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Inertia and mass
Newton’s laws of motion describe the relationship between the motion of an object and the forces acting on it.
His laws of motion were first compiled in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), published in 1687.
1st law of motion – Inertia
Every object continues in it’s state of rest, or of uniform velocity, as long as no net force acts on it.
2nd law of motion –
The acceleration experienced by an object will be proportional to the applied force, and inversely proportional to its mass.
3rd law of motion –
For every force there is an equal but opposite reaction force.
In this lesson we focus on his first law of motion.
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Every object continues in it’s state of rest, or of uniform velocity, as long as no net force acts on it.
If at rest, objects require force to start moving.
If moving, objects require a force to stop moving.
A quick summary.
An object at rest, stays at rest, unless accelerated by some external force.
Tow truck operators see this more often than you might imagine.
An object at rest, stays at rest, unless accelerated by some external force.
Elegantly illustrated by the leaves staying behind here (until gravity accelerates them!)
Animations showing Newton’s Law of Inertia
Two different definitions of mass
a measure of inertia (how much stuff resists being moved)
the quantity of matter (how much stuff is present in an object)
Don’t confuse mass with volume
Here are five cylinders of different metals:
they all are different volumes, yet all of equal mass.
Lead, copper, brass, zinc, and aluminum.
How is this possible?
Somehow, more matter can be crammed into the same volume with denser materials.
Less matter takes up the same volume in less-dense materials
Mass is not weight
Weight is how much a mass is pulled down by gravity.
This girl has the same mass on both worlds, yet her weight varies.
Mass is the quantity of matter in an object.
Weight is the force of gravity on an object.
One kilogram weighs (approximately) 10 Newtons
The gravity of Earth gives a downward acceleration to objects.
acceleration of gravity on earth = g
At Earth’s surface, measurements show that g = 9.8 m/s2
We often approximate this as g ≅ 10 m/s2
Because the object is being accelerated down, we feel this as “weight”.
Can we convert between mass and weight?
Strictly speaking – no, we can not.
Why not? Because the same mass will have different weights when placed on different planets.
Well, can we convert between mass and weight, assuming that the object is here on Earth?
Oh, that case is different. Yes, in that case we can convert between mass and weight.
Here’s a conversion that’s valid only on Earth.
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1 kg × g = 9.8 N (more exact)
1 kg × g ≅ 10N (approximation)
These approximate conversions are useful.
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1 kg of mass is about 10 Newtons of weight
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1/10 kg of mass is about 1 Newton of weight
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100 kg of mass is about a kN of weight
Learning standards
AAAS Benchmarks for Science Literacy, The Physical Setting
“Newton’s laws of motion are simple to state, and sometimes teachers mistake the ability of students to recite the three laws correctly as evidence that they understand them. The fact that it took such a long time, historically, to codify the laws of motion suggests that they are not self-evident truths, no matter how obvious they may seem to us once we understand them well. ”
Common Core ELA
CCSS.ELA-LITERACY.RST.9-10.4
Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9-10 texts and topics.
CCSS.ELA-LITERACY.RST.11-12.4
Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11-12 texts and topics.
CCSS.ELA-LITERACY.RST.11-12.9
Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.
New York Physics: The Physical Setting Core Curriculum
Key Idea 5: Energy and matter interact through forces that result in changes in motion.
5.1 Explain and predict different patterns of motion of objects (e.g., linear and uniform circular motion, velocity and acceleration, momentum and inertia)
5.1i According to Newton’s First Law, the inertia of an object is directly proportional to its mass. An object remains at rest or moves with constant velocity, unless acted upon by an unbalanced force
Appendix D: AP Physics Enduring Understandings
Enduring Understanding 1C:
Objects and systems have properties of inertial mass and gravitational mass that are experimentally verified to be the same. Inertial mass is the property of an object or a system that determines how its motion changes when it interacts with other objects or systems.
Math skills needed for physics
High School students are expected to know the content of the Massachusetts Mathematics Curriculum Framework, through grade 8. These are skills from the framework that students will need:
Construct and use tables and graphs to interpret data sets.
Solve simple algebraic expressions.
Perform basic statistical procedures to analyze the center and spread of data.
Measure with accuracy and precision (e.g., length, volume, mass, temperature, time)
Metric system: Convert within a unit (e.g., centimeters to meters).
Metric system: Use common prefixes such as milli-, centi-, and kilo-.
Use scientific notation, where appropriate.
Use ratio and proportion to solve problems.
Conversion from Metric-to-Imperial (English) and Imperial-to-Metric
Determine percent error from experimental and accepted values.
Use appropriate Metric units, e.g. mass (kg); length (m); time (s); force (N); speed (m/s), etc.
Use the Celsius and Kelvin temperature scales
8th grade math skills that students should have
8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi2).
8.EE Work with radicals and integer exponents.
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8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
- Understanding exponents (8-F.1)
- Evaluate exponents (8-F.2)
- Solve equations with variable exponents (8-F.3)
- Exponents with negative bases (8-F.4)
- Exponents with decimal and fractional bases (8-F.5)
- Understanding negative exponents (8-F.6)
- Evaluate negative exponents (8-F.7)
- Multiplication with exponents (8-F.8)
- Division with exponents (8-F.9)
- Multiplication and division with exponents (8-F.10)
- Power rule (8-F.11)
- Evaluate expressions using properties of exponents (8-F.12)
- Identify equivalent expressions involving exponents (8-F.13)
- Multiply monomials (8-BB.6)
- Divide monomials (8-BB.7)
- Multiply and divide monomials (8-BB.8)
- Powers of monomials (8-BB.9)
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8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
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8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
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8.EE Understand the connections between proportional relationships, lines, and linear equations.
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8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
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8.EE Analyze and solve linear equations and pairs of simultaneous linear equations.
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8.EE.7 Solve linear equations in one variable.
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8.EE.7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
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8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
8.F Use functions to model relationships between quantities.
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8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- Write equations for proportional relationships from tables (8-I.2)
- Find the constant of proportionality from a graph (8-I.4)
- Interpret graphs of proportional relationships (8-I.8)
- Write and solve equations for proportional relationships (8-I.9)
- Find the slope of a graph (8-Y.1)
- Find the slope from two points (8-Y.2)
- Find a missing coordinate using slope (8-Y.3)
- Write a linear equation from a graph (8-Y.8)
- Write a linear equation from two points (8-Y.10)
- Rate of change (8-Z.4)
- Constant rate of change (8-Z.5)
- Write a linear function from a table (8-Z.10)
- Write linear functions: word problems (8-Z.12)
8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
8.G Understand and apply the Pythagorean Theorem.
8.G Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
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8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
from https://www.ixl.com/standards/massachusetts/math/grade-8
Selected new skills students will learn in 9th grade physics.
Dimensional analysis
Determine the correct number of significant figures.
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
Apply ratios, rates, percentages, and unit conversions in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/mL, kg/m 3, acre-feet, etc.).
National Council of Teachers of Mathematics
Students need to develop an understanding of metric units and their relationships, as well as fluency in applying the metric system to real-world situations. Because some non-metric units of measure are common in particular contexts, students need to develop familiarity with multiple systems of measure, including metric and customary systems and their relationships.
National Science Teachers Association
The efficiency and effectiveness of the metric system has long been evident to scientists, engineers, and educators. Because the metric system is used in all industrial nations except the United States, it is the position of the National Science Teachers Association that the International System of Units (SI) and its language be incorporated as an integral part of the education of children at all levels of their schooling.
Math is different from physics
Mathematics does not need to bother itself with real-world observations. It exists independently of any and all real-world measurements. It exists in a mental space of axioms, operators and rules.
Physics depends on real-world observations. Any physics theory could be overturned by a real-world measurement.
None of maths can be overturned by a real-world measurement. None of geometry can be.
Physics starts from what could be described as a romantic or optimistic notion: that the universe can be usefully described in mathematical terms; and that humans have the mental ability to assemble, and even interpret, that mathematical description.
Maths need not concern itself with how the universe actually works. Perhaps there are no real numbers, one might think it is likely that there is only a countable number of possible measurements in this universe, and nothing can form a perfect triangle or point.
Maths, including geometry, is a perfect abstraction that need bear no relation to the universe as it is.
Physics, to have any meaning, must bear some sort of correspondence to the universe as it is.
Why-is-geometry-mathematics-and-not-physics? Physics StackExchange, by EnergyNumbers
Measuring data with smartphone apps
From Google: Science Journal transforms your device into a pocket-size science tool that encourages students to explore their world. As they conduct eye-opening experiments, they’ll record observations and make new, exciting discoveries.
Science Journal With Google: Intro and website
Science Journal by Google (Android app)
Science Journal by Google. iOS (Apple) app
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On Physics Central Tamela Maciel writes:
That smartphone you carry around in your pocket all day is a pretty versatile lab assistant. It is packed with internal sensors that measure everything from acceleration to sound volume to magnetic field strength. But I’ll wager most people don’t realize what their phones can actually do. Apps like SensorLog (iOS) or AndroSensor (Android) display and record raw data from the phone’s movement, any background noises, and even the number of satellites in the neighborhood. Watching this data stream across my screen, I’m reminded just how powerful a computer my phone really is. Wrapped into one, the smartphone is an accelerometer, compass, microphone, magnetometer, photon detector, and a gyroscope. Many phones can even measure things like temperature and air pressure.
http://physicsbuzz.physicscentral.com/2015/01/your-smartphone-can-do-physics.html
Apps
Physics Toolbox Sensor Suite (Google Android)
Physics Toolbox Sensor Suite (Apple iOS)
Useful for STEM education, academia, and industry, this app uses device sensor inputs to collect, record, and export data in comma separated value (csv) format through a shareable .csv file. Data can be plotted against elapsed time on a graph or displayed digitally. Users can export the data for further analysis in a spreadsheet or plotting tool. See http://www.vieyrasoftware.net for a variety of usage ideas
SENSORS
(1) G-Force Meter – ratio of Fn/Fg (x, y, z and/or total)
(2) Linear Accelerometer – acceleration (x, y, and/or z)
(3) Gyroscope – radial velocity (x, y, and/or z)
(4) Barometer – atmospheric pressure
(5) Roller Coaster – G-Force Meter, Linear Accelerometer, Gyroscope, and Barometer
(6) Hygrometer – relative humidity
(7) Thermometer – temperature
(8) Proximeter – periodic motion and timer (timer and pendulum modes)
(9) Ruler – distance between two points
(10) Magnetometer – magnetic field intensity (x, y, z and/or total)
(11) Compass – magnetic field direction and bubble level
(12) GPS – latitude, longitude, altitude, speed, direction, number of satellites
(13) Inclinometer – azimuth, roll, pitch
(14) Light Meter – light intensity
(15) Sound Meter – sound intensity
(16) Tone Detector – frequency and musical tone
(17) Oscilloscope – wave shape and relative amplitude
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PDF Labs to use with smartphone apps
https://mobilescience.wikispaces.com/Labs
| EnglishIntroduction | EspañolIntroducción |
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https://mobilescience.wikispaces.com/Labs
Simple Harmonic Motion, and measuring Period
Smartphone Physics in the Park
Here’s a simple physics experiment you can do at your local park.
By swinging on a swing and collecting a bit of data, you can measure the length of the swing – without ever pulling out a ruler.
1. To get started, download the free SPARKvue app (or another data logger app like SensorLog or AndroSensor). Open it up and have a play.
By clicking on the measurement you want to track and then clicking on ‘Show’, you will see an graph window open with a green play button in the corner.
Click the play button and the phone will start tracking acceleration over time.
To stop recording, click the play button again.
Save your data using the share icon above the graph.
2. Find a swing.
3. Fix your phone to the swing chain with tape – or hold it really still against your chest in portrait orientation with the screen facing your body.
Since I was a bit lazy, I opted for the latter option but this makes the final data a bit messier with all the inevitable extra movement.
You want portrait orientation in order to measure the acceleration along the direction of the swing chains.
This will tell us how the centripetal acceleration from the tension in the chains changes as you swing.
4. Start swinging and recording the Y-axis acceleration, without moving your legs or twisting your body. Collect data for about 20 seconds.
5. Stop recording and have a look at your lovely sinusoidal graph.
You could try to do the next step directly from this graph.
I wanted a bigger plot, so I saved the raw data and copied it into Excel.
Here are the first 20 seconds of my swing.
Plotting the centripetal (Y-axis) acceleration against time.
You can immediately see the sine wave pattern of the swing,
and the fact that the height of the peaks is decreasing over time.
This is because all pendulums have a bit of friction and gradually come to a halt.
Keep in mind that this plot shows the change in acceleration, not velocity or position.
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| Acceleration of a swing, as measured along the chain of a swing. Data collected with SPARKvue and graphed in Excel. Credit: author, Tamela Maciel |
6. Measure the period of the swing from the graph.
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| Direction of total velocity and acceleration for a simple pendulum. Credit: Ruryk via Wikimedia Commons |
To make sense of the peaks and troughs:
think about the point mid-swing when your speed is highest.
This is when you’re closest to the ground, zooming through the swing’s resting point.
It is at this point that the force or tension along the swing chain is highest, corresponding to a maximum peak on the graph.
The minimum peaks correspond to when you are at the highest point in the swing,
and you briefly come to a stop before zooming back down the other way.
Check out The Physics Classroom site for some handy diagrams of pendulum acceleration parallel and perpendicular to the string.
Once we know what the peaks represent,
we can see that the time between two peaks is half a cycle (period).
Therefore the time between every other peak is one period.
For slightly more accuracy, I counted out the time between 5 periods (shown on the graph)
and then divided by five to get an average period of 2.65 seconds per swing.
A simple pendulum has a period that depends only on its length, l,
and the constant acceleration due to gravity, g:
I measured T = 2.65 s and know that g = 9.8 m/s/s,
so I can solve for l, the length of the swing.
I get l = 1.74 meters or 5.7 feet.
This is a reasonable value, based on my local swing set, but of course I could always double check with a ruler.
Now a few caveats: my swing and my body are not a simple pendulum, which assumes a point mass on the end of a weightless string.
I have legs and arms that stick out away from my center of mass, and the chains of the swing definitely do have mass.
So this simple period equation is not quite correct for the swing (instead I should think about the physics of the physical pendulum).
But as a first approximation, the period equation gives a pretty reasonable answer.
http://physicsbuzz.physicscentral.com/2015/01/your-smartphone-can-do-physics.html
By the way , here are comments on the above graph:
Claim: “Your graph is wrong. You write at the peaks, where the acceleration is highest, that the velocity is highest and the mid-swing-point. That is wrong. There is also a turning point with lowest velocity. The highest velocity and the mid-swing-point is where the acceleration is 0.”
Response #1
Remember, the phone is only recording the y-component of the total acceleration. At the end points the where the acceleration, a, is at maximum, but is at right angles to the chains so the y-component is zero.
This coincides with the velocity reaching zero as well.
At the mid-point where the velocity reaches maximum, the x-component of the acceleration is zero and the y-component reaches its maximum.
There is no point where the total acceleration reaches zero, only the x-component.
Response #2
My phone was measuring only the y-component of the acceleration, which from the way I held it, was only along the direction of the chains.
The maximum acceleration or force along the chains happens at the mid-point of the swing.
The minimum acceleration along the chains happens at the turning point.
So the graph is correct for the y-component acceleration.
But it would be interesting to repeat the experiment measuring the acceleration in the x-component, where the graph would look somewhat different.
Other experiments to explore
Morelessons from Vieyra software
http://www.vieyrasoftware.net/browse-lessons
Smartphones in science teaching
Mobile sensor apps for learning physics: A Google Plus community
https://plus.google.com/communities/117493961647466126964
Article: Turn Your Smartphone into a Science Laboratory
http://static.nsta.org/files/tst1509_32.pdf
Using smartphone apps to take physics day to the next level
Placing the smartphone onto a record, playing on a turntable
To study angular motion
Smartphone app contest
http://physicsday.usu.edu/Information/ContestInfo/smartphone.asp
Many more ideas https://mobilescience.wikispaces.com/Ideas
Physics Toolbox Apps by Vieyra Software http://www.vieyrasoftware.net/browse-lessons
Belmont University Summer Science Camp
Physics with Phones, Dr. Scott Hawley http://hedges.belmont.edu/~shawley/PhonePhysics.pdf
References
Familiarizing Students with the Basics of a Smartphone’s Internal Sensors
Colleen Lanz Countryman, Phys. Teach. 52, 557 (2014)
http://dx.doi.org/10.1119/1.4902204
http://scitation.aip.org/content/aapt/journal/tpt/52/9/10.1119/1.4902204
Full text of article, in PDF format
http://scitation.aip.org/content/aapt/journal/tpt/52/3/10.1119/1.4865529
http://scitation.aip.org/content/aapt/journal/tpt/52/5/10.1119/1.4872422
http://iopscience.iop.org/0143-0807/35/4/045013/article
http://scitation.aip.org/content/aapt/journal/tpt/52/8/10.1119/1.4897595
Surface tension
Surface tension
New section: to be written
MythBusters: Buried Alive & Falling off of a bridge 10/2003
(1) If a person is falling off a bridge, can they save themselves by throwing a hammer ahead of them to break the surface tension of the water prior to their own impact?
How can cliff divers survive their dives?
http://physics.stackexchange.com/questions/9059/jumping-into-water
objects in motion
Kinematics is the study of objects in motion. It allow us to study displacement, velocity, time and acceleration. Check out our introductory lesson on this study of motion here:
https://kaiserscience.wordpress.com/physics/kinematics/
The new Hi-Fi debate and the science of sound
Music today is listened to almost exclusively through digital compression. The most common digital compression format is called MP3 (MPEG-2 Audio Layer III.) A competing digital compression format is FLAC (Free Lossless Audio Codec.)
Many audio enthusiasts believe that FLAC provides significantly more accurate sound reproduction, which can be heard by listeners. Most audio enthusiasts, however, hold that more is not always better, and that the FLAC format does not produce any audible benefits for listeners. PONO is a highly publicized FLAC-based digital music player, a high-tech MP3 player of sorts, that promises significantly better music reproduction.
Both FLAC proponents and skeptics use math and physics based arguments to explain their position. Here’s a brief overview, with links to articles that have more detail.
Our physics article on sound :
Sources of sound
The physics of sound.
On Cnet, Stephen Shankland writes about the science of sound, in the latest generation of audio devices:
Pono Music’s roaring success on Kickstarter, raising $4.3 million so far, shows that thousands of people believe better audio quality is worth paying for. The company — backed by star musician Neil Young and selling a $400 digital audio player along with accompanying music — promises people will hear a difference between Pono Music and ordinary music that’s “surprising and dramatic.” The company’s promise is based in part on music files that can contain more data than not only conventional MP3 files, but also compact discs.
… Just as some skeptics think 4K TVs is wasted on human eyes, which mostly can’t perceive an image quality improvement over mainstream HD 1080p under normal viewing conditions, others think CD audio technology that’s now more than three decades old is actually very well matched to human hearing abilities. For playback, they’re fine with two key aspects of CD audio encoding: its 16-bit dynamic range, which means audio is measured with a precision of 65,536 levels, and its 44.1kHz “sampling” frequency that means those levels are measured 44,100 times each second.
“From a scientific point of view, there’s no need to go beyond,” said Bernhard Grill, leader of Fraunhofer Institute’s audio and multimedia division and one of the creators of the MP3 and AAC audio compression formats. “It’s always nice to have higher numbers on the box, and 24 bits sounds better than 16 bits. But practically, I think people should much more worry about speakers and room acoustics.”
Pono’s recordings will range from CD-quality 16-bit/44.1kHz to 24-bit/192kHz “ultra-high resolution.” To house the data, Pono follows in the footsteps of the digital audiophile industry by sticking with a file format called FLAC (Free Lossless Audio Codec) that compresses files for smaller sizes but not to the degree of alternatives including MP3 and AAC that throw away some of the original data. The company also is betting its success on a player with better electronics and a catalog of HD music designed to let listeners hear music true to its original sound in the recording studio.
…The idea is that more data allows a higher dynamic range — the span between the loudest and quietest passages of music — and comes closer to the detail of live, original sound….
A prominent part of the case against high-resolution audio is a 2007 study by E. Brad Meyer and David Moran of the Boston Audio Society – that concluded listeners couldn’t tell the difference between SACD and DVD-A music on the one hand and CD-quality versions of the same recordings on the other.
Results of a blind audio test. By E. Brad Meyer and David R. Moran
In that experiment’s 554 tests, listeners correctly identified when a SACD or DVD-A recording compared to a CD only 49.8 percent of the time — in other words, they didn’t do better than randomly guessing.…
Another high-profile non-believer is Christopher “Monty” Montgomery, an engineer who writes codec software for the Xiph.Org Foundation and who works for Firefox developer Mozilla. The most prominent part of his effort is a video arguing that CD quality sound is good enough. Montgomery’s video, illustrated with lucid demonstrations and backed by a blog post, persuasively debunks misconceptions such as the idea that encoding music digitally reduces it to a series of jagged stairsteps instead of the original smooth curves.
24/192 Music Downloads …and why they make no sense:
Video on 24/192 music downloads: D/A and A/D | Digital Show and Tell (Monty Montgomery @ xiph.org)
Montgomery and his allies have yet to persuade everyone on two points, including the idea that 16-bit resolution and 44.1kHz is sufficient.
“Monty is wrong. Twenty-four bits does matter — but for a very small sliver of the music business,” said Mark Waldrep, an audio engineer who’s founder and chief executive of AIX Records and iTrax.com and who focuses on high-resolution audio — including efforts of his own to debunk some claims. And of the sampling frequency he said, “I’d rather err on having those frequencies in the signal rather than assuming we don’t need them.”
But Grill thinks any purported benefit would be lost in the real world. “The limiting factor is the loudspeaker, the room acoustics, and the human ear,” he said.
From “The Digital Myth: Why Digital Audio Sounds Better Than You Think”
By Gordon Reid
Now, perhaps the greatest myth in digital audio relates to the misconception that digital signals are shaped like staircases, and that much of their ‘brittleness’ is a consequence of the steps. This is nonsense. Digital signals are not shaped like anything — they are sequences of numbers. Unfortunately, the type of representation in diagram 8 has led many people to confuse graphics with reality.
Let’s be clear. When the samples in a digital signal are converted back into an analogue signal, they pass through a device called a reconstruction filter. This is the process that makes the Sampling Theorem work in the real world. If there are enough samples and they are of sufficient resolution, the signal that emerges is not only smooth but virtually identical to the analogue signal from which the samples were originally derived. Of course, it’s possible to design a poor reconstruction filter that introduces unwanted changes and artifacts but, again, this is an engineering consideration, not a deficiency in the concept itself.
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Source: Sound bite: Despite Pono’s promise, experts pan HD audio
Another great article on this topic
What is FLAC? The high-def MP3 explained. C|Net
Here is a detailed physics experiment showing an analysis of FLAC and MP3 audio files. The result is that there is no audible difference between any of these formats! Each is equally good. There are, however, significant problems in how iTunes engineers (and probably engineers from other companies) are choosing to compress original recordings. many times they make choices which negatively affect the music. However, those errors are independent of whether one ends up using MP3, FLAC or other formats.
Also see “There are no “stair steps” in digital audio ! What The Matrix can teach us about “resolution””
There are no “stair steps” in digital audio ! What The Matrix can teach us about “resolution”
The nature of reality
What is the ultimate nature of reality? This the core questions of physics, as well as of classical, rationalist philosophy. We now know that this question relates to interpretations of quantum mechanics.
“Those are the kind of questions in play when a physicist tackles the dry-sounding issue of, “what is the correct interpretation of quantum mechanics?” About 80 years after the original flowering of quantum theory, physicists still don’t agree on an answer. Although quantum mechanics is primarily the physics of the very small – of atoms, electrons, photons and other such particles – the world is made up of those particles. If their individual reality is radically different from what we imagine then surely so too is the reality of the pebbles, people and planets that they make up.”
The Many Interpretations of Quantum Mechanics, Graham P. Collins, Scientific American, November 19, 2007
To what can we compare our knowledge of the universe?
The allegory of Plato’s cave
The Allegory of the Cave was presented by the Greek philosopher Plato the Republic (380 BCE) He retells an analogy created by Socrates, about people who think that they know the true nature of reality – however, as the analogy progresses, we find that they have no idea what the real world is like at all.
The idea is that most people don’t actually understand our own real world – and that we never will without philosophical and scientific inquiry.
Socrates says to imagine a cave where people have been imprisoned from childhood. They are chained so that their legs and necks are fixed, forcing them to gaze at the wall in front of them, and not look around at the cave, each other, or themselves
Behind the prisoners is a fire, and between the fire and the prisoners is a raised walkway with a low wall, behind which people walk carrying objects or puppets “of men and other living things”
The masters walk behind the wall – so their bodies do not cast shadows for the prisoners to see. But the objects they carry cast shadows. The prisoners can’t see anything behind them : they only able see the shadows cast on the cave wall in front of them. The sounds of people talking echo off the wall, so the prisoners falsely believe these sounds come from the shadows.
The shadows constitute reality for the prisoners – because they have never seen anything else. They do not realize that what they see are shadows of objects in front of a fire, much less that these objects are inspired by real living things outside the cave
The philosopher (or scientist) is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all, for he can perceive the true form of reality – rather than the mere shadows seen by the prisoners.
Plato then supposes that one prisoner is freed: he turns to see the fire. The light would hurt his eyes and make it hard for him to see the objects that are casting the shadows. If he is told that what he saw before was not real but instead that the objects he is now struggling to see are, he would not believe it. In his pain the freed prisoner would turn away and run back to what he is accustomed to, the shadows of the carried objects.
Plato continues: “suppose…that someone should drag him…by force, up the rough ascent, the steep way up, and never stop until he could drag him out into the light of the sun.” The prisoner would be angry and in pain, and this would only worsen when the light of the sun overwhelms his eyes and blinds him.” The sunlight represents the new knowledge that the freed prisoner is experiencing.
Slowly, his eyes adjust to the light of the sun. First he can only see shadows. Gradually he can see reflections of people and things in water, and then later see the people and things themselves. Eventually he is able to look at the stars and moon at night until finally he can look upon the sun itself (516a). Only after he can look straight at the sun “is he able to reason about it” and what it is.
- adapted from “Allegory of the Cave.” Wikipedia, The Free Encyclopedia. 29 May. 2016. Web. 3 Jun. 2016
Another illustration of Plato’s cave.
Are the laws of physics really absolute?
One of the major goals of physics is to emerge from the relative ignorance of the cave, and venture out into an understanding of the real world – how our universe really works.
We have made remarkable progress in doing so – everything we have learned in classical physics over the last two millennia is part of the human adventure.
What we have learned is, in an important sense, “real.” Physics lets us ask specific questions and then use math to make specific answers. We then compare our predictions to the way that universe really works.
Yet we need to be careful – we could make the mistake of using physics equations as if they are absolutely true. Yes, they certainly are true in the sense that they work. But are these math equations the absolute truth themselves – or are they really emerging from a deeper phenomenon? See The laws of physics are emergent phenomenon.
Is nature a simulation?
The simulation hypothesis proposes that our reality is actually some kind of super detailed computer simulation. This hypothesis relies on the development of a simulated reality, a proposed technology that would seem realistic enough to convince its inhabitants. The hypothesis has been a central plot device of many science fiction stories and films.
Simulation hypothesis (Wikipedia)
Video Why Elon Musk says we’re living in a simulation: YouTube, Vox
Elon Musk thinks we’re characters in a computer simulation. He might be right.
Is the Universe a Simulation? Scientists Debate
Nick Bostrom: Are you living in a computer simulation?
Is the universe a hologram?
The holographic principle is a principle of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region—preferably a light-like boundary like a gravitational horizon.
First proposed by Gerard ‘t Hooft, it was given a precise string-theory interpretation by Leonard Susskind who combined his ideas with previous ones of ‘t Hooft and Charles Thorn.
As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. In a larger sense, the theory suggests that the entire universe can be seen as two-dimensional information on the cosmological horizon. – Wikipedia
Our Universe May Be a Giant Hologram
Study reveals substantial evidence of holographic universe
Space’The Holy Grail for Physicists’: First Evidence Universe is a Hologram Uncovered
To learn more about quantum mechanics
The Cosmic Code: Quantum Physics as the Language of Nature, Heinz R. Pagels
One of the best books on quantum mechanics for general readers. Heinz Pagels, an eminent physicist and science writer, discusses the core concepts without resorting to complicated mathematics. He covers the development of quantum physics. And although this is an intellectually challenging topics, he is one of the few popular physics writers to discuss the development and meaning of Bell’s theorem.
Quantum Reality: Beyond the New Physics, Nick Herbert
Herbert brings us from the “we’ve almost solved all of physics!” era of the early 1900s through the unexpected experiments which forced us to develop a new and bizarre model of the universe, quantum mechanics. He starts with unexpected results, such as the “ultraviolet catastrophe,” and then brings us on a tour of the various ways that modern physicists developed quantum mechanics.
And note that there isn’t just one QM theory – there are several! Werner Heisenberg initially developed QM using a type of math called matrix mechanics, while Erwin Schrödinger created an entirely different way of explaining things using wave mechanics. Yet despite their totally different math languages – we soon discovered that both ways of looking at the world were logically equivalent, and made the same predictions. Herbert discussed the ways that Paul Dirac and Richard Feynman saw QM, and he describes eight very different interpretations of quantum mechanics, all of which nonetheless are consistent with observation…
In Search of Schrödinger’s Cat: Quantum Physics and Reality, John Gribbon
“John Gribbin takes us step by step into an ever more bizarre and fascinating place, requiring only that we approach it with an open mind. He introduces the scientists who developed quantum theory. He investigates the atom, radiation, time travel, the birth of the universe, superconductors and life itself. And in a world full of its own delights, mysteries and surprises, he searches for Schrodinger’s Cat – a search for quantum reality – as he brings every reader to a clear understanding of the most important area of scientific study today – quantum physics.”
External links
The Many Interpretations of Quantum Mechanics, Scientific American
Tom’s Top 10 interpretations of quantum mechanics
Learning Standards
SAT Subject Test: Physics
Quantum phenomena, such as photons and photoelectric effect – Atomic, such as the Rutherford and Bohr models, atomic energy levels, and atomic spectra. Nuclear and particle physics, such as radioactivity, nuclear reactions, and fundamental particles.
AP Physics Curriculum Framework
Essential Knowledge 1.D.1: Objects classically thought of as particles can exhibit properties of waves.
a. This wavelike behavior of particles has been observed, e.g., in a double-slit experiment using elementary particles.
b. The classical models of objects do not describe their wave nature. These models break down when observing objects in small dimensions.
Learning Objective 1.D.1.1:
The student is able to explain why classical mechanics cannot describe all properties of objects by articulating the reasons that classical mechanics must be refined and an alternative explanation developed when classical particles display wave properties.
Essential Knowledge 1.D.2: Certain phenomena classically thought of as waves can exhibit properties of particles.
a. The classical models of waves do not describe the nature of a photon.
b. Momentum and energy of a photon can be related to its frequency and wavelength.
Content Connection: This essential knowledge does not produce a specific learning objective but serves as a foundation for other learning objectives in the course.
A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Electromagnetic radiation can be modeled as a wave of changing electric and magnetic fields or as particles called photons. The wave model is useful for explaining many features of electromagnetic radiation, and the particle model explains other features. Quantum theory relates the two models…. Knowledge of quantum physics enabled the development of semiconductors, computer chips, and lasers, all of which are now essential components of modern imaging, communications, and information technologies
Scaling and biophysics
From Math Bench Biology Modules:
Scaling examines how form and function change as organisms get larger – in other words, how do biological features scale across size? Do they change in meaningful ways as organisms get bigger or smaller? Of course, you can’t even ask these types of questions without having a way of measuring how relationships change mathematically.
Why study these relationships? Well, if you understand how form or functions change as organisms get bigger or smaller, it is possible to learn something fundamental about what underlies the processes or learn about what factors place evolutionary limits on organism growth and adaptations. For instance, determining at what size arthropods can no longer support the weight of their exoskeleton gives us clues about the limits of their growth.
Let’s use a concrete example so you’ll know what we mean. Here is some data on body size and metabolic rate for mammals….
- metabolic rate increases as animals get bigger. That’s because we are specifically interested in total energy consumed (here measured through oxygen consumption). Of course, bigger animals will use more oxygen than smaller ones (think about how big a breath a lion takes compared to a mouse).
- But look at the values adjusted for body size (the last value listed for each species). Mice use a lot more oxygen per gram than a lion. This means that lions use oxygen more efficiently than mice.
- As mammals get bigger, this increase in efficiency is not linear (notice how the steepness of the slope decreases as size increases).
- This means that metabolism does not scale linearly with body size.
“Who cares?” Well, it turns out that how metabolism (and other factors) scales with body size can give important information about which factors are most important in limiting these biological functions. If we can understand that, we understand a lot more about biology!
Math Bench Biology Modules, University of Maryland: Scaling and Power laws
– – –
How scaling affects biology
There are species of animals such as the deer and the elk that are closely related but of different size. Galileo took notice that the bones of the elk are not just proportionally thicker to the bones of the deer – but instead the elk’s bones are even much thicker.
The elk’s bone has to be much thicker to lower the stress in the bone below the breaking point of the bone. Even so, elk and all the other large vertebrates are still more likely of breaking their bones than the more active smaller animals.
http://www.dinosaurtheory.com/scaling.html
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External links
http://www.av8n.com/physics/scaling.htm
The Principle of Scale: A fundamental lesson they failed to teach us at school
The Biology of B-Movie Monsters, Michael C. LaBarbera
Scaling: Why Giants Don’t Exist, Michael Fowler, UVa 10/12/06
See the speed of sound
from http://nerdist.com/watch-the-speed-of-sound-ripple-through-queen-fans-at-live-aid-1985/
By Kyle Hill
Queen’s performance at Live Aid 1985 was only 20 minutes, but it lived on forever. Propped up by an infectiously enthusiastic Freddie Mercury and Brian May’s screaming guitar, the performance has gone on to be regarded as one of the best rock concerts of all time. If you haven’t seen it (above), take a little break to appreciate this supernova of a show.
The other thing you’ll notice is the crowd. They are in near-perfect unison, signing along and gesturing with Mercury’s mesmerizing gyrations. The audience was so in sync, in fact, that the only thing separating their movement was the speed of sound itself.
Watch the GIF below. Can you see the rapid, pulsing ripples that radiate through the fist-pumping masses? This is much faster than an organized wave like you’d see during a baseball game. No one is coordinating the movement, so what is going on?
A little math might help. The venue, Wembley Stadium, goes about 115 yards deep. The time it takes wave in the crowd to go from Mercury to the back of the stadium is maybe 0.3 seconds (a rough approximation). Dividing these two values results in a wave velocity of 340 meters per second. That’s almost exactly Mach 1, or the speed of sound.
Think about that! What you are actually seeing is thousands of people reacting reflexively the show, and what pops out is a wave moving at Mach 1. The people are a visual representation of Queen’s music–a unbridled manifestation of sound. It could only have happened at a show like this, yet another testament to Mercury and the band.
What are fields?
What are gravitational fields? Electric fields? Magnetic fields? the electromagnetic field?
What do physicists mean by the term “field”?
Let’s start by seeing how we describe stuff – anything – that exists in space.
Imagine our universe is flat. We could describe what’s going on at any point by defining a 2D grid, like graph paper.
We could use a 2D field to show the temperature at any location.
2-d field representing wind speed
But our universe is 3D.
That means we need 3 dimensions – 3 axes – to describe any point in space.
This image shows an empty universe, with nothing in it.
Now imagine a three dimensional grid extending through our world.
(indoor 3D climbing array by Croatian-Austrian artists Sven Jonke, Christoph Katzler and Nikola Radeljković.)
Now imagine this 3D field extending through all of space – and that this isn’t just a mathematical tool.
All of space is filled with several different fields.
These fields exist everywhere – on Earth and in space, between you and me.
They are within our bodies, and they extend to the visible boundaries of the universe
These fields are not a hypothesis or idea; they are absolutely real.
Our universe is made of two basic things:
particles (like protons, electrons, neutrons)
fields
Can we make these invisible fields visible?
Yes! Throw a handful of magnetic iron filings around a suspended magnet.
Magnify this image, look carefully:
We see each tiny pieces of metal pulled along the (otherwise invisible) magnetic field lines.
Now consider a horseshoe magnet – it’s a 3D object, with a 3D magnetic field invisibly emanating from it.
Toss a few thousand small iron filings at it – and suddenly those invisible fields become apparent!
At every single point in our universe there is an electric field.
And at every single point in our universe there is also a magnetic field, a gravitational field, and more!
Our whole planet creates, and is surrounded by, a magnetic field – and we can easily see its effects!
Just walk around with a compass, and this mysterious invisible field clearly grabs small slivers of metal, and orients them N/S.
That’s not a “concept” – these fields are real.
You can see the effect of invisible fields with the magnetometer built into your cell phone (yes, there’s an app for that!)
Physics Toolbox Magnetometer: google play
Now walk around indoors, and then outdoors – literally place you stand is filled with electric fields, magnetics fields.
Oh, and both are difference aspects of one greater field, the electro-magnetic field.
All of space, everywhere, is filled with this!
Oh, and there is more. You know how regular matter – atoms – has mass? How is that possible? Why do any atoms have mass at all? Why do atoms have the mass that they have, and why can’t atoms move at the speed of light?
While we won’t get into the details here, the answers to those questions come from the fact that there is yet another field permeating the entire universe: the Higgs field.
Here are visualizations of how particles moving through space interact with the Higgs field.
and
Putting it all together
As you move around our world, as space probes fly through outer space, it turns out that every point in space has multiple fields, at the same point, at the same time.
Each one of these fields has a separate value, which changes over time. Although we can’t see fields directly, we know they are real, and we certainly can “feel” them, measure their effect on particles as they pass through them.
If we had a God’s eye view of the universe, everywhere we look we would see these many different fields, changing in time.
Learning Standards
NGSS
HS-PS2-4 – Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.
Disciplinary Core Ideas PS2.B: Types of Interactions
Newton’s law of universal gravitation and Coulomb’s law provide the mathematical models to describe and predict the effects of gravitational and electrostatic forces between distant objects.
Forces at a distance are explained by fields (gravitational, electric, and magnetic) permeating space that can transfer energy through space. Magnets or electric currents cause magnetic fields; electric charges or changing magnetic fields cause electric fields.
HS-PS3-2. Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy associated with the relative positions of particles (objects).
DCI PS3.A: Definitions of Energy
These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration (relative position of the particles).
In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.
HS-PS2-4. Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.
HS-PS2-5. Plan and conduct an investigation to provide evidence that an electric current can produce a magnetic field and that a changing magnetic field can produce an electric current.
DCI PS2.B: Types of Interactions – Forces at a distance are explained by fields (gravitational, electric, and magnetic) permeating space that can transfer energy through space. Magnets or electric currents cause magnetic fields; electric charges or changing magnetic fields cause electric fields. (HS-PS2-4),(HS-PS2-5)
AP Physics Learning Objectives
Essential Knowledge 2.A.1: A vector field gives, as a function of position (and perhaps time), the value of a physical quantity that is described by a vector.
a. Vector fields are represented by field vectors indicating direction and magnitude.
b. When more than one source object with mass or electric charge is present, the field value can be determined by vector addition.
c. Conversely, a known vector field can be used to make inferences about the number, relative size, and location of sources.
Content Connection: This essential knowledge does not produce a specific learning objective but serves as a foundation for other learning objectives in the course.
Essential Knowledge 2.A.2: A scalar field gives, as a function of position (and perhaps time), the value of a physical quantity that is described by a scalar. In Physics 2, this should include electric potential.
a. Scalar fields are represented by field values.
b. When more than one source object with mass or charge is present, the scalar field value can be determined by scalar addition.
c. Conversely, a known scalar field can be used to make inferences about the number, relative size, and location of sources.
Content Connection: This essential knowledge does not produce a specific learning objective but serves as a foundation for other learning objectives in the course.
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