of nature allow us to understand all phenomenon in our physical universe, from the simplest (fire and candles) to the most complex (how stars work.)
During the holiday season many science teachers do something fun on the physics of Christmas (Google that; thousands of results.) Yet there are more religions than Christianity and more phenomenon related to holidays. In the spirit of science and multiculturalism here we can look at the physics and chemistry of Hanukkah.
What is Hanukkah about?
Hanukkah is a minor Jewish holiday. It doesn’t come from the Hebrew Bible but instead from the book of Maccabees, part of the Jewish apocrypha. It is also known as Hag ha’urim, the Festival of Lights.
Hanukkah is a Hebrew word meaning “dedication.” It refers to the eight-day celebration during which Jews commemorate the victory of the Maccabees over the Hellenistic Syrians in 165 B.C.E. and the subsequent rededication of the Temple in Jerusalem. Hanukkah is specifically about countering antisemitism and was the first successful war for religious freedom.
Celebrations center around the lighting of the hanukkiyah (menorah,) foods prepared in oil, including latkes (potato pancakes) and sufganiyot (jelly doughnuts), songs and games. – Intro to Hanukkah
The Hebrew name Maccabee means “hammer”, and referred first to a leader of the revolt, Judas, the third son of Mattathias.
Capillary action
During the holiday Jewish people light a Chanukah menorah מנורת חנוכה, also called a Ḥanukiyah חַנֻכִּיָּה.
The wick is above the oil, drawing fluid up the wick through capillary action. What exactly is capillary action?
Capillary action & molecule forces
Oil is drawn up through capillary action, also called wicking.
This is a tale of two competing forces:
There is an adhesive force between the oil molecules and the cotton molecules.
And there is an intermolecular/cohesive force between the oil molecules.
Cohesion = ability of like molecules to stick together
Adhesion = ability of dissimilar molecules to stick together
from Bioninja
When the adhesion force > cohesion force then the oil molecules are slowly pulled into the wick.
From Hyperphysics, Surface tension
The following explanation is adapted from the discussion by Sean Snider, on Quora.
A fluid such as heating oil will tend to flow upwards against gravity due to capillary motion.
The individual atoms in the oil will interact with the fiber atoms to cause adhesion.
The oil atoms will bump into the fiber atoms – and move upwards due to intermolecular forces.
The difference in charge between the two types of atoms causes them to repel in all directions, including up.
The oil atoms will keep moving up – unless the forces between them cause them to clump together so that intermolecular forces weaken.
In that case their collective mass is too much to repel the force of gravity.
Typically the density of the fiber itself prevents the oil particles from clumping enough to reach this threshold. Thus they continue to move upward.
This allows the oil to reach the top of the wick and burn.
Instead of the fiber burning quickly, the oil burns.
(Some of the fiber also burns, but much less quickly.)
Capillary action student activities
Wick lab/game! sciensation.org
Capillary action and diffusion lab
Lights, Camera, (Capillary) Action! Scientific American
Convection & temperature differentials
The heat from the flame warms up the small olive oil vessels, below.
Those vessels are often transparent.
That heat causes a temperature differential: warmer oil at the top and cooler oil at the bottom.
This would cause convection and/or turbulence in the fluid.
This should be visible if we record it with a high speed, high-resolution smartphone camera.
Convection, turbulence, and related topics are usually left out of high school physics curriculum, so this might be a fun way to introduce it.
Experiment: Add a drop of coloring into oil. Light the wick.
Then we can visually observe the convection currents.
Dreidel physics
A dreidel (Yiddish: דרײדל) or sevivon (Hebrew: סביבון) is a four-sided spinning top, played by children during the Jewish holiday of Hanukkah. Contrary to popular belief, this toy is not part of the Hanukkah story. It is a Jewish variant on the teetotum, a gaming toy found in many European cultures.
Through use and observation of a dreidel students may be inspired to understand how it works, which requires knowledge of angular momentum, rotational motion, gyroscopes, and precession.
One idea for class use is to record the motion with a high speed camera, and then play the footage back in slow motion, to reveal details of motion that would not be clearly visible to the naked eye.
Let’s take a look at Extreme High-Speed Dreidel Physics by Alexander R. Klotz:
… a dreidel is an example of a spinning top, a source of extremely difficult homework problems in undergraduate classical mechanics related to torque and angular momentum and rigid body motion and whatnot. I was chatting with a theorist I know who mentioned that it would be fun to calculate some of these spinning-top phenomena for the dreidel’s specific geometry (essentially a square prism with a hyperboloid or paraboloid base), and I suggested trying to compare it to high-speed footage [1000 frames per second] ….
Check out the article and videos here.
Related dreidel topics to investigate
What keeps spinning tops upright? Ask a Mathematician/Physicist
What is precession? It is a change in the orientation of the rotational axis of a rotating body. In geometry we would say that if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis.
Precession (Wikipedia)
Dreidels also follow the law of conservation of angular momentum. We learn more about that in Angular momentum
And a dreidel itself is similar to a gyroscope.
The statistics of dreidel motion
Are dreidels fair? In other words, does the average dreidel have an equal chance of turning up any one of its four sides? Dreidel Fairness Study
Ultra High Speed Physics.
You’re not a mad scientist unless you ask questions like “Imagine a game of dreidel with a 60-billion-RPM top….” Focus: The Fastest Spinners. APS Physics
How is the holiday spelled? ELA connections
Why write “Hanukkah” instead of “Chanukah” – surely one spelling is right and the other is wrong? The reason for the spelling confusion is the limitations of the English alphabet. Hanukkah is a Hebrew word (חנוכה)
That first Hebrew letter of this word, ח , has a guttural sound. This sound used to exist in ancient English but doesn’t exist in modern English. The modern pronunciation of this letter is a voiceless uvular fricative (/χ/)
As such there is no one correct-and-only way to transliterate this letter. Over the past 2 centuries four ways have developed:
KH – Khanukah (used in old fashioned translations of Yiddish)
CH – Chanukah
H – Hanukkah (the extra ‘k’ is added just to make it 8 letters long.)
H – Ḥanukah (notice the H with a dot under it.)
Each of these is equally valid.
History, art, and social justice connections
Hanukkah and the Maccabees have been a common theme in classical Christian art, sculpture, and music. The story of the Maccabees is a part of Western Civilization through both Jewish and Christian culture. In this article one can see the art, music, and sculpture of Hanukkah.
On a related social justice note, a big part of being anti-racist is listening to voices. Make space to learn from the lived experiences of our students, their families, and their communities. As such I would like to share this:
Hanukkah is about countering antisemitism: Be aware of Hanukkah Erasure.
Learning Standards
College Board Standards for College Success in Science
ESM-PE.1.2.1 Describe and contrast the processes of convection, conduction and radiation, and give examples of natural phenomena that demonstrate these processes.
ESM-PE.1.2.1c Use representations and models (e.g., a burning candle or a pot of boiling water) to demonstrate how convection currents drive the motion of fluids. Identify areas of uneven heating, relative temperature and density of fluids, and direction of fluid movement.
Next Generation Science Standards
MS-PS1-4. Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.
Massachusetts Science and Technology/Engineering Curriculum Framework
7.MS-PS3-6 (MA). Use a model to explain how thermal energy is transferred out of hotter regions or objects and into colder ones by convection, conduction, and radiation.
College, Career, and Civic Life (C3) Framework for Social Studies State Standards
College, Career, and Civic ready students:
D2.Rel.4.9-12: Describe and analyze examples of how religions are embedded in all aspects of culture and cannot only be isolated to the “private” sphere.
D2.Rel.12.9-12: Identify which religious individuals, communities, and institutions are represented in public discourse, and explain how some are obscured.
Transliteration of Hebrew letters
Library of Congress (USA) ALA-LC Romanization Tables
The nature of time
What is time?
What is time? Where does time come from?
In what way is time really something objective? (something actually out there?)
In what ways is time not objective? (so it would be just a way that humans use to describe our perception of the universe)
What is time?
Why does time never go backward?
The answer apparently lies not in the laws of nature, which hardly distinguish between past and future, but in the conditions prevailing in the early universe.
The Arrow of Time, Scientific American article. David Layzer
Is there a relationship between time and the second law of thermodynamics?
Before reading further, understand that these topics require at least some familiarity with the laws of Thermodynamics
“According to many, there might be a link between what we perceive as the arrow of time and a quantity called entropy…. [but] as far as we can tell, the second law of thermodynamics is true: entropy never decreases for any closed system in the Universe, including for the entirety of the observable Universe itself. It’s also true that time always runs in one direction only, forward, for all observers. What many don’t appreciate is that these two types of arrows — the thermodynamic arrow of entropy and the perceptive arrow of time — are not interchangeable.”
No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time, Starts With A Bang, Ethan Siegel, Forbes
No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time
Is time (and perhaps space,) quantized?
Ethan Siegel leads us in a fascination discussion:
The idea that space (or space and time, since they’re inextricably linked by Einstein’s theories of relativity) could be quantized goes way back to Heisenberg himself.
Famous for the Uncertainty Principle, which fundamentally limits how precisely we can measure certain pairs of quantities (like position and momentum), Heisenberg realized that certain quantities diverged, or went to infinity, when you tried to calculate them in quantum field theory….
It’s possible that the problems that we perceive now, on the other hand, aren’t insurmountable problems, but are rather artifacts of having an incomplete theory of the quantum Universe.
It’s possible that space and time are really continuous backgrounds, and even though they’re quantum in nature, they cannot be broken up into fundamental units. It might be a foamy kind of spacetime, with large energy fluctuations on tiny scales, but there might not be a smallest scale. When we do successfully find a quantum theory of gravity, it may have a continuous-but-quantum fabric, after all.
Are Space And Time Quantized? Maybe Not, Says Science
Even In A Quantum Universe, Space And Time Might Be Continuous, Not Discrete
Theoretical physics: The origins of space and time
Avogadro’s law
Previously in Chemistry one has learned about Avogadro’s hypothesis:
Equal volumes of any gas, at the same temperature and pressure, contain the same number of molecules.
Reasoning
(from Modern Chemistry, Davis, HRW)
In 1811, Avogadro found a way to explain Gay-Lussac’s simple ratios of combining volumes without violating Dalton’s idea of indivisible atoms. He did this by rejecting Dalton’s idea that reactant elements are always in monatomic form when they combine to form products. He reasoned that these molecules could contain more than one atom.
Avogadro also put forth an idea known today as Avogadro’s law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
It follows that at the same temperature and pressure, the volume of any given gas varies directly with the number of molecules.
Avogadro’s law also indicates that gas volume is directly proportional to the amount of gas, at a given temperature and pressure.
Note the equation for this relationship.
V = kn
Here, n is the amount of gas, in moles, and k is a constant.
Avogadro’s reasoning applies to the combining volumes for the reaction of hydrogen and oxygen to form water vapor.
Dalton had guessed that the formula of water was HO, because this formula seemed to be the most likely formula for such a common compound.
But Avogadro’s reasoning established that water must contain twice as many H atoms as O atoms, consistent with the formula H2O.
As shown below, the coefficients in a chemical reaction involving gases indicate the relative numbers of molecules, the relative numbers of moles, and the relative volumes.
The simplest hypothetical formula for oxygen indicated 2 oxygen atoms, which turns out to be correct. The simplest possible molecule of water indicated 2 hydrogen atoms and 1 oxygen atom per molecule, which is also correct.
Experiments eventually showed that all elements that are gases near room temperature, except the noble gases, normally exist as diatomic molecules.
As an equation
Avogadro’s Law – also known as Avogadro–Ampère law
when temperature and pressure are held constant:
volume of a gas is directly proportional to the # moles (or # particles) of gas
n1 / V1 = n2 / V2
or
What does this imply?
As # of moles of gas increases, the volume of the gas also increases.
As # of moles of gas is decreased, the volume also decreases.
Thus, # of molecules (or atoms) in a specific volume of ideal gas is independent of their size (or molar mass) of the gas.
Important! This is not a law of physics!
Rather, this is a generally useful rule, which is only valid when gas temperature and pressure is low enough for the atoms to usually be far apart from each other. As we begin to deal with more extreme cases, this rule doesn’t hold up.
At what point does Avogadro’s law not apply?
Example problems
These problems are from The Chem Team, Kinetic Molecular Theory and Gas Laws
Example #1: 5.00 L of a gas is known to contain 0.965 mol. If the amount of gas is increased to 1.80 mol, what new volume will result (at an unchanged temperature and pressure)?
Solution:
I’ll use V1n2 = V2n1
(5.00 L) (1.80 mol) = (x) (0.965 mol)
x = 9.33 L (to three sig figs)
Example #2: A cylinder with a movable piston contains 2.00 g of helium, He, at room temperature. More helium was added to the cylinder and the volume was adjusted so that the gas pressure remained the same. How many grams of helium were added to the cylinder if the volume was changed from 2.00 L to 2.70 L? (The temperature was held constant.)
Solution:
1) Convert grams of He to moles:
2.00 g / 4.00 g/mol = 0.500 mol
2) Use Avogadro’s Law:
V1 / n1 = V2 / n2
2.00 L / 0.500 mol = 2.70 L / x
x = 0.675 mol
3) Compute grams of He added:
0.675 mol – 0.500 mol = 0.175 mol
0.175 mol x 4.00 g/mol = 0.7 grams of He added
Example #3: A balloon contains a certain mass of neon gas. The temperature is kept constant, and the same mass of argon gas is added to the balloon. What happens?
(a) The balloon doubles in volume.
(b) The volume of the balloon expands by more than two times.
(c) The volume of the balloon expands by less than two times.
(d) The balloon stays the same size but the pressure increases.
(e) None of the above.
Solution:
We can perform a calculation using Avogadro’s Law:
V1 / n1 = V2 / n2
Let’s assign V1 to be 1 L and V2 will be our unknown.
Let us assign 1 mole for the amount of neon gas and assign it to be n1.
The mass of argon now added is exactly equal to the neon, but argon has a higher gram-atomic weight (molar mass) than neon. Therefore less than 1 mole of Ar will be added. Let us use 1.5 mol for the total moles in the balloon (which will be n2) after the Ar is added. (I picked 1.5 because neon weighs about 20 g/mol and argon weighs about 40 g/mol.)
1 / 1 = x / 1.5
x = 1.5
answer choice (c).
Example #4: A flexible container at an initial volume of 5.120 L contains 8.500 mol of gas. More gas is then added to the container until it reaches a final volume of 18.10 L. Assuming the pressure and temperature of the gas remain constant, calculate the number of moles of gas added to the container.
Solution:
V1 / n1 = V2 / n2
| 5.120 L | 18.10 L | |
| –––––––– | = | –––––– |
| 8.500 mol | x |
x = 30.05 mol <— total moles, not the moles added
30.05 – 8.500 = 21.55 mol (to four sig figs)
Notice the specification in the problem to determine moles of gas added. The Avogadro Law calculation gives you the total moles required for that volume, NOT the moles of gas added. That’s why the subtraction is there.
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Charles’s Law
Here we learn about Charles’s Law (also known as Charles and Gay-Lussac’s Law.)
What does it do? It describes how gases tend to expand when they are heated.
This is an example of algebra in the real world:
A gas’s volume is proportional to its temperature.
(This is only true when measuring temperature on an absolute temperature scale.)
In algebra, this relationship can be written as:
-> Gas expands as the temperature increases
-> Gas contracts as the temperature decreases
This relationship can be written as:
Important! This is not a law of physics!
Rather, this is a generally useful rule, which is only valid when gas temperature and pressure is low enough for the atoms to usually be far apart from each other.
As we begin to deal with more extreme cases, this rule doesn’t hold up.
Let’s see this in action!
Origin
Named after Jacques Alexandre César Charles (1746 – 1823) a French inventor, scientist, mathematician, and balloonist.
Just so we’re all clear on this, he was kind of a mad scientist. And I say that with the utmost approval!
Contemporary illustration of the first flight by Prof. Jacques Charles with Nicolas-Louis Robert, December 1, 1783. Viewed from the Place de la Concorde to the Tuileries Palace (destroyed in 1871)
Apps
Charles’s law app
Learning standards
Massachusetts Science and Technology/Engineering Curriculum Framework
8.MS-PS1-4. Develop a model that describes and predicts changes in particle motion, relative spatial arrangement, temperature, and state of a pure substance when thermal energy is added or removed.
Next Generation Science Standards
MS-PS1-4. Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.
College Board Standards
Objective C.1.5 States of Matter
C-PE.1.5.2 Explain why gases expand to fill a container of any size, while liquids flow and spread out to fill the bottom of a container and solids hold their own shape. Justification includes a discussion of particle motion and the attractions between the particles.
C-PE.1.5.3 Investigate the behavior of gases. Investigation is performed in terms of volume (V ), pressure (P ), temperature (T ) and amount of gas (n) by using the ideal gas law both conceptually and mathematically.
Common Core Math
Analyze proportional relationships and use them to solve real-world and mathematical problems.
CCSS.MATH.CONTENT.7.RP.A.2
Recognize and represent proportional relationships between quantities.
CCSS.MATH.CONTENT.7.RP.A.2.A
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
CCSS.MATH.CONTENT.7.RP.A.2.B
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Pascal’s Principle
Pressure applied to an enclosed, incompressible, static fluid is transmitted undiminished to all parts of the fluid.
Hydraulic systems operate according to Pascal’s law.
- Define pressure.
- State Pascal’s principle.
- Understand applications of Pascal’s principle.
- Derive relationships between forces in a hydraulic system.
image from littlewhitecoats.blogspot.com
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Learning Standards
tba
Bernoulli’s equation
Bernoulli’s equation
This is the law of conservation of energy as applied to flowing fluids. We can explain to department heads and parents that basics ideas – such as conservation of energy – appear everywhere in life, everywhere in science, so it is important for us to see examples of how they play out, such as in Bernoulli’s equation.
Online textbook
The Most General Applications of Bernoulli’s Equation
Viscosity and Laminar Flow; Poiseuille’s Law
Motion of an Object in a Viscous Fluid
Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
Applications of the Bernoulli effect
How Do Airplanes Fly? Using Newton’s laws
image from http://www.thaitechnics.com
Resources
NASA How the Bernoulli equation works in rockets
Khan Academy What-is-Bernoulli’s-equation
Bernoulli’s Equation OpenStax College
Energyeducation.ca Bernoulli’s equation
Apps
lmnoeng.com Bernoulli equation calculator
Endmemo.com Bernoulli equation calculator
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS3-1. Use algebraic expressions and the principle of energy conservation to calculate the change in energy of one component of a system when the change in energy of the other component(s) of the system, as well as the total energy of the system including any energy entering or leaving the system, is known.
Disciplinary Core Idea Progression Matrix
PS3.A and 3.B: The total energy within a physical system is conserved. Energy transfer within and between systems can be described and predicted in terms of energy associated with the motion or configuration of particles (objects)
NGSS leaves out critical guidance on importance of teaching about vectors
As we all know the NGSS are more about skills than content. Confusingly, though, they ended up also listing core content topics as well – yet they left out kinematics and vectors, the basic tools needed for physics in the first place.
The NGSS also dropped the ball by often ignoring the relationship of math to physics. They should have noted which math skills are needed to master each particular area.
Hypothetically, they could have had offered options: For each subject, note the math skills that would be needed to do problem solving in this area, for
* a standard (“college prep”) level high school class
* a lower level high school class, perhaps along the lines of what we call “Conceptual Physics” (still has math, but less.)
* the highest level of high school class, the AP Physics level. And the AP study guides already offer what kinds of math one needs to do problem solving in each area.
Yes, the NGSS does have a wonderful introduction to this idea, (quoted below) – but when we look at the actual NGSS standards they don’t mention these skills.
In some school districts this has caused confusion, and even led to some administrators demanding that physics be taught without these essential techniques (i.e. kinematic equations, conceptual understanding of 2D motion, kinematic analysis of 2D motion, vectors, etc.)
To help back up teachers in the field I put together these standards for vectors, from both science and mathematics standards.
– Robert Kaiser
Learning Standards
Massachusetts Science Curriculum Framework (pre 2016 standards)
1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).
NGSS
Science and Engineering Practices: Using Mathematics and Computational Thinking
Mathematical and computational thinking in 9–12 builds on K–8 experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions.
- Apply techniques of algebra and functions to represent and solve scientific and engineering problems.
Although there are differences in how mathematics and computational thinking are applied in science and in engineering, mathematics often brings these two fields together by enabling engineers to apply the mathematical form of scientific theories and by enabling scientists to use powerful information technologies designed by engineers. Both kinds of professionals can thereby accomplish investigations and analyses and build complex models, which might otherwise be out of the question. (NRC Framework, 2012, p. 65)
Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions. Other applications of mathematics in science and engineering include logic, geometry, and at the highest levels, calculus…. Mathematics is a tool that is key to understanding science. As such, classroom instruction must include critical skills of mathematics. The NGSS displays many of those skills through the performance expectations, but classroom instruction should enhance all of science through the use of quality mathematical and computational thinking.
Common Core Standards for Mathematics (CCSM)
High School: Number and Quantity » Vector & Matrix Quantities. Represent and model with vector quantities.
Represent and model with vector quantities.
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
Perform operations on vectors.
(+) Add and subtract vectors.
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
Understand vector subtraction v – w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
(+) Multiply a vector by a scalar.
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
- Become fluent in generating equivalent expressions for simple algebraic expressions and in solving linear equations and inequalities.
- Develop fluency operating on polynomials, vectors, and matrices using by-hand operations for the simple cases and using technology for more complex cases.
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