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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.
-
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.
Diffraction
(adapted from Giancoli Physics)
Waves spread as they travel. When waves encounter an obstacle, they bend around it and pass into the region behind it. This phenomenon is called diffraction.
The amount of diffraction depends on the λ (wavelength) of the wave and on the size of the obstacle:
(a) λ is much larger than the object. Wave bends around object almost as if it is not there.
(b) and (c) the λ is shorter than the size of the object. There’s more of a “shadow” region behind the obstacle where we might not expect the waves to penetrate — but they do, at least a little.
(d) the obstacle is the same as in part (c) but the λ is longer. More diffraction around object.
Rule: Only when λ is smaller than the size of the object will there be a shadow region.
Water waves diffracting around an island
And then the next step
Sound waves can diffract in unusual and unexpected ways. See our article on anomalous sounds
Even light itself can diffract! See our article on light’s wave nature.
Fresnel diffraction
French scientist, Augustin-Jean Fresnel,
Discovering Fresnel diffraction: The Greatest Mistake In The History Of Physics
Example – diffraction in Boston Harbor
from bostonfoodandwhine.com
As part of the Central Artery/Tunnel project – the Big Dig – Applied Coastal Research and Engineering did research on wave diffraction in Boston Harbor, around Spectacle Island.
…A detailed beach nourishment design was developed for the southern shoreline of Spectacle Island, which is located within Boston Harbor… The propagation of waves from Massachusetts Bay into Boston Harbor was modeled using the refraction/diffraction model REF/DIF1. This model predicts the transformation of waves in areas where bathymetry is irregular and where diffraction is important, such as at Spectacle Island. The resulting wave heights, periods, and directions were used as input to both longshore and cross-shore sediment transport models. These models were employed to simulate the performance of several different beach fill designs…
Beach Nourishment Design for Spectacle Island
This map is from mass.gov/eea/images/dcr
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Recognize that electromagnetic waves can travel through empty space (without a medium) as compared to mechanical waves that require a medium
HS-PS4-3. Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described either by a wave model or a particle model, and that for some situations one model is more useful than the other. [Emphasis is on how the experimental evidence supports the claim and how a theory is generally modified in light of new evidence. Examples of a phenomenon could include resonance, interference, diffraction, and photoelectric effect.]
Anomalous sounds
Here’s an actual news story: “Loud booms heard across Southern New Hampshire: Source of the noise still unclear.”
Nashua police say they don’t know what caused several loud “booms” Saturday afternoon that were heard across Southern New Hampshire. Many reports came from Nashua and surrounding towns, but the sounds were reported as far north as Manchester and as far south as Westford, Massachusetts. Some who heard it in Nashua said they felt their houses shake. Police and fire departments said they have not been alerted to any incidents related to the noise in the area. The cause is still unclear.
– WMUR 9 News. (An ABC affiliated TV station) 2/10/18
How is it possible that such loud, possibly building shaking sounds could be heard in some parts of this town – yet in other parts of the city other residents reported no sound? Also, in a town next door no reports have yet surfaced of anyone hearing them – yet in a town after that, some residents also reported these booming sound.
The answer? It’s complicated, but basically:
(a) there are a wide variety of ways that sounds are produced – including some bizarre ways that most people have never heard of
(b) Sound waves don’t always move in a straight path like many people imagine; changing temperature/density of the air can cause sound waves to bend and diffract, so:
(b1) sound can sometimes travel much further distances than one would expect
(b2) sound can come from a location very different from what “seems obvious” just by listening
(b3) local wind can mask sound, so the same loud sound might be heard in one neighborhood, yet be undetectable by people just a mile away.
Basic idea
Sound doesn’t move in a straight line: It spreads out radially, and then – because of a phenomenon known as diffraction – it can even bend around obstacles.
Source: Hyperphysics, Diffraction of sound, http://hyperphysics.phy-astr.gsu.edu/
“If the air above the earth is warmer than that at the surface, sound will be bent back downward toward the surface by refraction.” – Hyperphysics
Normally, only sound initially directed toward the listener can be heard, but refraction can bend sound downward – effectively amplifying the sound.
This can occur over cool lakes.
Sounds also can bounce off of objects, and come to our ears from a direction different than the original source.
ABD Engineering writes:
…wind alters sound propagation by the mechanism of refraction; that is, wind bends sound waves. Wind nearer to the ground moves more slowly than wind at higher altitudes, due to surface characteristics such as hills, trees, and man-made structures that interfere with the wind.
This wind gradient, with faster wind at higher elevation and slower wind at lower elevation causes sound waves to bend downward when they are traveling to a location downwind of the source and to bend upward when traveling toward a location upwind of the source.
Waves bending downward means that a listener standing downwind of the source will hear louder noise levels than the listener standing upwind of the source.
Temperature gradients in the atmosphere. On a typical sunny afternoon, air is warmest near the ground and temperature decreases at higher altitudes. This temperature gradient causes sound waves to refract upward, away from the ground and results in lower noise levels being heard at the listener’s position.
In the evening, this temperature gradient will reverse, resulting in cooler temperatures near the ground. This condition, often referred to is a temperature inversion will cause sound to bend downward toward the ground and results in louder noise levels at the listener position.
How Weather Affects an Outdoor Noise Study by ABD Engineering and Design
Cheung Kai-chung, from Physics World (Hong Kong), (Translation by Janny Leung) offers this explanation
Sound wave will be refracted to the ground when traveling with the wind.
Sound wave will be refracted upwards when traveling against the wind.
Can wind mask even loud sounds?
A discussion to consider, from Physics forums, includes this phenomenon: “Yes. I have a freeway about 10 blocks South of my house. I can hear the traffic very clearly with no wind, or a South wind. If there is even a slight North wind, the traffic noise becomes almost inaudible. If there is a brisk North wind (over 15 MPH), the sound is completely gone.”
https://www.physicsforums.com/threads/does-wind-affect-how-far-sound-can-travel.149392/
Sound refraction due to cold air:
Also this “…if the air close to the ground is colder than the air above it then sound waves traveling upwards will be bent downwards. This is called Refraction. These refracted sound waves can act to amplify the sound to someone standing far away.”
http://sciencewows.ie/blog/does-sound-travel-faster-in-warm-or-cold-air/
Sound seems amplified when traveling over water.
In School-for-Champions we read
“If you are sitting in a boat, a sound coming from the shore will seem louder than the same sound heard by a person on land. Sound seems to be amplified when it travels over water. The reason is that the water cools the air above its surface, which then slows down the sound waves near the surface. This causes refraction or bending of the sound wave, such that more sound reaches the boat passenger. Sound waves skimming the surface of the water can add to the amplification effect, if the water is calm.”
See their full lesson here School-for-champions.com: Sound_amplified_over_water
Can snow on the ground affect sound?
“When the ground has a thick layer of fresh, fluffy snow, sound waves are readily absorbed at the surface of the snow. However, the snow surface can become smooth and hard as it ages or if there have been strong winds. Then the snow surface will actually help reflect sound waves. Sounds seem clearer and travel farther under these circumstances.” – Colorado State Climatologist Nolan Doesken
Related topic: The Hum is a phenomenon, or collection of phenomena, involving widespread reports of a persistent and invasive low-frequency humming, rumbling, or droning noise not audible to all people.
“Hums” have been widely reported by national media in the UK and the United States. The Hum is sometimes prefixed with the name of a locality where the problem has been particularly publicized: e.g., the “Bristol Hum” or the “Taos Hum”. It is unclear whether it is a single phenomenon; different causes have been attributed. ”
Human reactions to infrasound –
https://en.wikipedia.org/wiki/Infrasound#Human_reactions
Skyquakes or mystery booms are unexplained reports of a phenomenon that sounds like a cannon or a sonic boom coming from the sky. They have been heard in several locations around the world. –
https://en.wikipedia.org/wiki/Skyquake
The microwave auditory effect, also known as the microwave hearing effect or the Frey effect, consists of audible clicks (or, with speech modulation, spoken words[citation needed]) induced by pulsed/modulated microwave frequencies. The clicks are generated directly inside the human head without the need of any receiving electronic device. The effect was first reported by persons working in the vicinity of radar transponders during World War II. (Wikipedia)
References
Our first article.
How Weather Affects an Outdoor Noise Study by ABD Engineering and Design
This following discussion has helpful images.
A discussion to consider, from Physics forums, includes this phenomenon:
“Yes. I have a freeway about 10 blocks South of my house. I can hear the traffic very clearly with no wind, or a South wind. If there is even a slight North wind, the traffic noise becomes almost inaudible. If there is a brisk North wind (over 15 MPH), the sound is completely gone.”
https://www.physicsforums.com/threads/does-wind-affect-how-far-sound-can-travel.149392/
Also this “…if the air close to the ground is colder than the air above it then sound waves traveling upwards will be bent downwards. This is called Refraction. These refracted sound waves can act to amplify the sound to someone standing far away.”
http://sciencewows.ie/blog/does-sound-travel-faster-in-warm-or-cold-air/
Sound seems amplified when traveling over water
https://www.school-for-champions.com/science/sound_amplified_over_water.htm#.WoBbQ5M-fVo
Diffraction of sound waves
https://katrinasiron21.wordpress.com/properties-of-sound-waves/diffraction-of-sound-waves/
Temperature inversion and sound waves
http://kxan.com/blog/2015/02/13/why-does-sound-carry-farther-on-cold-calm-mornings/
Also look into: Humans hearing infra sound waves
“Colorado State Climatologist Nolan Doesken says: “When the ground has a thick layer of fresh, fluffy snow, sound waves are readily absorbed at the surface of the snow. However, the snow surface can become smooth and hard as it ages or if there have been strong winds. Then the snow surface will actually help reflect sound waves. Sounds seem clearer and travel farther under these circumstances.””
Related topic: The Hum is a phenomenon, or collection of phenomena, involving widespread reports of a persistent and invasive low-frequency humming,rumbling, or droning noise not audible to all people. Hums have been widely reported by national media in the UK and the United States. The Hum is sometimes prefixed with the name of a locality where the problem has been particularly publicized: e.g., the “Bristol Hum” or the “Taos Hum”. It is unclear whether it is a single phenomenon; different causes have been attributed. ”
Human reactions to infrasound – https://en.wikipedia.org/wiki/Infrasound#Human_reactions
Skyquakes or mystery booms are unexplained reports of a phenomenon that sounds like a cannon or a sonic boom coming from the sky. They have been heard in several locations around the world. – https://en.wikipedia.org/wiki/Skyquake
Learning Standards
Skeptical analysis of unexplained phenomenon.
The Massachusetts STEM Curriculum Framework addresses “Understandings about the Nature of Science”
Scientific inquiry is characterized by a common set of values that include: logical thinking, precision, open-mindedness, objectivity, skepticism, replicability of results, and honest and ethical reporting of findings.
Science disciplines share common rules of evidence used to evaluate explanations about natural systems. Science includes the process of coordinating patterns of evidence with current theory.
Most scientific knowledge is quite durable but is, in principle, subject to change based on new evidence and/or reinterpretation of existing evidence.
The “College Board Standards for College Success: Science” addresses these same skeptical inquiry methods in Standard SP.1: Scientific Questions and Predictions. Asking scientific questions that can be tested empirically and structuring these questions in the form of testable predictions.
Students recognize, formulate, justify and revise scientific questions that can be addressed by science in order to construct explanations.
Students make and justify predictions concerning natural phenomena. Predictions and justifications are based on observations of the world, on knowledge of the discipline and on empirical evidence.
Students determine which data from a specific investigation can be used as evidence to address a scientific question or to support a prediction or an explanation, and distinguish credible data from noncredible data in terms of quality.
Students construct explanations that are based on observations and measurements of the world, on empirical evidence and on reasoning grounded in the theories, principles and concepts of the discipline.
The “Benchmarks for Science Literacy” (AAAS) addresses these same skeptical inquiry methods:
In science, a new theory rarely gains widespread acceptance until its advocates can show that it is borne out by the evidence, is logically consistent with other principles that are not in question, explains more than its rival theories, and has the potential to lead to new knowledge. 12A/H3** (SFAA)
Scientists value evidence that can be verified, hypotheses that can be tested, and theories that can be used to make predictions. 12A/H4** (SFAA)
Curiosity motivates scientists to ask questions about the world around them and seek answers to those questions. Being open to new ideas motivates scientists to consider ideas that they had not previously considered. Skepticism motivates scientists to question and test their own ideas and those that others propose. 12A/H5*
SAT subject test in Physics: Waves and optics
• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
China’s Floating City Mirage
China’s Floating City – Was this a real mirage, a misinterpretation of a reflection, or a hoax?
from “Floating Cities are Generally not Fata Morgana Mirage.” Discussion by Mick West, Oct 20, 2015, on Metabunk.org.
A video is being widely shared on social media (and the “weird news” sections of more traditional media) claiming to show the image of an impossibly large city rising above the fog in the city of Foshan (佛山), Guangdong province, China. Here is a composite image from the video.
Some have said this is an example of a fata morgana, a type of mirage where light is bent though the atmosphere in such a way to create the illusion of buildings on the horizon.
This is utterly impossible in this case, as fata morgana only creates a very thin strip of such an illusion very close to the horizon, and appears small and far away. It does not create images high in the sky.
Besides, a fata morgana might create the illusion of buildings by stretching landscape features, or it might distort existing buildings. But what it cannot do it create a perfect image of existing nearby buildings, complete with windows.
It is important to note that no expert has actually looked at this video and said it was a fata morgana.
The second and more common type of “floating city” illusions is with buildings that are simply rising up out of clouds or low fog, and hence appear to be floating above them. This has led to “floating city” stories in the past, with this recent example, also from China.
This is simply a photo of building across the river, but when cropped it appears like they are floating, which led to all kinds of wild stories of “ghost cities”.
This actually came from mistranslations of the original news reports, where local people (who knew exactly what they were looking at) were simply marveling at how pretty the scene looked, with the buildings appearing to float above clouds.
Could the Foshan video be of real buildings obscured by clouds? It does not appear so. Look at some real buildings in Foshan (and keep in mind it’s not entirely clear if Foshan is the actual setting of either the top or the bottom of the video.
Consider what it would take for these buildings to appear like they do in the video, with the road beneath them. The scale is simply impossible. The image has to be composited somehow, and the possibilities are:
-
Computer generated buildings spliced into the video of the road.
-
Two different videos spliced together
-
The video is shot though glass, and the buildings are behind the camera, or to the side (with the glass at around 45°, like a half open window/door)
It’s unfortunate that many people leap for the “fata morgana” or other mirage explanation when it’s quite clear that this is far too high in the sky to be anything like that.
Resources
https://www.metabunk.org/floating-cities-are-generally-not-fata-morgana-mirages.t6922/
http://www.cnn.com/2015/10/20/world/china-floating-city-video-feat/index.html
https://www.snopes.com/floating-city-china/
An Introduction to Mirages, Andrew T. Young
Fata Morgana between the Continental Divide and the Missouri River
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS4-3. Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described by either a wave model or a particle model, and that for some situations involving resonance, interference, diffraction, refraction, or the photoelectric effect, one model is more useful than the other.
A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Core Idea PS4: Waves and Their Applications in Technologies for Information Transfer
When a wave passes an object that is small compared with its wavelength, the wave is not much affected; for this reason, some things are too small to see with visible light, which is a wave phenomenon with a limited range of wavelengths corresponding to each color. When a wave meets the surface between two different materials or conditions (e.g., air to water), part of the wave is reflected at that surface and another part continues on, but at a different speed. The change of speed of the wave when passing from one medium to another can cause the wave to change direction or refract. These wave properties are used in many applications (e.g., lenses, seismic probing of Earth).
The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing. The reflection, refraction, and transmission of waves at an interface between two media can be modeled on the basis of these properties.
All electromagnetic radiation travels through a vacuum at the same speed, called the speed of light. Its speed in any given medium depends on its wavelength and the properties of that medium. At the surface between two media, like any wave, light can be reflected, refracted (its path bent), or absorbed. What occurs depends on properties of the surface and the wavelength of the light.
SAT Subject Area Test in Physics
Waves and optics:
- Reflection and refraction, such as Snell’s law and changes in wavelength and speed
- Ray optics, such as image formation using pinholes, mirrors, and lenses
Fair use: This website is educational. Materials within it are being used in accord with the Fair Use doctrine, as defined by United States law.
§107. Limitations on Exclusive Rights: Fair Use
Notwithstanding the provisions of section 106, the fair use of a copyrighted work, including such use by reproduction in copies or phone records or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use, the factors to be considered shall include:
the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
the nature of the copyrighted work;
the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
the effect of the use upon the potential market for or value of the copyrighted work. (added pub. l 94-553, Title I, 101, Oct 19, 1976, 90 Stat 2546)
How do point particles create atoms with size?
This article is archived for use with my students from Ask Ethan: If Matter Is Made Of Point Particles, Why Does Everything Have A Size?
Forbes, Stars With a Bang, by Ethan Siegel 9/16/17
Proton Structure Brookhaven National Laboratory
The big idea of atomic theory is that, at some smallest, fundamental level, the matter that makes up everything can be divided no further. Those ultimate building blocks would be literally ἄ-τομος, or un-cuttable.
As we’ve gone down to progressively smaller scales, we’ve found that molecules are made of atoms, which are made of protons, neutrons, and electrons, and that protons and neutrons can be further split into quark and gluons. Yet even though quarks, gluons, electrons, and more appear to be truly point-like, all the matter made out of them has a real, finite size. Why is that? That’s what Brian Cobb wants to know:
Many sources state that quarks are point particles… so one would think that objects composed of them — in this instance, neutrons — would also be points. Is my logic flawed? Or would they be bound to each other in such a way that they would cause the resulting neutron to have angular size?
Let’s take a journey down to the smallest scales, and find out what’s truly going on.
Magdalena Kowalska / CERN / ISOLDE team
If we take a look at matter, things behave similar to how we expect they should, in the macroscopic world, down to about the size of molecules: nanometer (10-9meter) scales. On smaller scales than that, the quantum rules that govern individual particles start to become important.
Single atoms, with electrons orbiting a nucleus, come in at about the size of an Angstrom: 10-10 meters. The atomic nucleus itself, made up of protons and neutrons, is 100,000 times smaller than the atoms in which they are found: a scale of 10-15 meters. Within each individual proton or neutron, quarks and gluons reside.
While molecules, atoms, and nuclei all have sizes associated with them, the fundamental particles they’re made out of — quarks, gluons, and electrons — are truly point-like.
E. Siegel / Beyond The Galaxy
The way we determine whether something is point-like or not is simply to collide whatever we can with it at the highest possible energies, and to look for evidence that there’s a composite structure inside.
In the quantum world, particles don’t just have a physical size, they also have a wavelength associated with them, determined by their energy. Higher energy means smaller wavelength, which means we can probe smaller and more intricate structures. X-rays are high-enough in energy to probe the structure of atoms, with images from X-ray diffraction and crystallography shedding light on what molecules look like and how individual bonds look.
Imperial College London
At even higher energies, we can get even better resolution. Particle accelerators could not only blast atomic nuclei apart, but deep inelastic scattering revealed the internal structure of the proton and neutron: the quarks and gluons lying within.
It’s possible that, at some point down the road, we’ll find that some of the particles we presently think are fundamental are actually made of smaller entities themselves. At the present point, however, thanks to the energies reached by the LHC, we know that if quarks, gluons, or electrons aren’t fundamental, their structures must be smaller than 10-18 to 10-19 meters. To the best of our knowledge, they’re truly points.
Brookhaven National Laboratory
So how, then, are the things made out of them larger than points? It’s the interplay of (up to) three things: Forces, Particle properties, and Energy.
The quarks that we know don’t just have an electric charge, but also (like the gluons) have a color charge. While the electric charge can be positive or negative, and while like charges repel while opposites attract, the force arising from the color charges — the strong nuclear force — is always attractive. And it works, believe it or not, much like a spring does.
Warning: Analogy ahead!
Here we go:
How did the Proton Get Its Spin? Brookhaven National Laboratory
Above: The internal structure of a proton, with quarks, gluons, and quark spin shown. The nuclear force acts like a spring, with negligible force when unstretched but large, attractive forces when stretched to large distances
When two color-charged objects are close together, the force between them drops away to zero, like a coiled spring that isn’t stretched at all.
When quarks are close together, the electrical force takes over, which often leads to a mutual repulsion.
But when the color-charged objects are far apart, the strong force gets stronger. Like a stretched spring, it works to pull the quarks back together.
Based on the magnitude of the color charges and the strength of the strong force, along with the electric charges of each of the quarks, that’s how we arrive at the size of the proton and the neutron: where the strong and electromagnetic forces roughly balance.
APS/Alan Stonebraker
The three valence quarks of a proton contribute to its spin, but so do the gluons, sea quarks and antiquarks, and orbital angular momentum as well. The electrostatic repulsion and the attractive strong nuclear force, in tandem, are what give the proton its size.
On slightly larger scales, the strong force holds protons and neutrons together in an atomic nucleus, overcoming the electrostatic repulsion between the individual protons. This nuclear force is a residual effect of the strong nuclear force, which only works over very short distances.
Because individual protons and neutrons themselves are color-neutral, the exchange is mediated by virtual, unstable particles known as pions, which explains why nuclei beyond a certain size become unstable; it’s too difficult for pions to be exchanged across larger distances. Only in the case of neutron stars does the addition of gravitational binding energy suppress the nucleus’ tendency to rearrange itself into a more stable configuration.
Wikimedia Commons user Manishearth
And on the scale of the atom itself, the key is that the lowest-energy configuration of any electron bound to a nucleus isn’t a zero-energy state, but is actually a relatively high-energy one compared to the electron’s rest mass.
This quantum configuration means that the electron itself needs to zip around at very high speeds inside the atom; even though the nucleus and the electron are oppositely charged, the electron won’t simply hit the nucleus and remain at the center.
Instead, the electron exists in a cloud-like configuration, zipping and swirling around the nucleus (and passing through it) at a distance that’s almost a million times as great as the size of the nucleus itself.
The energy levels and electron wavefunctions that correspond to different states within a hydrogen atom, although the configurations are extremely similar for all atoms. The energy levels are quantized in multiples of Planck’s constant, but the sizes of the orbitals and atoms are determined by the ground-state energy and the electron’s mass.
There are some fun caveats that allow us to explore how these sizes change in extreme conditions. In extremely massive planets, the atoms themselves begin to get compressed due to large gravitational forces, meaning you can pack more of them into a small space.
Jupiter, for example, has three times the mass of Saturn, but is only about 20% larger in size. If you replace an electron in a hydrogen atom with a muon, an unstable electron-like particle that has the same charge but 206 times the mass, the muonic hydrogen atom will be only 1/206th the size of normal hydrogen.
And a Uranium atom is actually larger in size than the individual protons-and-neutrons would be if you packed them together, due to the long-range nature of the electrostatic repulsion of the protons, compared to the short-range nature of the strong force.
Image credit: Calvin Hamilton.
The planets of the Solar System, shown to the scale of their physical sizes, show a Saturn that’s almost as large as Jupiter. However, Jupiter is 3 times as massive, indicating that its atoms are substantially compressed due to gravitational pressure.
By having different forces at play of different strengths, you can build a proton, neutron, or other hadron of finite size out of point-like quarks. By combining protons and neutrons, you can build nuclei of larger sizes than their individual components, bound together, would give you. And by binding electrons to the nucleus, you can build a much larger structure, all owing to the fact that the zero-point energy of an electron bound to an atom is much greater than zero.
In order to get a Universe filled with structures that take up a finite amount of space and have a non-zero size, you don’t need anything more than zero-dimensional, point-like building blocks. Forces, energy, and the quantum properties inherent to particles themselves are more than enough to do the job.
__________________________________________
Ethan Siegel is the founder and primary writer of Starts With A Bang!
This website is educational. Materials within it are being used in accord with the Fair Use doctrine, as defined by United States law.
§107. Limitations on Exclusive Rights: Fair Use
Notwithstanding the provisions of section 106, the fair use of a copyrighted work, including such use by reproduction in copies or phone records or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use, the factors to be considered shall include:
the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
the nature of the copyrighted work;
the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
the effect of the use upon the potential market for or value of the copyrighted work. (added pub. l 94-553, Title I, 101, Oct 19, 1976, 90 Stat 2546)
KaiserScience 