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Ferris wheel physics
A Ferris wheel is a large structure consisting of a rotating upright wheel, with multiple passenger cars.
The cars are attached to the rim in such a way that as the wheel turns, they are kept upright by gravity.
The original Ferris Wheel was designed and constructed by George Washington Gale Ferris Jr. as a landmark for the 1893 World’s Columbian Exposition in Chicago.
The generic term Ferris wheel is now used for all such structures, which have become the most common type of amusement ride at state fairs in the United States.
Forces in the wheel
The wheel keeps its circular shape by the tension of the spokes, pulling upward against the lower half of the framework and downward against the huge axle.
Also see
Classical relativity
This animation shows simultaneous views of a ball tossed up and then caught by a ferris wheel rider –
It shows this from one inertial POV and from two non-inertial POVs.
P. Fraundorf writes
Although Newton’s predictions are easier to track from the inertial point of view, it turns out that they still work locally in accelerated frames and curved spacetime if we consider “geometric accelerations and forces” that act on every ounce of an object’s being and can be made to disappear by a suitable vantage point change.
Created by P. Fraundorf, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Net work done on you while on the wheel
if you are on a ferris wheel that is rotating, the total work done by all the forces acting on your is zero.
https://www.physicsforums.com/threads/ferris-wheel-work-done-by-net-force.715905/
External resources
https://www.real-world-physics-problems.com/ferris-wheel-physics.html
https://physics.stackexchange.com/questions/205918/centripetal-force-on-a-ferris-wheel
How products are made: http://www.madehow.com/Volume-6/Ferris-Wheel.html
AP Physics problems: Ferris wheels and rotational motion
Build A Big Wheel, by Try Engineering, Lesson plan
AP Physics problem solving
http://faculty.washington.edu/boynton/114AWinter08/LectureNotes/Le8.pdf
How records work
How record work (private for now)
https://kaiserscience.wordpress.com/physics/waves/how-records-work/
Miniaturization
Many sci-fi stories depend upon a technology called miniaturization. Isaac Asimov’s classic Fantastic Voyage; his more scientifically rigorous sequel, Fantastic Voyage II; DC Comics featuring The Atom, and Marvel Comics featuring Antman and The Wasp.
Is miniaturization real? Could it be real? What would be the results if it was real?
Scene from the 1966 movie Fantastic Voyage. The medical ship, inside in a blood vessel, is under attack from antibodies!
Miniaturization in movies and TV
1940’s movie – Dr. Cyclops. People are reduced to less than a foot in size by the titular mad scientist, and are subjugated to his whims.
1957 movie – The Incredible Shrinking Man inspired a boom in science fiction films that made use of size-alteration.
1961 The Atom, a Silver Age comic book character, the Atom, Dr. Ray Palmer, created by DC Comics.
1960’s Ant-Man, Marvel Comics superhero.
1966 Fantastic Voyage
1976 Dr. Shrinker, from the ABC network’s The Krofft Supershow
1987 Innerspace starring stars Dennis Quaid, Martin Short and Meg Ryan.
1989 Honey, I Shrunk the Kids, 1997 Honey, We Shrunk Ourselves
2015 Ant-Man, and 2018 Ant-Man and the Wasp
2016 – DC’s Legends of Tomorrow (featuring The Atom)
What would happen if we compressed someone?
Neil deGrasse Tyson shows us the real physics.
Although he probably shouldn’t write any more comics 😉
By Clay Yount, Hamlet’s Danish, 12/9/2014
Physics: How would one try to do this?
There are no practical ways to actually do this. However, science fiction stories speculate on how this could be done.
Interestingly, sustained thought and speculation on science fiction technologies has allowed scientists to develop real-world technologies.
A. Compression / increasing density
“Why are you so certain miniaturization is impossible?”
“If you reduce a man to the dimensions of a fly, then all the mass of a man would be crowded into the volume of a fly. You’d end up with a density of something like -” he paused to think – “a hundred and fifty thousand times that of platinum. “
From Fantastic Voyage II
B. Removing atoms
“But what if the mass were reduced in proportion?” – “Then you end up with one atom in the miniaturized man for every three million in the original. The miniaturized man would not only have the size of a fly but the brainpower of a fly as well. “
From Fantastic Voyage II:
C. Changing Planck’s constant
This is a major science-plot point in Fantastic Voyage II (1988)
“And if the atoms are reduced, too?”
“If it is miniaturized atoms you are speaking of, then Planck’s constant, which is an absolutely fundamental quantity in our Universe, forbids it. Miniaturized atoms would be too small to fit into the graininess of the Universe. “
“And if I told you that Planck’s constant was reduced as well, so that a miniaturized man would be encased in a field in which the graininess of the Universe was incredibly finer than it is under normal conditions?”
“Then I wouldn’t believe you. “
“Without examining the matter? You would refuse to believe it as a result of preconceived convictions, as your colleagues refuse to believe you?”
And at this, Morrison was, for a moment, silent….
…Well over half an hour had passed before Morrison felt convinced that the objects he could see outside the ship were shrinking and were receding perceptibly toward their normal size.
Morrison said, “I am thinking of a paradox.”
“What’s that?” said Kalinin, yawning. She had obviously taken her own advice about the advisability of relaxing.
“The objects outside the ship seemed to grow larger as we shrink. Ought not the wavelengths of light outside the ship also grow larger, becoming longer in wavelength, as we shrink? Should we not see everything outside turn reddish, since there can scarcely be enough ultraviolet outside to expand and replace the shorter-wave visible light?”
Kalinin said, “If you could see the light waves outside, that would indeed be how they would appear to you. But you don’t. You see the light waves only after they’ve entered the ship and impinged upon your retina. And as they enter the ship, they come under the influence of the miniaturization field and automatically shrink in wavelength, so that you see those wavelengths inside the ship exactly as you would see them outside.”
“If they shrink in wavelength, they must gain energy.”
“Yes, if Planck’s constant were the same size inside the miniaturization field as it is outside. But Planck’s constant decreases inside the miniaturization field — that is the essence of miniaturization. The wavelengths, in shrinking, maintain their relationship to the shrunken Planck’s constant and do not gain energy. An analogous case is that of the atoms. They also shrink and yet the interrelationships among atoms and among the subatomic particles that make them up remain the same to us inside the ship as they would seem to us outside the ship.”
“But gravity changes. It becomes weaker in here.”
“The strong interaction and the electroweak interaction come under the umbrella of the quantum theory. They depend on Planck’s constant. As for gravitation?” Kalinin shrugged. “Despite two centuries of effort, gravitation has never been quantized. Frankly, I think the gravitational change with miniaturization is evidence enough that gravitation cannot be quanitzed, that it is fundamentally nonquantum in nature.”
“I can’t believe that,” said Morrison. “Two centuries of failure can merely mean we haven’t managed to get deep enough into the problem yet. Superstring theory nearly gave us out unified field at last.” (It relieved him to discuss the matter. Surely he couldn’t do so if his brain were heating in the least.)
“Nearly doesn’t count,” said Kalinin. “Still, Shapirov aagreed with you, I think. It was his notion that once we tied Planck’s constant to the speed of light, we would not only have the practical effect of miniaturizing and deminiaturizing in an essentially energy-free manner, but that we would have the theoretical effect of being able to work out the connection between quantum theory and relativity and finally have a good unified field theory. And probably a simpler one than we could have imagined possible, he would say.”
“Maybe,” said Morrison. He didn’t know enough to comment beyond that.
Surely this is complete fantasy, correct? Well, probably. But there is some room in physics to believe that the constants of nature, even Planck’s constant, may quite be constant:
Could Fundamental Constants Be Neither Fundamental nor Constant?
Are Nature’s Laws Really Universal? Dr. Michael Murphy, Centre for Astrophysics and Supercomputing,
Swinburne University of Technology
The Variability of the ‘Fundamental Constants’
The Constants of Nature: From Alpha to Omega – The Numbers That Encode the Deepest Secrets of the Universe (book,) John D. Barrow
D. Nanotechnology as miniaturization
“…The ideas and concepts behind nanoscience and nanotechnology started with a talk entitled “There’s Plenty of Room at the Bottom” by physicist Richard Feynman at an American Physical Society meeting at the California Institute of Technology (CalTech) on December 29, 1959, long before the term nanotechnology was used.
In his talk, Feynman described a process in which scientists would be able to manipulate and control individual atoms and molecules. Over a decade later, in his explorations of ultraprecision machining, Professor Norio Taniguchi coined the term nanotechnology. It wasn’t until 1981, with the development of the scanning tunneling microscope that could “see” individual atoms, that modern nanotechnology began.”
Nano.gov What is nanotechnology?
Nanotechnology isn’t so impossible. We already have developed techniques to image and pick atoms. one atom at a time, through technologies such as atomic force microscopy and Scanning tunneling microscopy.
References
Miniaturization: Technovelgy article
Excerpt from Fantastic Voyage: II A Novel By Isaac Asimov, 1988
Learning Standards
Next Generation Science Standards: Science & Engineering Practices
● Ask questions that arise from careful observation of phenomena, or unexpected results, to clarify and/or seek additional information.
● Ask questions that arise from examining models or a theory, to clarify and/or seek additional information and relationships.
● Ask questions to determine relationships, including quantitative relationships, between independent and dependent variables.
● Ask questions to clarify and refine a model, an explanation, or an engineering problem.
● Evaluate a question to determine if it is testable and relevant.
● Ask questions that can be investigated within the scope of the school laboratory, research facilities, or field (e.g., outdoor environment) with available resources and, when appropriate, frame a hypothesis based on a model or theory.
● Ask and/or evaluate questions that challenge the premise(s) of an argument, the interpretation of a data set, or the suitability of the design
MA 2016 Science and technology
Appendix I Science and Engineering Practices Progression Matrix
Science and engineering practices include the skills necessary to engage in scientific inquiry and engineering design. It is necessary to teach these so students develop an understanding and facility with the practices in appropriate contexts. The Framework for K-12 Science Education (NRC, 2012) identifies eight essential science and engineering practices:
1. Asking questions (for science) and defining problems (for engineering).
2. Developing and using models.
3. Planning and carrying out investigations.
4. Analyzing and interpreting data.
5. Using mathematics and computational thinking.
6. Constructing explanations (for science) and designing solutions (for engineering).
7. Engaging in argument from evidence.
8. Obtaining, evaluating, and communicating information.
Scientific inquiry and engineering design are dynamic and complex processes. Each requires engaging in a range of science and engineering practices to analyze and understand the natural and designed world. They are not defined by a linear, step-by-step approach. While students may learn and engage in distinct practices through their education, they should have periodic opportunities at each grade level to experience the holistic and dynamic processes represented below and described in the subsequent two pages… http://www.doe.mass.edu/frameworks/scitech/2016-04.pdf
Facts and Fiction of the Schumann Resonance
This has been excerpted from Facts and Fiction of the Schumann Resonance,by Brian Dunning, Skeptoid Podcast #352
It’s increasingly hard to find a web page dedicated to the sales of alternative medicine products or New Age spirituality that does not cite the Schumann resonances as proof that some product or service is rooted in science. … Today we’re going to see what the Schumann resonances actually are, how they formed and what they do, and see if we can determine whether they are, in fact, related to human health.
In physics, Schumann resonances are the name given to the resonant frequency of the Earth’s atmosphere, between the surface and the densest part of the ionosphere.
Image from nasa.gov/mission_pages/sunearth/news/gallery
They’re named for the German physicist Winfried Otto Schumann (1888-1974) who worked briefly in the United States after WWII, and predicted that the Earth’s atmosphere would resonate certain electromagnetic frequencies.
[What is a resonant frequency? Here is a common example. When you blow on a glass bottle at a certain frequency, you can get the bottle to vibrate at the same frequency]
from acs.psu.edu/drussell/Demos/BeerBottle/beerbottle.html
This glass bottle has a resonant frequency of about 196 Hz.
That’s the frequency of sound waves that most efficiently bounce back and forth between the sides of the bottle, at the speed of sound, propagating via the air molecules.
Electromagnetic radiation – like light, and radio waves – is similar, except the waves travel at the speed of light, and do not require a medium like air molecules.
The speed of light is a lot faster than the speed of sound, but the electromagnetic waves have a lot further to go between the ground and the ionosphere than do the sound waves between the sides of the bottle.
This atmospheric electromagnetic resonant frequency is 7.83 Hz, which is near the bottom of the ELF frequency range, or Extremely Low Frequency.
The atmosphere has its own radio equivalent of someone blowing across the top of the bottle: lightning.
Lightning is constantly flashing all around the world, many times per second; and each bolt is a radio source. This means our atmosphere is continuously resonating with a radio frequency of 7.83 Hz, along with progressively weaker harmonics at 14.3, 20.8, 27.3 and 33.8 Hz.
These are the Schumann resonances.
It’s nothing to do with the Earth itself, or with life, or with any spiritual phenomenon;
it’s merely an artifact of the physical dimensions of the space between the surface of the Earth and the ionosphere.
Every planet and moon that has an ionosphere has its own set of Schumann resonances defined by the planet’s size.
Biggest point: this resonated radio from lightning is a vanishingly small component of the electromagnetic spectrum to which we’re all naturally exposed.
The overwhelming source is the sun, blasting the Earth with infrared, visible light, and ultraviolet radiation.
All natural sources from outer space, and even radioactive decay of naturally occurring elements on Earth, produce wide-spectrum radio noise. Those resonating in the Schumann cavity are only a tiny, tiny part of the spectrum.
Nevertheless, because the Schumann resonance frequencies are defined by the dimensions of the Earth, many New Age proponents and alternative medicine advocates have come to regard 7.83 Hz as some sort of Mother Earth frequency, asserting the belief that it’s related to life on Earth.
The most pervasive of all the popular fictions surrounding the Schumann resonance is that it is correlated with the health of the human body.
There are a huge number of products and services sold to enhance health or mood, citing the Schumann resonance as the foundational science.
A notable example is the Power Balance bracelets. Tom O’Dowd, formerly the Australian distributor, said that the mylar hologram resonated at 7.83 Hz.
When the bracelet was placed within the body’s natural energy field, the resonance would [supposedly] “reset” your energy field to that frequency.
Well, there were a lot of problems with that claim.
First of all, 7.83 Hz has a wavelength of about 38,000 kilometers. This is about the circumference of the Earth, which is why its atmospheric cavity resonates at that frequency. 38,000 kilometers is WAY bigger than a bracelet!
There’s no way that something that tiny could resonate such an enormous wavelength. O’Dowd’s sales pitch was implausible, by a factor of billions, to anyone who understood resonance.
This same fact also applies to the human body. Human beings are so small, relative to a radio wavelength of 38,000 kilometers, that there’s no way our anatomy could detect or interact with such a radio signal in any way.
Proponents of binaural beats cite the Schumann frequency as well. These are audio recordings which combine two slightly offset frequencies to produce a third phantom beat frequency that is perceived from the interference of the two.
Some claim to change your brain’s encephalogram, which they say is a beneficial thing to do. Brain waves range from near zero up to about 100 Hz during normal activity, with a typical reading near the lower end of the scale.
This happens to overlap 7.83 — suggesting the aforementioned pseudoscientific connection between humans and the Schumann resonance — but with a critical difference. An audio recording is audio, not radio. It’s the physical oscillation of air molecules, not the propagation of electromagnetic waves. The two have virtually nothing to do with each other.
[Other salespeople claim] that our bodies’ energy fields need to interact with the Schumann resonance, but can’t because of all the interference from modern society [and so they try to sell devices that supposedly connect our body to the Schumann resonance.]
It’s all complete and utter nonsense. Human bodies do not have an energy field: in fact there’s not even any such thing as an energy field. Fields are constructs in which some direction or intensity is measured at every point: gravity, wind, magnetism, some expression of energy.
Energy is just a measurement; it doesn’t exist on its own as a cloud or a field or some other entity. The notion that frequencies can interact with the body’s energy field is, as the saying goes, so wrong it’s not even wrong.
Another really common New Age misconception about the Schumann resonance is that it is the resonant frequency of the Earth. But there’s no reason to expect the Earth’s electromagnetic resonant frequency to bear any similarity to the Schumann resonance.
Furthermore, the Earth probably doesn’t even have a resonant electromagnetic frequency. Each of the Earth’s many layers is a very poor conductor of radio; combined all together, the Earth easily absorbs just about every frequency it’s exposed to. If you’ve ever noticed that your car radio cuts out when you drive through a tunnel, you’ve seen an example of this.
Now the Earth does, of course, conduct low-frequency waves of other types. Earthquakes are the prime example of this. The Earth’s various layers propagate seismic waves differently, but all quite well. Seismic waves are shockwaves, a physical oscillation of the medium. Like audio waves, these are unrelated to electromagnetic radio waves.
Each and every major structure within the Earth — such as a mass of rock within a continent, a particular layer of magma, etc. — does have its own resonant frequency for seismic shockwaves, but there is (definitively) no resonant electromagnetic frequency for the Earth as a whole.
So our major point today is that you should be very skeptical of any product that uses the Schumann resonance as part of a sales pitch.
The Earth does not have any particular frequency. Life on Earth is neither dependent upon, nor enhanced by, any specific frequency.
Source: skeptoid.com/episodes/4352
Resonance
Resonance: The increase in the amplitude of an oscillation of a system under the influence of a periodic force whose frequency is close to that of the system’s natural frequency.
There are several types of resonance
Mechanical resonance
tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system’s natural frequency of vibration (its resonance frequency or resonant frequency) than it does at other frequencies.
This may cause violent swaying motions and potentially catastrophic failure in improperly constructed structures including bridges, buildings and airplanes.
Examples include:
Musical instruments (acoustic resonance)
Musical instruments are set into vibrational motion at their natural frequency when a person hits, strikes, strums, plucks or somehow disturbs the object.
Each natural frequency of the object is associated with one of the many standing wave patterns by which that object could vibrate. The natural frequencies of a musical instrument are sometimes referred to as the harmonics of the instrument.
Physics Classroom – Sounds – Lesson 5 – Resonance
Clocks
Most clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal.
Tidal resonance
Seen at the Bay of Fundy in Canada.
Objects can shatter at resonant frequencies
A wineglass breaking when someone sings a loud note at exactly the right pitch.
Resonance in weather systems
Rossby waves, also known as planetary waves, are a type of wave naturally occurring in rotating fluids (gas or liquid.) Here on Earth are they are giant meanders in high-altitude winds; they have a major influence on weather.
It has been proposed that a number of regional weather extremes in the Northern Hemisphere associated with blocked atmospheric circulation patterns may have been caused by quasiresonant amplification of Rossby waves.
Examples include the 2013 European floods, the 2012 China floods, the 2010 Russian heat wave, the 2010 Pakistan floods and the 2003 European heat wave.
Rossyby Wave, Wikipedia
Orbital resonance
The motion of one object orbiting a star, or planet, can influence the motion of another object. Eventually, over time the motions of the objects can become in resonance with each other.
Orbital resonance can occur in many ways:
Here we see an asteroid sometimes called a quasi-satellite. It has its own orbit around the Sun, but over time this has developed a 1:1 resonance with Earth’s orbit.
The most well known quasi-satellite of Earth is the asteroid Cruithne, discovered in 1986. It is 5 KM in diameter. If you were “above” our Sun, looking down into the solar system then you would see it’s orbit and Earth’s orbit like this:
IMAGE FROM WIKIPEDIA
But from the point of view of people here on Earth, it appears to be trailing us, making a horseshoe-shaped orbit.
That’s not a moon as such, and it doesn’t even orbit us. But due to the oddities of orbital mechanics it appears to be behind us in space, orbiting empty space!
IMAGE FROM WIKIPEDIA
Resonance in the rings of Saturn
TBA
Ring dynamics, Stars and Planets, ASTR 221
The Forces that Sculpt Saturn’s Rings….
Resonance Moon and Rings, NASA Science
Staggering Structure, NASA Science
Planetary Rings, Lumen Learning
Resonance in electrical circuits
Circuits involving capacitors and inductors can demonstrate resonance.
A collapsing magnetic field from the inductor generates an electric current in its windings;
this current charges the capacitor,
hen the discharging capacitor provides an electric current that builds the magnetic field in the inductor.
This process is repeated continually. An analogy is a mechanical pendulum, and both are a form of simple harmonic oscillator.
Symbols: resistor – R, inductor – L, capacitor – C
Resonance in RLC circuits
An RLC circuit consists of a resistor, an inductor, and a capacitor.
The circuit forms a harmonic oscillator for current; it resonates similarly to an LC circuit.
The main difference (due to the presence of the resistor) is that any oscillation induced in the circuit decays over time if it is not kept going by a source.
This effect of the resistor is called damping.
The presence of the resistance reduces the peak resonant frequency of damped oscillation, although the resonant frequency for driven oscillations remains the same as an LC circuit.
Some resistance is always unavoidable in real circuits, even if a resistor is not specifically included as a separate component.
A pure LC circuit is an ideal that exists only in theory.
An important application for this type of circuit is tuning, such as in radio receivers or television sets. They are used to select a narrow range of frequencies from the ambient radio waves.
Intuitor.com The Physics of Resonance
Video LC Inductor-Capacitor Resonating Circuits by by Eugene Khutoryansky
Mechanical structure susceptible to damage from resonance
RedGrittyBrick, a physicist writing on skeptics.stackexchange.com, notes that a bridge can be susceptible to mechanical resonance:
Mechanical structures usually have one or more frequencies at which some part of the structure oscillates. A tuning fork has a well-defined natural frequency of oscillation. More complex structures may have a dominant natural frequency of oscillation.
If some mechanical inputs (such as the pressure of feet walking in unison) have a frequency that is close to a natural frequency of the structure, these inputs will tend to initiate and, over a short time, increase the oscillating movements of the structure. Like pushing a child’s swing at the right time.
One example is London’s Millennium Bridge which was closed shortly after opening because low-frequency vibrations in the bridge were causing large groups of pedestrians to simultaneously shift their weight and reinforcing the oscillation. Dampers were fitted.
Skeptics.stackexchange Does a column of marching soldiers have to break their rhythm while crossing a bridge to prevent its collapse?
Related topics (local)
Nikola Tesla and wireless power transmission
Facts and Fiction of the Schumann Resonance: On this website
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS4-5. Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. Examples of principles of wave behavior include resonance, photoelectric effect, and constructive and destructive interference.
Uses of imaginary numbers
What are imaginary numbers?
Elsewhere in math class you have learned about the definition and use imaginary numbers.
This resource is specifically about the usefulness and meaning of imaginary numbers.
It assumes that you already know what imaginary numbers are and how to use them.
But sure, since you, here’s a good refresher – Ask Dr. Math: What is an imaginary number? What is i?
And here’s another explanation: Better Explained: A Visual, Intuitive Guide to Imaginary Numbers
Are they “real” in some sense?
In what sense are imaginary numbers just as real as “real” numbers? People used to say the same thing about fractions! People argued that either something is a number or it isn’t – how can one possibly have part of a number?
Later, people said the same thing about irrational numbers.
And for quite a long time, people said the same thing about the number 0 – people argued that there couldn’t possible be a number without value.
Yet today everyone agrees that fractions, irrational numbers, and zero are all “real.”
How it possible that people didn’t “believe in” those numbers before, but they do now? Because we introduce people to these numbers and show how they all work together in a well-defined, useful system (“mathematics”.)
So the same could be true for imaginary numbers – what if we showed people how imaginary numbers filled in a gap in our math system?
Consider this function 𝑓(𝑥) = 𝑥2 + 1
Here is this function’s plot in the real x-y plane:
Now according to the Fundamental Theorem of Algebra we should have n-roots for n-th degree polynomial. Yet when we consider the graph for this function it doesn’t appear to intersect the x-axis right.
Well, the thing is, we are not seeing it correctly and have not included a fundamental set of numbers : Complex Numbers which have both real and imaginary part but don’t get confused yet as both the parts are quite real.
The below GIF plots the the function in the complex plane The vertical axis that comes out of the paper is the imaginary axis, NOT the Z-axis.
The explanation in the paragraphs above comes from math.stackexchange
from “Imaginary Numbers are Real,” Welch labs
How can one show that imaginary numbers exist? In the same way that people showed that fractions exist. Exactly the same argument shows that imaginary numbers exist:
How can one show that imaginary numbers really exist?
Here’s a great video showing how imaginary numbers can be thought of as just as real as other numbers:
Imaginary numbers are not some wild invention, they are a natural part of our number system.
How are imaginary numbers used?
I. Alternating current circuits
“The handling of the impedance of an AC circuit with multiple components quickly becomes unmanageable if sines and cosines are used to represent the voltages and currents.”
“A mathematical construct which eases the difficulty is the use of complex exponential functions. “
.
II. Engineering – damped oscillators
Many objects have simple harmonic motion, aka oscillation. Objects move back and forth, and the “pull back” force is related to how far the object is pulled from the center.
This motion doesn’t last forever. Due to friction, the motion slowly dampens, or dies away, over time. This is called damped oscillation.
There are mechanical vibrations in any structures, such as bridges, overpasses, tunnel walls, and floors of shopping malls and buildings.
Here’s a practical example of a problem that requires imaginary numbers in math to produce an engineering solution:
“An existing mid-rise office building included a gymnasium on the second floor. Floors above the gym level were occupied as offices by different tenants. Vibration complaints were reported by the tenants on the fourth floor at two different locations.
In essence, vibrations generated at second floor were traveling up through the columns and producing unacceptable vibrations at the fourth floor. The task was to verify the reported vibration complaints analytically, and then propose vibration mitigation measures.”
Vertical vibration transmission from a gym, Floor Vibration Expert, Boston, MA
Here is an (exaggerated) analysis of how oscillation in bridge structures.
The same is true for studying a plucked violin or guitar string,
And of course the same kind of analysis is used for studying damped oscillations in car shock absorbers, pendulums, bungee jumping, etc.
The engineering of any of these involves equations that use imaginary numbers.
See Real World Example: Oscillating Springs (Math Warehouse)
III. Useful in some parts of Economics
Image from St. Lawrence University, Mathematics-Economics Combined Major
“Complex numbers and complex analysis do show up in Economic research. For example, many models imply some difference-equation in state variables such as capital, and solving these for stationary states can require complex analysis.”
and
“The application of complex numbers had been attempted in the past by various economists, especially for explaining economic dynamics and business fluctuations in economic system
In fact, the cue was taken from electrical systems. Oscillations in economic activity level gets represented by sinusoidal curves The concept of Keynesian multiplier and the concept of accelerator were combined in models to trace the path of economic variables like income, employment etc over time. This is where complex numbers come in.”
{This explanation by sensekonomikx, Yahoo Answers, Complex numbers in Economics}
Why use imaginary math for real numbers?
Electrical engineers and economists study real world objects and get real world answers, yet they use complex functions with imaginary numbers. Couldn’t we just use “regular” math?
Image from Imaginary Numbers Are Real, Welch Labs
Answer:
Imaginary numbers transform complex equations in the real X-Y axis into simpler functions in the “imaginary” plane.
This lets us transform complicated problems into simpler ones.
Here is an explanation from “Ask Dr. Math” (National Council of Teachers of Mathematics.)
Also
We sometimes just use imaginary numbers because they can be easier to use: Engineers and physicists use the complex exponential 𝑒𝑗𝜔𝑡 instead of sines and cosines.
Why? This notation makes differential equations much easier to deal with.
That’s why we use imaginary numbers when studying electrical impedance.
Why is impedance represented as a complex number rather than a vector?
Other examples of real world uses
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html
Careers That Use Complex Numbers, by Stephanie Dube Dwilson
Imaginary numbers in real life: Ask Dr. Math
Imaginary numbers, Myron Berg, Dickinson State Univ.
The universe physically seems to run on complex numbers
If we look only at things in our everyday life – objects with masses larger than atoms, and moving at speeds far lower than the speed of light – then we can pretend that the entire word is made of solid objects (particles) following more or less “common sense” rules – the classical laws of physics.
But there’s so much more to our universe – and when we look carefully, we find that literally all of our classical laws of physics are only approximations of a more general, and often bizarre law – the laws of quantum mechanics. And QM laws follow a math that uses complex numbers!
When you have time, look at our intro to the development of QM and at deeper, high school level look at what QM really is .
Scott Aaronson writes about a central, hard to believe feature of quantum mechanics:
“Nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.”
He points out that this weird reality seems to be a basic feature of the universe itself
“This transformation is just a mirror reversal of the plane. That is, it takes a two-dimensional Flatland creature and flips it over like a pancake, sending its heart to the other side of its two-dimensional body.
But how do you apply half of a mirror reversal without leaving the plane? You can’t! If you want to flip a pancake by a continuous motion, then you need to go into … dum dum dum … THE THIRD DIMENSION.
More generally, if you want to flip over an N-dimensional object by a continuous motion, then you need to go into the (N+1)st dimension.
But what if you want every linear transformation to have a square root in the same number of dimensions? Well, in that case, you have to allow complex numbers. So that’s one reason God might have made the choice She did.”
– PHYS771 Quantum Computing Since Democritus, Lecture 9: Quantum. Aaronson is Professor of Computer Science at The University of Texas at Austin.
Imaginary Numbers May Be Essential for Describing Reality
A new thought experiment indicates that quantum mechanics doesn’t work without strange numbers that turn negative when squared.
Charlie Wood, Quanta Magazine , 3/3/2021
A group of quantum theorists designed an experiment whose outcome depends on whether nature has an imaginary side. Provided that quantum mechanics is correct — an assumption few would quibble with — the team’s argument essentially guarantees that complex numbers are an unavoidable part of our description of the physical universe.
“These complex numbers, usually they’re just a convenient tool, but here it turns out that they really have some physical meaning,” said Tamás Vértesi, a physicist at the Institute for Nuclear Research at the Hungarian Academy of Sciences who, years ago, argued the opposite. “The world is such that it really requires these complex” numbers, he said.
Read Imaginary numbers could be needed to describe reality, new studies find, Ben Turner, Live Science, 12/21/2021
Quantum theory based on real numbers can be experimentally falsified, Marc-Olivier Renou et al. Nature volume 600, pages625–629 (2021)
Testing real quantum theory in an optical quantum network, Phys. Rev. Lett. Zheng-Da Li, et al.
Are negative probabilities real?
In 1942, Paul Dirac wrote a paper “The Physical Interpretation of Quantum Mechanics” where he introduced the concept of negative energies and negative probabilities:
“Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money.”
The idea of negative probabilities later received increased attention in physics and particularly in quantum mechanics. Richard Feynman argue that no one objects to using negative numbers in calculations: although “minus three apples” is not a valid concept in real life, negative money is valid.
Similarly he argued how negative probabilities as well as probabilities above unity possibly could be useful in probability calculations.
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Wikipedia, Negative Probabilities, 3/18
John Baez ( mathematical physicist at U. C. Riverside in California) writes
The physicists Dirac and Feynman, both bold when it came to new mathematical ideas, both said we should think about negative probabilities. What would it mean to say something had a negative chance of happening?
I haven’t seen many attempts to make sense of this idea… or even work with this idea. Sometimes in math it’s good to temporarily put aside making sense of ideas and just see if you can develop rules to consistently work with them. For example: the square root of -1. People had to get good at using it before they understood what it really was: a rotation by a quarter turn in the plane. Here’s an interesting attempt to work with negative probabilities:
• Gábor J. Székely, Half of a coin: negative probabilities, Wilmott Magazine (July 2005), p.66–68
He uses rigorous mathematics to study something that sounds absurd: half a coin. Suppose you make a bet with an ordinary fair coin, where you get 1 dollar if it comes up heads and 0 dollars if it comes up tails. Next, suppose you want this bet to be the same as making two bets involving two separate ‘half coins’. Then you can do it if a half coin has infinitely many sides numbered 0,1,2,3, etc., and you win n dollars when side number n comes up….
… and if the probability of side n coming up obeys a special formula…
and if this probability can be negative whenever n is even!
This seems very bizarre, but the math is solid, even if the problem of interpreting it may drive you insane.
By the way, it’s worth remembering that for a long time mathematicians believed that negative numbers made no sense. As late as 1758 the British mathematician Francis Maseres claimed that negative numbers “… darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.”
So opinions on these things can change. By the way: experts on probability theory will like Székely’s use of ‘probability generating functions’. Experts on generating functions and combinatorics will like how the probabilities for the different sides of the half-coin coming up involve the Catalan numbers.
Learning standards
Massachusetts Mathematics Curriculum Framework 2017
Number and Quantity Content Standards: The Complex Number System
A. Perform arithmetic operations with complex numbers.
B. Represent complex numbers and their operations on the complex plane.
C. Use complex numbers in polynomial identities and equations.
Common Core Mathematics
High School: Number and Quantity » The Complex Number System
CCSS.MATH.CONTENT.HSN.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
CCSS.MATH.CONTENT.HSN.CN.A.2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CCSS.MATH.CONTENT.HSN.CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Resources
Lectures on the history of physics
Galileo and Einstein: Lectures on the history of physics
Michael Fowler – University of Virginia Physics
Physics of Batman: The Dark Knight
Let’s assume that the memory fiber used in “The Dark Knight” is real.
In the movie it is used to change the shape of a cape into wings with the application of an electrical current.
No such material yet exists, but materials scientists are getting close.
If this kind of fabric existed, would it work? What kind of forces would this put on the human body?
(Remember: For every force there is an equal and opposite force – this is one of Newton’s laws.)
http://www.popsci.com/entertainment-%2526-gaming/article/2008-08/physics-batman
Adapted from “The Physics of Batman: The Dark Knight – High Dive”, Adam Weiner, 08.15.2008
Let’s start with the basic situation: Batman spreads the cape-wings & moves into a circular path.
Therefore his motion goes from vertical to mostly horizontal.
The force of air resistance increases dramatically when he expands these wings.
This force turns his linear path into a circular path.
This inward pointing force is a centripetal force.
Law of physics: No object travels in a circular path (Newton’s 1st law), unless some force continually pulls it radially inward.
The balance of inertia and a radially inward force can create circular motion.
Centripetal force depends on the radius of the curve (r) and the radial velocity (v)
F = mv2/r
When a glider – or a Batwing – is bent into the wind, one can use the force to deflect the glider, plane or Batman.
Red arrow to upper right = “lift” (due to the wind hitting the wings)
Red arrow down = weight
Horizontal green arrow is the horizontal component of lift (aka centripetal force)
Vertical green arrow is the vertical component of lift. (If it is big enough then one can glide for long periods of time)
What about Newton’s 3rd law of motion?
To hold his arms out, Batman has to exert the same force back on the air.
So while he moves in a circle, we can calculate the force that will be exerted on Batman’s arms.
circle radius = 20 meters
man + equipment mass = 80 kg
speed remains constant during this turn
Let’s estimate the force on Batman’s arms as he sweeps through the bottom of the arc.
F = weight + centripetal force
F = m g + m v2/r = m ( g + v2/r )
= 80 kg (9.8 m/s2 + [40 m/s]2 /20 m) = 7200 N
= about 1600 pounds
This means that Batman has to hold 800 pounds on each arm!
Imagine lying on your back, on a workout bench, holding your arms out and having 800 pounds of weights placed on each one! This is probably impossible for someone to do without super-strength.
Perhaps there is a way out of this. Maybe there are some hinges that connect the wings to the Bat suit. If so, then these hinges could be doing some of the supporting, rather than Batman’s arms.
Cartoon Laws of Physics
Cartoon Law I
Any body suspended in space will remain in space until made aware of its situation.
Daffy Duck steps off a cliff, expecting further pastureland. He loiters in midair, soliloquizing flippantly, until he chances to look down. At this point, the familiar principle of 32 feet per second per second takes over.
Cartoon Law II
Any body in motion will tend to remain in motion until solid matter intervenes suddenly.
Whether shot from a cannon or in hot pursuit on foot, cartoon characters are so absolute in their momentum that only a telephone pole or an outsize boulder retards their forward motion absolutely. Sir Isaac Newton called this sudden termination of motion the stooge’s surcease.
Cartoon Law III
Any body passing through solid matter will leave a perforation conforming to its perimeter.
Also called the silhouette of passage, this phenomenon is the speciality of victims of directed-pressure explosions and of reckless cowards who are so eager to escape that they exit directly through the wall of a house, leaving a cookie-cutout-perfect hole. The threat of skunks or matrimony often catalyzes this reaction.
Cartoon Law IV
The time required for an object to fall twenty stories is greater than or equal to the time it takes for whoever knocked it off the ledge to spiral down twenty flights to attempt to capture it unbroken.
Such an object is inevitably priceless, the attempt to capture it inevitably unsuccessful.
Cartoon Law V
All principles of gravity are negated by fear.
Psychic forces are sufficient in most bodies for a shock to propel them directly away from the earth’s surface. A spooky noise or an adversary’s signature sound will induce motion upward, usually to the cradle of a chandelier, a treetop, or the crest of a flagpole. The feet of a character who is running or the wheels of a speeding auto need never touch the ground, especially when in flight.
Cartoon Law VI
As speed increases, objects can be in several places at once.
This is particularly true of tooth-and-claw fights, in which a character’s head may be glimpsed emerging from the cloud of altercation at several places simultaneously. This effect is common as well among bodies that are spinning or being throttled. A ‘wacky’ character has the option of self- replication only at manic high speeds and may ricochet off walls to achieve the velocity required.
Cartoon Law VII
Certain bodies can pass through solid walls painted to resemble tunnel entrances; others cannot.
This trompe l’oeil inconsistency has baffled generations, but at least it is known that whoever paints an entrance on a wall’s surface to trick an opponent will be unable to pursue him into this theoretical space. The painter is flattened against the wall when he attempts to follow into the painting. This is ultimately a problem of art, not of science.
Cartoon Law VIII
Any violent rearrangement of feline matter is impermanent.
Cartoon cats possess even more deaths than the traditional nine lives might comfortably afford. They can be decimated, spliced, splayed, accordion-pleated, spindled, or disassembled, but they cannot be destroyed. After a few moments of blinking self pity, they reinflate, elongate, snap back, or solidify.
Corollary: A cat will assume the shape of its container.
Cartoon Law IX
Everything falls faster than an anvil.
Cartoon Law X
For every vengea nce there is an equal and opposite revengeance.
This is the one law of animated cartoon motion that also applies to the physical world at large. For that reason, we need the relief of watching it happen to a duck instead.
Cartoon Law Amendment A
A sharp object will always propel a character upward.
When poked (usually in the buttocks) with a sharp object (usually a pin), a character will defy gravity by shooting straight up, with great velocity.
Cartoon Law Amendment B
The laws of object permanence are nullified for “cool” characters.
Characters who are intended to be “cool” can make previously nonexistent objects appear from behind their backs at will. For instance, the Road Runner can materialize signs to express himself without speaking.
Cartoon Law Amendment C
Explosive weapons cannot cause fatal injuries.
They merely turn characters temporarily black and smoky.
Cartoon Law Amendment D
Gravity is transmitted by slow-moving waves of large wavelengths.
Their operation can be wittnessed by observing the behavior of a canine suspended over a large vertical drop. Its feet will begin to fall first, causing its legs to stretch. As the wave reaches its torso, that part will begin to fall, causing the neck to stretch. As the head begins to fall, tension is released and the canine will resume its regular proportions until such time as it strikes the ground.
Cartoon Law Amendment E
Dynamite is spontaneously generated in “C-spaces” (spaces in which cartoon laws hold).
The process is analogous to steady-state theories of the universe which postulated that the tensions involved in maintaining a space would cause the creation of hydrogen from nothing. Dynamite quanta are quite large (stick sized) and unstable (lit). Such quanta are attracted to psychic forces generated by feelings of distress in “cool” characters (see Amendment B, which may be a special case of this law), who are able to use said quanta to their advantage. One may imagine C-spaces where all matter and energy result from primal masses of dynamite exploding. A big bang indeed.
© 1997 William Geoffrey Shotts. Last update: Thursday, December 4, 1997
Mousetrap racer build project
Your task is to build a mousetrap powered car!
It can be built from wood, paper, plastic, metal, erector sets, pens, rulers, old toys, Legos, and other materials.
We need a fair comparison between race cars. Therefore it must be powered by only 1 mousetrap.
You may not modify the mousetrap, such as by over-winding the metal coil, because that would unfairly increase its potential energy storage.
A rat trap, or trap for any other animal, is not safe or acceptable.
2 people may collaborate to make 1 car.
If you do not have your car on the day that it is due, you lose 5 points per day.
I suggest working in groups, making your own local mousetrap racer “factory”. This approach is easier and more fun.
Clearly print your names somewhere on the car!
Giving time to do this
Day 1 – We introduce the project, discuss the physics and engineering principles, show some videos and photos.
Day 2 – (Which could be any day that fits our class schedule) – Have students bring in the building materials they have procured so far. Also, as a teacher I will help make materials available in class. Both teacher and some volunteer students will show in class how to assemble a mousetrap racer. The way that it is shown in class is not the only way to do it.
Day 3 – Classroom build. Students individually or in pairs work on the mousetrap racer. First start off with a brief review of physics principles – storing energy as PE, simple machines, how mechanical devices can transform PE into kinetic energy, etc.
Day 4 – Run the mousetrap racers! Find a long hallway with a smooth floor. We will have competitions:
(A) Fastest: Which car goes to the finish line in the shortest amount of time?
(B) Furthest distance: Which car goes the furthest?
Much information on mouse trap racers is available online. However, you may not use a kit to build your racer.
Instructables (several ideas here)
Mousetrap cars and kits from Doc Fizzix. Great for ideas
Gallery of great mousetrap racers. from UCI Summer Science Institute
What is a mousetrap powered car? How does it work?
It is a vehicle powered by a mousetrap spring. We tie one end of a string to the tip of a mousetrap’s snapper arm, and the other end of the string has a loop that is designed to “catch” a hook that is glued to a drive axle.
Once the loop is placed over the axle hook, the string is wound around the drive axle by turning the wheels in the opposite direction to the vehicle intended motion.
As the string is wound around the axle, the lever arm is pulled closer to the drive axle causing the mousetrap’s spring to “wind-up” and store energy.
When the drive wheels are released, the string is pulled off the drive axle by the mousetrap, causing the wheels to rotate.
How do you build a mouse trap powered racer?
There is no one “right way” to build a mousetrap powered vehicle. The first step to making a good mouse trap powered car is simple: put something together and find out how it works.
Once you have something working you can begin to isolate the variables that are affecting the performance and learn to adjust to improve your results.
Build, test, have fun spectacular failures, and improve, just like SpaceX rockets.
What’s the difference between a FAST Racer and a LONG distance traveler?
When you build a mouse-trap car for distance, you want a small energy consumption per second or a small power usage. Smaller power outputs will produce less wasted energy and have greater efficiency.
When you build a vehicle for speed, you want to use your energy quickly or at a high power output.
We change the power ratio of a vehicle by changing one or all of the following:
* where the string attaches to the mouse-trap’s lever arm
* the drive wheel diameter
* the drive axle diameter.
The amount of energy released by using a short lever arm or a long lever arm is the same, but the length of the lever arm will determine the rate at which the energy is released and this is called the power output.
Long lever arms decrease the pulling force and power output but increase the pulling distance.
Short lever arms increase the pulling force and the power output by decrease the pulling distance but increasing the speed.
Building for speed
If you are building a mouse-trap car for speed, you will want to maximize the power output to a point just before the wheels begin to spin-out on the floor. Maximum power output means more energy is being transferred into energy of motion in a shorter amount of time. Greater acceleration can be achieved by having a short length lever arm and/or by having a small axle to wheel ratio.
Building for distance
Minimize the power output or transfer stored energy into energy of motion at a slow rate. This usually means having a long lever arm and a large axle-to-wheel ratio.
If you make the lever arm too long, you may not have enough torque through the entire pulling distance to keep the vehicle moving, in which case you will have to attach the string to a lower point or change the axle-to wheel ratio.
Supplies
Most parts can be scavenged from toys, or recycled materials. You may also consider stores such as Michael’s Art Supply, Home Depot, or A. C. Moore. Mousetraps are available in 2 packs, for less than $2, from supermarkets.
Learning Standards
Next Generation Science Standards
DCI – Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as, within the system, energy is continually transferred from one object to another and between its various possible forms.
Conservation of energy means that the total change of energy in any system is always equal to the total energy transferred into or out of the system.
Energy cannot be created or destroyed, but it can be transported from one place to another and transferred between systems.
Mathematical expressions, which quantify how the stored energy in a system depends on its configuration (e.g., relative positions of charged particles, compression of a spring) and how kinetic energy depends on mass and speed, allow the concept of conservation of energy to be used to predict and describe system behavior.
The availability of energy limits what can occur in any system.
Next Generation Science Standards: Science – Engineering Design (6-8)
• Evaluate competing design solutions using a systematic process to determine how well they meet the criteria and constraints of the problem.
Massachusetts Science and Technology/Engineering Curriculum Framework
HS-ETS4-5(MA). Explain how a machine converts energy, through mechanical means, to do work. Collect and analyze data to determine the efficiency of simple and complex machines.
HS-PS3-3. Design and evaluate a device that works within given constraints to convert one form of energy into another form of energy.
• Emphasis is on both qualitative and quantitative evaluations of devices.
• Examples of devices could include Rube Goldberg devices, wind turbines, solar cells, solar ovens, and generators.
Appendix VIII Value of Crosscutting Concepts and Nature of Science in Curricula
Cause and Effect: Mechanism and Explanation. Events have causes, sometimes simple, sometimes multifaceted. A major activity of science and engineering is investigating and explaining causal relationships and the mechanisms by which they are mediated. Such mechanisms can then be tested across given contexts and used to predict and explain events in new contexts or design solutions.
KaiserScience 