## Emmy Noether

Emmy Noether

(article to be written)

Amalie Emmy Noether (1882 – 1935) was a German mathematician known for her landmark contributions to abstract algebra and theoretical physics. She was described by Pavel Alexandrov, Albert Einstein, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics. In physics, Noether’s theorem explains the connection between symmetry and conservation laws.

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Articles

http://www.thephysicsmill.com/2014/03/09/international-womens-day-spotlight-emmy-noether/

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## Magnetism MCAS topics

See the lesson here Kaiserscience -> Physics -> Electromagnetism -> “Magnetism-and-electricity”

## MCAS questions

Which of the following forces allow a battery-powered motor to generate mechanical energy? (2014)

A. magnetic and static B. electric and magnetic

C. static and gravitational D. electric and gravitational

———————————————————

Which of the following statements describes an electric generator? (2013)

A. A magnet is rotated through a coil of wire to produce an electric current.

B. Electric potential in a rotating coil of wire creates a permanent magnet.

C. An electrical current causes a coil of wire to rotate in a magnetic field.

D. Forces from a permanent magnet allow a coil of wire to rotate.

———————————————————

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This next one is from 2010

———————————————————

Which of the following would cause the galvanometer needle to move?

A. wrapping additional wire around the tube

B. uncoiling the wire wrapped around the tube

C. moving a magnet back and forth inside the tube

D. moving an aluminum block up and down inside the tube

———————————————————

### This next one is from 2009

Precise measuring instruments require shock absorbers to eliminate small vibrations that can affect the results of an experiment. One type of shock absorber that can be used is an electromagnet that repels a magnetic platform placed above it. Which of the following setups would provide the greatest lift to the platform?

## Scientific calculators < $10

This might seem amazing, but today one can purchase scientific calculators for less than $10.

http://budgetlightforum.com/node/34682

Learning Standards

MASSACHUSETTS CURRICULUM FRAMEWORK FOR MATHEMATICS

CONCEPTUAL CATEGORY: Number and Quantity

Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer exponents.

Guiding Principle 3: Technology Technology is an essential tool that should be used strategically in mathematics education. Technology enhances the mathematics curriculum in many ways. Tools such as measuring instruments, manipulatives (such as base ten blocks and fraction pieces), scientific and graphing calculators, and computers with appropriate software, if properly used, contribute to a rich learning environment for developing and applying mathematical concepts. However, appropriate use of calculators is essential; calculators should not be used as a replacement for basic understanding and skills. Elementary students should learn how to perform the basic arithmetic operations independent of the use of a calculator.4 Although the use of a graphing calculator can help middle and secondary students to visualize properties of functions and their graphs, graphing calculators should be used to enhance their understanding and skills rather than replace them. Teachers and students should consider the available tools when presenting or solving a problem. Students should be familiar with tools appropriate for their grade level to be able to make sound decisions about which of these tools would be helpful.

Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

## Data needs an interpretation to have meaning

Lesson: “Data has no meaning without a physical interpretation”

Content objectives:

1. SWBAT to identify trends in data (apparent linear plots; apparent linear data plus noise; and simple harmonic motion.)

2. SWBAT to id

Thesis: raw data doesn’t tells us anything physical phenomenon. We always first need to know what physical phenomenon we are analyzing, before we can interpret it.

Tier III vocabulary: Simple harmonic motion

Launch: Students are given graph paper, and data. Plot the given ata points, and connect the dots in a way that they think is logical.

Question: Justify *why* you connected the dots in that way. Why not in some other way?

Direct Instruction/guided practice

Part A. Teacher instructions:

Print out a sine wave (attached.)

Draw a straight line across it, from upper right to lower left.

The line will intersect the sine wave at 7 points.

Overlay graph paper on top of this, and plot these 7 points.

Tag six more points from the sine wave, that are not on the original straight line.

These points should be at the wave’s maxima, minima, and zeroes, and other points.

Determine the Cartesian coordinates for them,

Give students graph papers, and at first, only 7 data points. Additional data points come afterwards.

If one were to plot only these 7 points, they would appear as a straight line. A naive reading of the raw data would lead one (mistakenly) to believe that we are studying some kind of linear phenomenon.Give examples of linear phenomenon.

One at a time, give new data points, ask them to re-draw their graph each time

Part A Student Instructions

Use the graph that you created for the Do now.

Add the additional data points to this graph.

What function (line, curve, etc) best fits all of this this data? (both old and new data points.)

Draw the line/curve that best fits.

Part B: Examples of data not involving motion: Size of objects from 10^1 meters, to 10^20 meters (human-size up to galactic structures.) The Scale of The Universe (interactive applet)

Independent /collaborative work:

Part A: Justify your choice: What real world motion would produce such a function? Think-Pair-Share

After the discussion, the teacher reveals what produces such data: SHM, Simple Harmonic Motion:

Summative question, tying this all together: Why couldn’t most students plot the data correctly, even after the final data points were added? Answer: Unless you know what kind of phenomenon you are studying, you have no idea whether the data is supposed to be linear, harmonic, exponential, etc. Data – bt itself – has no meaning without a physical interpretation.

Part B: Last night they built a data table for this part of the lesson. As we use “The Scale of The Universe” (interactive applet) they’ll fill in sizes of objects at all scales.

Closure:

Query multiple students: Where do you experience SHM in your own life?

Possible answers: Moving back-and-forth on a swing, pendulum of a clock, automobile suspension system

Homework:

Textbook: Physics: Principles and Problems (Glencoe)

Read Chap. 2, Section 1, “Representing Motion”, and Section 2, “Where and When? Coordinate Systems”

Write definition for highlighted “new vocabulary” words. Page 36. Section 1 Review, #1-4.

## Learning standards

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)

Dimension 1: Scientific and Engineering Practices: Practice 4: Analyzing and Interpreting Data.

“Once collected, data must be presented in a form that can reveal any patterns and relationships and that allows results to be communicated to others. Because raw data as such have little meaning, a major practice of scientists is to organize and interpret data through tabulating, graphing, or statistical analysis. Such analysis can bring out the meaning of data—and their relevance—so that they may be used as evidence.”

## Escape Slide Parachute

MythBusters and the Scientific method

Episode 37, “Escape Slide Parachute (Story of Vesna Vulović)

Textbook: Chapter 3, Accelerated Motion. Free Fall.

https://kaiserscience.wordpress.com/physics/physics-in-films/mythbusters/

- What is the myth?

A person can survive free-fall from over 20,000 feet

How far is this in meters?Convert feet to meters. 1 foot = 0.31 meters.

20,000 feet x

__0.31 meters__= 6200 m = 6.2 x 10^{3}m

feet

- Why would people believe that myth is true?
(Discuss)

Newspaper articles: Vesna Vulović, a Serbia flight attendant, survived the explosion of a DC-9, from 30,000 feet, in 1972.

- Testable predictions: A crash-test dummy, dropped from a high altitude, within an airplane, will experience survivable damage
- How do they test their prediction?

* Purchase used airplane

* cut away the rear section of the plane.

* Strap a crash-test dummy into flight attendant’s seat.

* Lift section up, thousands of feet high, with a helicopter.

* Drop section in free-fall. - Evaluation of the results?

* Buster had a terrible amount of damage.

* A person in that position would not survive this drop.

* Yet it has been confirmed that Vesna Vulović (Serbia flight attendant) survived the explosion of a DC-9, from 30,000 feet, in 1972.

* The section of the plane that the MythBusters dropped was only partially destroyed; other parts were still in shape.* Conclusion: Since we know such a fall is thus theoretically survivable, something was not adequate about our experiment:

* Did they need to form a new hypothesis: Yes – they should re-test this with multiple crash dummies. They can be seated in various sections of the airplane. And the entire test could be repeated several times.

## Theoretical physics: The origins of space and time

### Many researchers believe that physics will not be complete until it can explain not just the behaviour of space and time, but where these entities come from.

Zeeya Merali, Nature, 28 August 2013

“Imagine waking up one day and realizing that you actually live inside a computer game,” says Mark Van Raamsdonk, describing what sounds like a pitch for a science-fiction film. But for Van Raamsdonk, a physicist at the University of British Columbia in Vancouver, Canada, this scenario is a way to think about reality. If it is true, he says, “everything around us — the whole three-dimensional physical world — is an illusion born from information encoded elsewhere, on a two-dimensional chip”. That would make our Universe, with its three spatial dimensions, a kind of hologram, projected from a substrate that exists only in lower dimensions.

This ‘holographic principle’ is strange even by the usual standards of theoretical physics. But Van Raamsdonk is one of a small band of researchers who think that the usual ideas are not yet strange enough. If nothing else, they say, neither of the two great pillars of modern physics — general relativity, which describes gravity as a curvature of space and time, and quantum mechanics, which governs the atomic realm — gives any account for the existence of space and time. Neither does string theory, which describes elementary threads of energy.

Van Raamsdonk and his colleagues are convinced that physics will not be complete until it can explain how space and time emerge from something more fundamental — a project that will require concepts at least as audacious as holography. They argue that such a radical reconceptualization of reality is the only way to explain what happens when the infinitely dense ‘singularity’ at the core of a black hole distorts the fabric of space-time beyond all recognition, or how researchers can unify atomic-level quantum theory and planet-level general relativity — a project that has resisted theorists’ efforts for generations.

“All our experiences tell us we shouldn’t have two dramatically different conceptions of reality — there must be one huge overarching theory,” says Abhay Ashtekar, a physicist at Pennsylvania State University in University Park.

Finding that one huge theory is a daunting challenge. Here, *Nature* explores some promising lines of attack — as well as some of the emerging ideas about how to test these concepts.

NIK SPENCER/NATURE; Panel 4 adapted from Budd, T. & Loll, R. Phys. Rev. D 88, 024015 (2013)

## Gravity as thermodynamics

One of the most obvious questions to ask is whether this endeavour is a fool’s errand. Where is the evidence that there actually is anything more fundamental than space and time?

A provocative hint comes from a series of startling discoveries made in the early 1970s, when it became clear that quantum mechanics and gravity were intimately intertwined with thermodynamics, the science of heat.

In 1974, most famously, Stephen Hawking of the University of Cambridge, UK, showed that quantum effects in the space around a black hole will cause it to spew out radiation as if it was hot. Other physicists quickly determined that this phenomenon was quite general. Even in completely empty space, they found, an astronaut undergoing acceleration would perceive that he or she was surrounded by a heat bath. The effect would be too small to be perceptible for any acceleration achievable by rockets, but it seemed to be fundamental. If quantum theory and general relativity are correct — and both have been abundantly corroborated by experiment — then the existence of Hawking radiation seemed inescapable.

A second key discovery was closely related. In standard thermodynamics, an object can radiate heat only by decreasing its entropy, a measure of the number of quantum states inside it. And so it is with black holes: even before Hawking’s 1974 paper, Jacob Bekenstein, now at the Hebrew University of Jerusalem, had shown that black holes possess entropy.

But there was a difference. In most objects, the entropy is proportional to the number of atoms the object contains, and thus to its volume. But a black hole’s entropy turned out to be proportional to the surface area of its event horizon — the boundary out of which not even light can escape. It was as if that surface somehow encoded information about what was inside, just as a two-dimensional hologram encodes a three-dimensional image.

In 1995, Ted Jacobson, a physicist at the University of Maryland in College Park, combined these two findings, and postulated that every point in space lies on a tiny ‘black-hole horizon’ that also obeys the entropy–area relationship. From that, he found, the mathematics yielded Einstein’s equations of general relativity — but using only thermodynamic concepts, not the idea of bending space-time^{1}.

“This seemed to say something deep about the origins of gravity,” says Jacobson. In particular, the laws of thermodynamics are statistical in nature — a macroscopic average over the motions of myriad atoms and molecules — so his result suggested that gravity is also statistical, a macroscopic approximation to the unseen constituents of space and time.

In 2010, this idea was taken a step further by Erik Verlinde, a string theorist at the University of Amsterdam, who showed^{2} that the statistical thermodynamics of the space-time constituents — whatever they turned out to be — could automatically generate Newton’s law of gravitational attraction.

And in separate work, Thanu Padmanabhan, a cosmologist at the Inter-University Centre for Astronomy and Astrophysics in Pune, India, showed^{3} that Einstein’s equations can be rewritten in a form that makes them identical to the laws of thermodynamics — as can many alternative theories of gravity. Padmanabhan is currently extending the thermodynamic approach in an effort to explain the origin and magnitude of dark energy: a mysterious cosmic force that is accelerating the Universe’s expansion.

Testing such ideas empirically will be extremely difficult. In the same way that water looks perfectly smooth and fluid until it is observed on the scale of its molecules — a fraction of a nanometre — estimates suggest that space-time will look continuous all the way down to the Planck scale: roughly 10^{−35} metres, or some 20 orders of magnitude smaller than a proton.

But it may not be impossible. One often-mentioned way to test whether space-time is made of discrete constituents is to look for delays as high-energy photons travel to Earth from distant cosmic events such as supernovae and γ-ray bursts. In effect, the shortest-wavelength photons would sense the discreteness as a subtle bumpiness in the road they had to travel, which would slow them down ever so slightly.

Giovanni Amelino-Camelia, a quantum-gravity researcher at the University of Rome, and his colleagues have found^{4} hints of just such delays in the photons from a γ-ray burst recorded in April. The results are not definitive, says Amelino-Camelia, but the group plans to expand its search to look at the travel times of high-energy neutrinos produced by cosmic events. He says that if theories cannot be tested, “then to me, they are not science. They are just religious beliefs, and they hold no interest for me.”

Other physicists are looking at laboratory tests. In 2012, for example, researchers from the University of Vienna and Imperial College London proposed^{5} a tabletop experiment in which a microscopic mirror would be moved around with lasers. They argued that Planck-scale granularities in space-time would produce detectable changes in the light reflected from the mirror (see Naturehttp://doi.org/njf; 2012).

## Loop quantum gravity

Even if it is correct, the thermodynamic approach says nothing about what the fundamental constituents of space and time might be. If space-time is a fabric, so to speak, then what are its threads?

One possible answer is quite literal. The theory of loop quantum gravity, which has been under development since the mid-1980s by Ashtekar and others, describes the fabric of space-time as an evolving spider’s web of strands that carry information about the quantized areas and volumes of the regions they pass through^{6}. The individual strands of the web must eventually join their ends to form loops — hence the theory’s name — but have nothing to do with the much better-known strings of string theory. The latter move around in space-time, whereas strands actually are space-time: the information they carry defines the shape of the space-time fabric in their vicinity.

Because the loops are quantum objects, however, they also define a minimum unit of area in much the same way that ordinary quantum mechanics defines a minimum ground-state energy for an electron in a hydrogen atom. This quantum of area is a patch roughly one Planck scale on a side. Try to insert an extra strand that carries less area, and it will simply disconnect from the rest of the web. It will not be able to link to anything else, and will effectively drop out of space-time.

One welcome consequence of a minimum area is that loop quantum gravity cannot squeeze an infinite amount of curvature onto an infinitesimal point. This means that it cannot produce the kind of singularities that cause Einstein’s equations of general relativity to break down at the instant of the Big Bang and at the centres of black holes.

In 2006, Ashtekar and his colleagues reported^{7} a series of simulations that took advantage of that fact, using the loop quantum gravity version of Einstein’s equations to run the clock backwards and visualize what happened before the Big Bang. The reversed cosmos contracted towards the Big Bang, as expected. But as it approached the fundamental size limit dictated by loop quantum gravity, a repulsive force kicked in and kept the singularity open, turning it into a tunnel to a cosmos that preceded our own.

This year, physicists Rodolfo Gambini at the Uruguayan University of the Republic in Montevideo and Jorge Pullin at Louisiana State University in Baton Rouge reported^{8} a similar simulation for a black hole. They found that an observer travelling deep into the heart of a black hole would encounter not a singularity, but a thin space-time tunnel leading to another part of space. “Getting rid of the singularity problem is a significant achievement,” says Ashtekar, who is working with other researchers to identify signatures that would have been left by a bounce, rather than a bang, on the cosmic microwave background — the radiation left over from the Universe’s massive expansion in its infant moments.

Loop quantum gravity is not a complete unified theory, because it does not include any other forces. Furthermore, physicists have yet to show how ordinary space-time would emerge from such a web of information. But Daniele Oriti, a physicist at the Max Planck Institute for Gravitational Physics in Golm, Germany, is hoping to find inspiration in the work of condensed-matter physicists, who have produced exotic phases of matter that undergo transitions described by quantum field theory. Oriti and his colleagues are searching for formulae to describe how the Universe might similarly change phase, transitioning from a set of discrete loops to a smooth and continuous space-time. “It is early days and our job is hard because we are fishes swimming in the fluid at the same time as trying to understand it,” says Oriti.

## Causal sets

Such frustrations have led some investigators to pursue a minimalist programme known as causal set theory. Pioneered by Rafael Sorkin, a physicist at the Perimeter Institute in Waterloo, Canada, the theory postulates that the building blocks of space-time are simple mathematical points that are connected by links, with each link pointing from past to future. Such a link is a bare-bones representation of causality, meaning that an earlier point can affect a later one, but not vice versa. The resulting network is like a growing tree that gradually builds up into space-time. “You can think of space emerging from points in a similar way to temperature emerging from atoms,” says Sorkin. “It doesn’t make sense to ask, ‘What’s the temperature of a single atom?’ You need a collection for the concept to have meaning.”

In the late 1980s, Sorkin used this framework to estimate^{9} the number of points that the observable Universe should contain, and reasoned that they should give rise to a small intrinsic energy that causes the Universe to accelerate its expansion. A few years later, the discovery of dark energy confirmed his guess. “People often think that quantum gravity cannot make testable predictions, but here’s a case where it did,” says Joe Henson, a quantum-gravity researcher at Imperial College London. “If the value of dark energy had been larger, or zero, causal set theory would have been ruled out.”

## Causal dynamical triangulations

That hardly constituted proof, however, and causal set theory has offered few other predictions that could be tested. Some physicists have found it much more fruitful to use computer simulations. The idea, which dates back to the early 1990s, is to approximate the unknown fundamental constituents with tiny chunks of ordinary space-time caught up in a roiling sea of quantum fluctuations, and to follow how these chunks spontaneously glue themselves together into larger structures.

The earliest efforts were disappointing, says Renate Loll, a physicist now at Radboud University in Nijmegen, the Netherlands. The space-time building blocks were simple hyper-pyramids — four-dimensional counterparts to three-dimensional tetrahedrons — and the simulation’s gluing rules allowed them to combine freely. The result was a series of bizarre ‘universes’ that had far too many dimensions (or too few), and that folded back on themselves or broke into pieces. “It was a free-for-all that gave back nothing that resembles what we see around us,” says Loll.

*randomly generated two dimensional universes*.

But, like Sorkin, Loll and her colleagues found that adding causality changed everything. After all, says Loll, the dimension of time is not quite like the three dimensions of space. “We cannot travel back and forth in time,” she says. So the team changed its simulations to ensure that effects could not come before their cause — and found that the space-time chunks started consistently assembling themselves into smooth four-dimensional universes with properties similar to our own^{10}.

Intriguingly, the simulations also hint that soon after the Big Bang, the Universe went through an infant phase with only two dimensions — one of space and one of time. This prediction has also been made independently by others attempting to derive equations of quantum gravity, and even some who suggest that the appearance of dark energy is a sign that our Universe is now growing a fourth spatial dimension. Others have shown that a two-dimensional phase in the early Universe would create patterns similar to those already seen in the cosmic microwave background.

## Holography

Meanwhile, Van Raamsdonk has proposed a very different idea about the emergence of space-time, based on the holographic principle. Inspired by the hologram-like way that black holes store all their entropy at the surface, this principle was first given an explicit mathematical form by Juan Maldacena, a string theorist at the Institute of Advanced Study in Princeton, New Jersey, who published^{11} his influential model of a holographic universe in 1998. In that model, the three-dimensional interior of the universe contains strings and black holes governed only by gravity, whereas its two-dimensional boundary contains elementary particles and fields that obey ordinary quantum laws without gravity.

Hypothetical residents of the three-dimensional space would never see this boundary, because it would be infinitely far away. But that does not affect the mathematics: anything happening in the three-dimensional universe can be described equally well by equations in the two-dimensional boundary, and vice versa.

In 2010, Van Raamsdonk studied what that means when quantum particles on the boundary are ‘entangled’ — meaning that measurements made on one inevitably affect the other^{12}. He discovered that if every particle entanglement between two separate regions of the boundary is steadily reduced to zero, so that the quantum links between the two disappear, the three-dimensional space responds by gradually dividing itself like a splitting cell, until the last, thin connection between the two halves snaps. Repeating that process will subdivide the three-dimensional space again and again, while the two-dimensional boundary stays connected. So, in effect, Van Raamsdonk concluded, the three-dimensional universe is being held together by quantum entanglement on the boundary — which means that in some sense, quantum entanglement and space-time are the same thing.

Or, as Maldacena puts it: “This suggests that quantum is the most fundamental, and space-time emerges from it.”

Nature 500,516–519 (29 August 2013) doi:10.1038/500516a

http://www.nature.com/news/theoretical-physics-the-origins-of-space-and-time-1.13613

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__________________________________________________

## Power (electrical)

If you look carefully at a stereo, hair dryer, or other household appliance, you find that most devices list a “power rating” that tells how many watts the appliance uses. In this section you will learn what these power ratings mean, and how to figure out the electricity costs of using various appliances.

The three electrical quantities

We have now learned three important electrical quantities:

Paying for electricity

Electric bills sent out by utility companies don’t charge by the volt, the amp, or the ohm. You may have noticed that electrical appliances in your home usually include another unit – the watt. Most appliances have a label that lists the number of watts or kilowatts. You may have purchased 60-watt light bulbs, or a 900-watt hair dryer, or a 1500-watt toaster oven. Electric companies charge for the energy you use, which depends on how many watts each appliance consumes in a given month.

A watt is a unit of power

The watt is a unit of power. Power, in the scientific sense, has a precise meaning. Power is the rate at which energy is flowing. Energy is measured in joules. Power is measured in joules per second. One joule per second is equal to one watt. A 100-watt light bulb uses 100 joules of energy every second. Where does the electrical power go?

Electrical power can be easily transformed into many different forms. An electric

motor takes electrical power and makes mechanical power. A light bulb turns electrical power into light and a toaster oven turns the power into heat. The same unit (watts) applies to all forms of energy flow, including light, motion, electrical, thermal, or many others.

Power in a circuit can be measured using the tools we already have. Remember

that one watt equals an energy flow of one joule per second.

Amps = flow of 1 coulomb of charge per second

Volts = an energy of 1 joule of energy / coulomb of charge

If these two quantities are multiplied together, you will find that the units of

coulombs cancel out, leaving the equation we want for power.

Watts equal joules/second, so we can calculate electrical power in a circuit by

multiplying voltage times current.

# P = VI

power measured in watts; voltage in volts; current in amps

A larger unit of power is sometimes needed.

A 1500-watt toaster oven may be labeled 1.5 kW.

kilowatt (kW) is equal to 1000 watts, or 1000 joules per second.

Horsepower – another common unit of power often seen on electric motors

1 horsepower = 746 watts.

Electric motors you find around the house range in

size from 1/25th of a horsepower (30 watts) for a small electric fan to 2 horsepower (1492 watts) for an electric saw.