Having Had No Predecessor to Imitate, He Had No Successor Capable of Imitating Him | Fantastic Anachronism

9 weeks ago by nhaliday

on Homer

essay
critique
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literature
history
iron-age
the-classics
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canon
language
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9 weeks ago by nhaliday

Build your own X: project-based programming tutorials | Hacker News

november 2019 by nhaliday

https://news.ycombinator.com/item?id=21430321

https://www.reddit.com/r/programming/comments/8j0gz3/build_your_own_x/

hn
commentary
repo
paste
programming
minimum-viable
frontier
allodium
list
links
roadmap
accretion
quixotic
🖥
interview-prep
system-design
move-fast-(and-break-things)
graphics
SIGGRAPH
vr
p2p
project
blockchain
cryptocurrency
bitcoin
bots
terminal
dbs
virtualization
frontend
web
javascript
frameworks
libraries
facebook
pls
c(pp)
python
dotnet
jvm
ocaml-sml
haskell
networking
systems
metal-to-virtual
deep-learning
os
physics
mechanics
simulation
automata-languages
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huge-data-the-biggest
strings
computer-vision
multi
reddit
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detail-architecture
https://www.reddit.com/r/programming/comments/8j0gz3/build_your_own_x/

november 2019 by nhaliday

Mars Direct | West Hunter

september 2019 by nhaliday

Send Mr Bezos. He even looks like a Martian.

--

Throw in Zuckerberg and it’s a deal…

--

We could send twice as many people half-way to Mars.

--

I don’t think that the space station has been worth anything at all.

As for a lunar base, many of the issues are difficult and one ( effects of low-gee) is probably impossible to solve.

I don’t think that there are real mysteries about what is needed for a kind-of self-sufficient base – it’s just too hard and there’s not much prospect of a payoff.

That said, there may be other ways of going about this that are more promising.

--

Venus is worth terraforming: no gravity problems. Doable.

--

It’s not impossible that Mars might harbor microbial life – with some luck, life with a different chemical basis. That might be very valuable: there are endless industrial processes that depend upon some kind of fermentation.

Why, without acetone fermentation, there might not be a state of Israel.

--

If we used a reasonable approach, like Orion, I think that people would usefully supplement those robots.

https://westhunt.wordpress.com/2019/01/11/the-great-divorce/

Jeff Bezos isn’t my favorite guy, but he has ability and has built something useful. And an ugly, contested divorce would be harsh and unfair to the children, who have done nothing wrong.

But I don’t care. The thought of tens of billions of dollars being spent on lawyers and PIs offer the possibility of a spectacle that will live forever, far wilder than the antics of Nero or Caligula. It could make Suetonius look like Pilgrim’s Progress.

Have you ever wondered whether tens of thousands of divorce lawyers should be organized into legions or phalanxes? This is our chance to finally find out.

west-hunter
scitariat
commentary
current-events
trump
politics
troll
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expansionism
frontier
cost-benefit
ideas
speculation
roots
deep-materialism
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wild-ideas
gravity
barons
amazon
facebook
sv
tech
government
debate
critique
physics
mechanics
robotics
multi
lol
law
responsibility
drama
beginning-middle-end
direct-indirect
--

Throw in Zuckerberg and it’s a deal…

--

We could send twice as many people half-way to Mars.

--

I don’t think that the space station has been worth anything at all.

As for a lunar base, many of the issues are difficult and one ( effects of low-gee) is probably impossible to solve.

I don’t think that there are real mysteries about what is needed for a kind-of self-sufficient base – it’s just too hard and there’s not much prospect of a payoff.

That said, there may be other ways of going about this that are more promising.

--

Venus is worth terraforming: no gravity problems. Doable.

--

It’s not impossible that Mars might harbor microbial life – with some luck, life with a different chemical basis. That might be very valuable: there are endless industrial processes that depend upon some kind of fermentation.

Why, without acetone fermentation, there might not be a state of Israel.

--

If we used a reasonable approach, like Orion, I think that people would usefully supplement those robots.

https://westhunt.wordpress.com/2019/01/11/the-great-divorce/

Jeff Bezos isn’t my favorite guy, but he has ability and has built something useful. And an ugly, contested divorce would be harsh and unfair to the children, who have done nothing wrong.

But I don’t care. The thought of tens of billions of dollars being spent on lawyers and PIs offer the possibility of a spectacle that will live forever, far wilder than the antics of Nero or Caligula. It could make Suetonius look like Pilgrim’s Progress.

Have you ever wondered whether tens of thousands of divorce lawyers should be organized into legions or phalanxes? This is our chance to finally find out.

september 2019 by nhaliday

The Physics of Information Processing Superobjects: Daily Life Among the Jupiter Brains

nibble pdf study article essay ratty bostrom physics lower-bounds interdisciplinary computation frontier singularity civilization communication time phys-energy thermo entropy-like lens intelligence futurism philosophy software hardware enhancement no-go data scale magnitude network-structure structure complex-systems concurrency density bits retention mechanics electromag quantum quantum-info speed information-theory measure chemistry gravity relativity the-world-is-just-atoms dirty-hands skunkworks gedanken ideas hard-tech nitty-gritty intricacy len:long spatial whole-partial-many frequency neuro internet web trivia cocktail humanity composition-decomposition instinct reason illusion the-self psychology cog-psych dennett within-without signal-noise coding-theory quotes scifi-fantasy fiction giants death long-short-run janus eden-heaven efficiency finiteness iteration-recursion cycles nietzschean big-peeps examples

april 2018 by nhaliday

nibble pdf study article essay ratty bostrom physics lower-bounds interdisciplinary computation frontier singularity civilization communication time phys-energy thermo entropy-like lens intelligence futurism philosophy software hardware enhancement no-go data scale magnitude network-structure structure complex-systems concurrency density bits retention mechanics electromag quantum quantum-info speed information-theory measure chemistry gravity relativity the-world-is-just-atoms dirty-hands skunkworks gedanken ideas hard-tech nitty-gritty intricacy len:long spatial whole-partial-many frequency neuro internet web trivia cocktail humanity composition-decomposition instinct reason illusion the-self psychology cog-psych dennett within-without signal-noise coding-theory quotes scifi-fantasy fiction giants death long-short-run janus eden-heaven efficiency finiteness iteration-recursion cycles nietzschean big-peeps examples

april 2018 by nhaliday

Eternity in six hours: intergalactic spreading of intelligent life and sharpening the Fermi paradox

march 2018 by nhaliday

We do this by demonstrating that traveling between galaxies – indeed even launching a colonisation project for the entire reachable universe – is a relatively simple task for a star-spanning civilization, requiring modest amounts of energy and resources. We start by demonstrating that humanity itself could likely accomplish such a colonisation project in the foreseeable future, should we want to, and then demonstrate that there are millions of galaxies that could have reached us by now, using similar methods. This results in a considerable sharpening of the Fermi paradox.

pdf
study
article
essay
anthropic
fermi
space
expansionism
bostrom
ratty
philosophy
xenobio
ideas
threat-modeling
intricacy
time
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🔬
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questions
paradox
risk
physics
engineering
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volo-avolo
dirty-hands
ai
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robotics
duplication
iteration-recursion
von-neumann
data
scale
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hard-tech
ems
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phys-energy
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analysis
spock
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speed
gedanken
nibble
march 2018 by nhaliday

How do you measure the mass of a star? (Beginner) - Curious About Astronomy? Ask an Astronomer

december 2017 by nhaliday

Measuring the mass of stars in binary systems is easy. Binary systems are sets of two or more stars in orbit about each other. By measuring the size of the orbit, the stars' orbital speeds, and their orbital periods, we can determine exactly what the masses of the stars are. We can take that knowledge and then apply it to similar stars not in multiple systems.

We also can easily measure the luminosity and temperature of any star. A plot of luminocity versus temperature for a set of stars is called a Hertsprung-Russel (H-R) diagram, and it turns out that most stars lie along a thin band in this diagram known as the main Sequence. Stars arrange themselves by mass on the Main Sequence, with massive stars being hotter and brighter than their small-mass bretheren. If a star falls on the Main Sequence, we therefore immediately know its mass.

In addition to these methods, we also have an excellent understanding of how stars work. Our models of stellar structure are excellent predictors of the properties and evolution of stars. As it turns out, the mass of a star determines its life history from day 1, for all times thereafter, not only when the star is on the Main Sequence. So actually, the position of a star on the H-R diagram is a good indicator of its mass, regardless of whether it's on the Main Sequence or not.

nibble
q-n-a
org:junk
org:edu
popsci
space
physics
electromag
measurement
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gravity
cycles
oscillation
temperature
visuo
plots
correlation
metrics
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measure
methodology
We also can easily measure the luminosity and temperature of any star. A plot of luminocity versus temperature for a set of stars is called a Hertsprung-Russel (H-R) diagram, and it turns out that most stars lie along a thin band in this diagram known as the main Sequence. Stars arrange themselves by mass on the Main Sequence, with massive stars being hotter and brighter than their small-mass bretheren. If a star falls on the Main Sequence, we therefore immediately know its mass.

In addition to these methods, we also have an excellent understanding of how stars work. Our models of stellar structure are excellent predictors of the properties and evolution of stars. As it turns out, the mass of a star determines its life history from day 1, for all times thereafter, not only when the star is on the Main Sequence. So actually, the position of a star on the H-R diagram is a good indicator of its mass, regardless of whether it's on the Main Sequence or not.

december 2017 by nhaliday

Fisica ingenua o Fisica di senso comune

november 2017 by nhaliday

naive physics

cited by Lucio Russo

you can plug pdf into google translate here: https://translate.google.com/?tr=f&hl=en

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pdf
presentation
physics
mechanics
psychology
cog-psych
heuristic
science
west-hunter
scitariat
lens
the-classics
slides
foreign-lang
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multi
intuition
feynman
giants
neurons
biases
error
gotchas
instinct
init
teaching
tutoring
cited by Lucio Russo

you can plug pdf into google translate here: https://translate.google.com/?tr=f&hl=en

november 2017 by nhaliday

[chao-dyn/9907004] Quasi periodic motions from Hipparchus to Kolmogorov

november 2017 by nhaliday

The evolution of the conception of motion as composed by circular uniform motions is analyzed, stressing its continuity from antiquity to our days.

nibble
preprint
papers
math
physics
mechanics
space
history
iron-age
mediterranean
the-classics
science
the-trenches
fourier
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composition-decomposition
series
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waves
org:mat
november 2017 by nhaliday

The Moon And Tides

november 2017 by nhaliday

Why does the Moon produce TWO water tides on the Earth and not just one?

"It is intuitively easy to understand why the gravitational pull of the Moon should produce a water tide on the Earth in the part of the ocean closest to the moon along the line connecting the center of the Moon with the center of the Earth. But in fact not one but TWO water tides are produced under which the Earth rotates every day to produce about two high tides and two low tides every day. How come?

It is not the gravitational force that is doing it, but the change in the gravitational force across the body of the Earth. If you were to plot the pattern of the Moon's 'tidal' gravitational force added to the Earth's own gravitational force, at the Earth's surface, you would be able to resolve the force vectors at different latitudes and longitudes into a radial component directed towards the Earth's center, and a component tangential to the Earth's surface. On the side nearest the moon, the 'differential' gravitational force is directed toward the Moon showing that for particles on the Earth's surface, they are being tugged slightly towards the Moon because the force of the Moon is slightly stronger at the Earth's surface than at the Earth's center which is an additional 6300 kilometers from the Moon. On the far side of the Earth, the Moon is tugging on the center of the Earth slightly stronger than it is on the far surface, so the resultant force vector is directed away from the Earth's center.

The net result of this is that the Earth gets deformed into a slightly squashed, ellipsoidal shape due to these tidal forces. This happens because if we resolve the tidal forces at each point on the Earth into a local vertical and horizontal component, the horizontal components are not zero, and are directed towards the two points along the line connecting the Earth and the Moon's centers. These horizontal forces cause rock and water to feel a gravitational force which results in the flow of rock and water into the 'tidal bulges'. There will be exactly two of these bulges. At exactly the positions of the tidal bulges where the Moon is at the zenith and at the nadir positions, there are no horizontal tidal forces and the flow stops. The water gets piled up, and the only effect is to slightly lower the weight of the water along the vertical direction.

Another way of thinking about this is that the gravitational force of the Moon causes the Earth to accelerate slightly towards the Moon causing the water to get pulled towards the Moon faster than the solid rock on the side nearest the Moon. On the far side, the solid Earth 'leaves behind' some of the water which is not as strongly accelerated towards the Moon as the Earth is. This produces the bulge on the 'back side' of the Earth."- Dr. Odenwald's ASK THE ASTRONOMER

org:junk
nibble
space
physics
mechanics
cycles
navigation
gravity
marginal
oceans
explanation
faq
objektbuch
rhythm
"It is intuitively easy to understand why the gravitational pull of the Moon should produce a water tide on the Earth in the part of the ocean closest to the moon along the line connecting the center of the Moon with the center of the Earth. But in fact not one but TWO water tides are produced under which the Earth rotates every day to produce about two high tides and two low tides every day. How come?

It is not the gravitational force that is doing it, but the change in the gravitational force across the body of the Earth. If you were to plot the pattern of the Moon's 'tidal' gravitational force added to the Earth's own gravitational force, at the Earth's surface, you would be able to resolve the force vectors at different latitudes and longitudes into a radial component directed towards the Earth's center, and a component tangential to the Earth's surface. On the side nearest the moon, the 'differential' gravitational force is directed toward the Moon showing that for particles on the Earth's surface, they are being tugged slightly towards the Moon because the force of the Moon is slightly stronger at the Earth's surface than at the Earth's center which is an additional 6300 kilometers from the Moon. On the far side of the Earth, the Moon is tugging on the center of the Earth slightly stronger than it is on the far surface, so the resultant force vector is directed away from the Earth's center.

The net result of this is that the Earth gets deformed into a slightly squashed, ellipsoidal shape due to these tidal forces. This happens because if we resolve the tidal forces at each point on the Earth into a local vertical and horizontal component, the horizontal components are not zero, and are directed towards the two points along the line connecting the Earth and the Moon's centers. These horizontal forces cause rock and water to feel a gravitational force which results in the flow of rock and water into the 'tidal bulges'. There will be exactly two of these bulges. At exactly the positions of the tidal bulges where the Moon is at the zenith and at the nadir positions, there are no horizontal tidal forces and the flow stops. The water gets piled up, and the only effect is to slightly lower the weight of the water along the vertical direction.

Another way of thinking about this is that the gravitational force of the Moon causes the Earth to accelerate slightly towards the Moon causing the water to get pulled towards the Moon faster than the solid rock on the side nearest the Moon. On the far side, the solid Earth 'leaves behind' some of the water which is not as strongly accelerated towards the Moon as the Earth is. This produces the bulge on the 'back side' of the Earth."- Dr. Odenwald's ASK THE ASTRONOMER

november 2017 by nhaliday

Stability of the Solar System - Wikipedia

november 2017 by nhaliday

The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is chaotic,[1] and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.[2]

The Solar System is stable in human terms, and far beyond, given that it is unlikely any of the planets will collide with each other or be ejected from the system in the next few billion years,[3] and the Earth's orbit will be relatively stable.[4]

Since Newton's law of gravitation (1687), mathematicians and astronomers (such as Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold and Jürgen Moser) have searched for evidence for the stability of the planetary motions, and this quest led to many mathematical developments, and several successive 'proofs' of stability of the Solar System.[5]

...

The planets' orbits are chaotic over longer timescales, such that the whole Solar System possesses a Lyapunov time in the range of 2–230 million years.[3] In all cases this means that the position of a planet along its orbit ultimately becomes impossible to predict with any certainty (so, for example, the timing of winter and summer become uncertain), but in some cases the orbits themselves may change dramatically. Such chaos manifests most strongly as changes in eccentricity, with some planets' orbits becoming significantly more—or less—elliptical.[7]

Is the Solar System Stable?: https://www.ias.edu/ideas/2011/tremaine-solar-system

Is the Solar System Stable?: https://arxiv.org/abs/1209.5996

nibble
wiki
reference
article
physics
mechanics
space
gravity
flux-stasis
uncertainty
robust
perturbation
math
dynamical
math.DS
volo-avolo
multi
org:edu
org:inst
papers
preprint
time
data
org:mat
The Solar System is stable in human terms, and far beyond, given that it is unlikely any of the planets will collide with each other or be ejected from the system in the next few billion years,[3] and the Earth's orbit will be relatively stable.[4]

Since Newton's law of gravitation (1687), mathematicians and astronomers (such as Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold and Jürgen Moser) have searched for evidence for the stability of the planetary motions, and this quest led to many mathematical developments, and several successive 'proofs' of stability of the Solar System.[5]

...

The planets' orbits are chaotic over longer timescales, such that the whole Solar System possesses a Lyapunov time in the range of 2–230 million years.[3] In all cases this means that the position of a planet along its orbit ultimately becomes impossible to predict with any certainty (so, for example, the timing of winter and summer become uncertain), but in some cases the orbits themselves may change dramatically. Such chaos manifests most strongly as changes in eccentricity, with some planets' orbits becoming significantly more—or less—elliptical.[7]

Is the Solar System Stable?: https://www.ias.edu/ideas/2011/tremaine-solar-system

Is the Solar System Stable?: https://arxiv.org/abs/1209.5996

november 2017 by nhaliday

forces - The Time That 2 Masses Will Collide Due To Newtonian Gravity - Physics Stack Exchange

october 2017 by nhaliday

If two particles of dust are placed in an empty universe 1 light year apart from each other, how long will it take for them to collide due to the effects of gravity?: https://www.reddit.com/r/theydidthemath/comments/3rum1p/request_if_two_particles_of_dust_are_placed_in_an/

How long for 2 particles to collide due to gravity?: https://www.physicsforums.com/threads/how-long-for-2-particles-to-collide-due-to-gravity.698767/

nibble
q-n-a
overflow
physics
mechanics
gravity
tidbits
time
multi
reddit
social
discussion
elegance
How long for 2 particles to collide due to gravity?: https://www.physicsforums.com/threads/how-long-for-2-particles-to-collide-due-to-gravity.698767/

october 2017 by nhaliday

I can throw a baseball a lot further than a ping pong ball. I cannot throw a bowling ball nearly as far as a baseball. Is there an "optimal" weight for a ball to throw it as far as possible? : answers

september 2017 by nhaliday

If there are two balls with the same size, they will have the same drag force when traveling at the same speed.

Smaller balls will have less wetted area, and therefore less drag force acting on them

A ball with more mass will decelerate less given the same amount of drag.

The human hand has difficulty holding objects that are too large or too small.

I think that a human's throw is limited by the speed of the hand at the moment of release -- the object can't move faster than your hand when it's released.

A ball with more mass will also be more difficult for a human to throw. Thier arm will rotate slower and the object will have less velocity.

As such, you want the smallest ball that a human can comfortably hold, that is heavy for its size but still light with respect to a human's perspective. Bonus points for drag reduction tech.

Golf balls are surprisingly heavy given their size, and the dimples are designed to convert a laminar boundary layer into a turbulent one. Turbulent boundary layers grip the surface better, delaying flow separation, which is likely the most significant contribution to parasitic drag.

TL; DR: probably a golf ball.

nibble
reddit
social
discussion
q-n-a
physics
mechanics
fluid
street-fighting
biomechanics
extrema
optimization
atmosphere
curiosity
explanation
Smaller balls will have less wetted area, and therefore less drag force acting on them

A ball with more mass will decelerate less given the same amount of drag.

The human hand has difficulty holding objects that are too large or too small.

I think that a human's throw is limited by the speed of the hand at the moment of release -- the object can't move faster than your hand when it's released.

A ball with more mass will also be more difficult for a human to throw. Thier arm will rotate slower and the object will have less velocity.

As such, you want the smallest ball that a human can comfortably hold, that is heavy for its size but still light with respect to a human's perspective. Bonus points for drag reduction tech.

Golf balls are surprisingly heavy given their size, and the dimples are designed to convert a laminar boundary layer into a turbulent one. Turbulent boundary layers grip the surface better, delaying flow separation, which is likely the most significant contribution to parasitic drag.

TL; DR: probably a golf ball.

september 2017 by nhaliday

Ptolemy's Model of the Solar System

september 2017 by nhaliday

It follows, from the above discussion, that the geocentric model of Ptolemy is equivalent to a heliocentric model in which the various planetary orbits are represented as eccentric circles, and in which the radius vector connecting a given planet to its corresponding equant revolves at a uniform rate. In fact, Ptolemy's model of planetary motion can be thought of as a version of Kepler's model which is accurate to first-order in the planetary eccentricities--see Cha. 4. According to the Ptolemaic scheme, from the point of view of the earth, the orbit of the sun is described by a single circular motion, whereas that of a planet is described by a combination of two circular motions. In reality, the single circular motion of the sun represents the (approximately) circular motion of the earth around the sun, whereas the two circular motions of a typical planet represent a combination of the planet's (approximately) circular motion around the sun, and the earth's motion around the sun. Incidentally, the popular myth that Ptolemy's scheme requires an absurdly large number of circles in order to fit the observational data to any degree of accuracy has no basis in fact. Actually, Ptolemy's model of the sun and the planets, which fits the data very well, only contains 12 circles (i.e., 6 deferents and 6 epicycles).

org:junk
org:edu
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space
mechanics
history
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september 2017 by nhaliday

Lecture 14: When's that meteor arriving

september 2017 by nhaliday

- Meteors as a random process

- Limiting approximations

- Derivation of the Exponential distribution

- Derivation of the Poisson distribution

- A "Poisson process"

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org:junk
org:edu
exposition
lecture-notes
physics
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space
earth
probability
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closure
additive
limits
approximation
tidbits
acm
binomial
multiplicative
- Limiting approximations

- Derivation of the Exponential distribution

- Derivation of the Poisson distribution

- A "Poisson process"

september 2017 by nhaliday

Resonance in a Pendulum - YouTube

september 2017 by nhaliday

The vibration of any given washer is able to transmit its energy only to another washer with exactly the same frequency. Since the length of a pendulum determines its frequency of vibration, each pendulum can only set another pendulum vibrating if it has the same length.

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video
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synchrony
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increase-decrease
concrete
ground-up
dirty-hands
phys-energy
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september 2017 by nhaliday

Resonance - Wikipedia

september 2017 by nhaliday

Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies.

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wiki
reference
article
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september 2017 by nhaliday

Reynolds number - Wikipedia

september 2017 by nhaliday

The Reynolds number is the ratio of inertial forces to viscous forces within a fluid which is subjected to relative internal movement due to different fluid velocities, in what is known as a boundary layer in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.[6]

Re = ρuL/μ

(inertial forces)/(viscous forces)

= (mass)(acceleration) / (dynamic viscosity)(velocity/distance)(area)

= (ρL^3)(v/t) / μ(v/L)L^2

= Re

NB: viscous force/area ~ μ du/dy is definition of viscosity

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history
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Re = ρuL/μ

(inertial forces)/(viscous forces)

= (mass)(acceleration) / (dynamic viscosity)(velocity/distance)(area)

= (ρL^3)(v/t) / μ(v/L)L^2

= Re

NB: viscous force/area ~ μ du/dy is definition of viscosity

september 2017 by nhaliday

newtonian gravity - Newton's original proof of gravitation for non-point-mass objects - Physics Stack Exchange

september 2017 by nhaliday

This theorem is Proposition LXXI, Theorem XXXI in the Principia. To warm up, consider the more straightforward proof of the preceding theorem, that there's no inverse-square force inside of a spherical shell:

picture

The crux of the argument is that the triangles HPI and LPK are similar. The mass enclosed in the small-but-near patch of sphere HI goes like the square of the distance HP, while the mass enclosed in the large-but-far patch of sphere KL, with the same solid angle, goes like the square of the distance KP. This mass ratio cancels out the distance-squared ratio governing the strength of the force, and so the net force from those two patches vanishes.

For a point mass outside a shell, Newton's approach is essentially the same as the modern approach:

picture

One integral is removed because we're considering a thin spherical shell rather than a solid sphere. The second integral, "as the semi-circle AKB revolves about the diameter AB," trivially turns Newton's infinitesimal arcs HI and KL into annuli.

The third integral is over all the annuli in the sphere, over 0≤ϕ≤τ/20≤ϕ≤τ/2 or over R−r≤s≤R+rR−r≤s≤R+r. This one is a little bit hairy, even with the advantage of modern notation.

Newton's clever trick is to consider the relationship between the force due to the smaller, nearer annulus HI and the larger, farther annulus KL defined by the same viewing angle (in modern notation, dθdθ). If I understand correctly he argues again, based on lots of similar triangles with infinitesimal angles, that the smaller-but-nearer annulus and the larger-but-farther annulus exert the same force at P. Furthermore, he shows that the force doesn't depend on the distance PF, and thus doesn't depend on the radius of the sphere; the only parameter left is the distance PS (squared) between the particle and the sphere's center. Since the argument doesn't depend on the angle HPS, it's true for all the annuli, and the theorem is proved.

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geometry
spatial
picture

The crux of the argument is that the triangles HPI and LPK are similar. The mass enclosed in the small-but-near patch of sphere HI goes like the square of the distance HP, while the mass enclosed in the large-but-far patch of sphere KL, with the same solid angle, goes like the square of the distance KP. This mass ratio cancels out the distance-squared ratio governing the strength of the force, and so the net force from those two patches vanishes.

For a point mass outside a shell, Newton's approach is essentially the same as the modern approach:

picture

One integral is removed because we're considering a thin spherical shell rather than a solid sphere. The second integral, "as the semi-circle AKB revolves about the diameter AB," trivially turns Newton's infinitesimal arcs HI and KL into annuli.

The third integral is over all the annuli in the sphere, over 0≤ϕ≤τ/20≤ϕ≤τ/2 or over R−r≤s≤R+rR−r≤s≤R+r. This one is a little bit hairy, even with the advantage of modern notation.

Newton's clever trick is to consider the relationship between the force due to the smaller, nearer annulus HI and the larger, farther annulus KL defined by the same viewing angle (in modern notation, dθdθ). If I understand correctly he argues again, based on lots of similar triangles with infinitesimal angles, that the smaller-but-nearer annulus and the larger-but-farther annulus exert the same force at P. Furthermore, he shows that the force doesn't depend on the distance PF, and thus doesn't depend on the radius of the sphere; the only parameter left is the distance PS (squared) between the particle and the sphere's center. Since the argument doesn't depend on the angle HPS, it's true for all the annuli, and the theorem is proved.

september 2017 by nhaliday

Physics 152: Gravity, Fluids, Waves, Heat

september 2017 by nhaliday

lots of good lecture notes with pictures, worked examples, and simulations

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examples
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visualization
visual-understanding
ground-up
fluid
waves
oscillation
thermo
stat-mech
p:whenever
accretion
math.CA
hi-order-bits
nitty-gritty
linearity
spatial
space
entropy-like
temperature
proofs
yoga
plots
september 2017 by nhaliday

Centers of gravity in non-uniform fields - Wikipedia

september 2017 by nhaliday

In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity. This is a very good approximation for smaller bodies near the surface of Earth, so there is no practical need to distinguish "center of gravity" from "center of mass" in most applications, such as engineering and medicine.

In a non-uniform field, gravitational effects such as potential energy, force, and torque can no longer be calculated using the center of mass alone. In particular, a non-uniform gravitational field can produce a torque on an object, even about an axis through the center of mass. The center of gravity seeks to explain this effect. Formally, a center of gravity is an application point of the resultant gravitational force on the body. Such a point may not exist, and if it exists, it is not unique. One can further define a unique center of gravity by approximating the field as either parallel or spherically symmetric.

The concept of a center of gravity as distinct from the center of mass is rarely used in applications, even in celestial mechanics, where non-uniform fields are important. Since the center of gravity depends on the external field, its motion is harder to determine than the motion of the center of mass. The common method to deal with gravitational torques is a field theory.

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In a non-uniform field, gravitational effects such as potential energy, force, and torque can no longer be calculated using the center of mass alone. In particular, a non-uniform gravitational field can produce a torque on an object, even about an axis through the center of mass. The center of gravity seeks to explain this effect. Formally, a center of gravity is an application point of the resultant gravitational force on the body. Such a point may not exist, and if it exists, it is not unique. One can further define a unique center of gravity by approximating the field as either parallel or spherically symmetric.

The concept of a center of gravity as distinct from the center of mass is rarely used in applications, even in celestial mechanics, where non-uniform fields are important. Since the center of gravity depends on the external field, its motion is harder to determine than the motion of the center of mass. The common method to deal with gravitational torques is a field theory.

september 2017 by nhaliday

Why was the Catholic Church so opposed to heliocentrism (for example, in the Renaissance)? Why did they not simply claim that God lived in the Sun, so we go around Him? - Quora

september 2017 by nhaliday

The main reason the Catholic Church opposed the teaching of heliocentrism as a fact was that it was contrary to the science of the time.

Amongst the modern myths about early science is the persistent idea that the opposition to heliocentrism was one of "science" versus "religion". According to this story, early modern astronomers like Copernicus and Galileo "proved" the earth went around the sun and the other scientists of the time agreed. But the Catholic Church clung to a literal interpretation of the Bible and rejected this idea purely out of a fanatical faith, insisting that the earth had to be the centre of the cosmos because man was the pinnacle of all creation. Pretty much everything in this popular story is wrong.

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Amongst the modern myths about early science is the persistent idea that the opposition to heliocentrism was one of "science" versus "religion". According to this story, early modern astronomers like Copernicus and Galileo "proved" the earth went around the sun and the other scientists of the time agreed. But the Catholic Church clung to a literal interpretation of the Bible and rejected this idea purely out of a fanatical faith, insisting that the earth had to be the centre of the cosmos because man was the pinnacle of all creation. Pretty much everything in this popular story is wrong.

september 2017 by nhaliday

Why is Earth's gravity stronger at the poles? - Physics Stack Exchange

september 2017 by nhaliday

The point is that if we approximate Earth with an oblate ellipsoid, then the surface of Earth is an equipotential surface,11 see e.g. this Phys.SE post.

Now, because the polar radius is smaller than the equatorial radius, the density of equipotential surfaces at the poles must be bigger than at the equator.

Or equivalently, the field strength22 gg at the poles must be bigger than at the equator.

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Now, because the polar radius is smaller than the equatorial radius, the density of equipotential surfaces at the poles must be bigger than at the equator.

Or equivalently, the field strength22 gg at the poles must be bigger than at the equator.

september 2017 by nhaliday

Does your weight change between the poles and the equator? (Intermediate) - Curious About Astronomy? Ask an Astronomer

september 2017 by nhaliday

You are right, that because of centrifugal force you will weigh a tiny amount less at the Equator than at the poles. Try not to think of centrifugal force as a force though; what's really going on is that objects which are in motion like to go in a straight line and so it takes some force to make them go round in a circle. (Centrifugal force is a fictitious force that shows up in the equations of motion for an object in a rotating reference frame - such as on Earth's Equator.)

So some of the force of gravity (centripetal force) is being used to make you go around in a circle at the Equator (instead of flying off into space) while at the pole this is not needed. The centripetal acceleration at the Equator is given by four times pi squared times the radius of the Earth divided by the period of rotation squared (4×π2×R/T2). Earth's period of rotation is a sidereal day (86164.1 seconds, slightly less than 24 hours), and the equatorial radius of the Earth is about 6378 km. This means that the centripetal acceleration at the Equator is about 0.03 m/s2 (metres per second squared). Compare this to the acceleration due to gravity which is about 9.8 m/s2 and you can see how tiny an effect this is - you would weigh about 0.3% less at the equator than at the poles!

There is an additional effect due to the oblateness of the Earth. The Earth is not exactly spherical but rather is a little bit like a "squashed" sphere (technically, an oblate spheroid), with the radius at the Equator slightly larger than the radius at the poles. (This shape can be explained by the effect of centrifugal acceleration on the material that makes up the Earth, exactly as described above.) This has the effect of slightly increasing your weight at the poles (since you are close to the centre of the Earth and the gravitational force depends on distance) and slightly decreasing it at the equator.

Taking into account both of the above effects, the gravitational acceleration is 9.78 m/s2 at the equator and 9.83 m/s2 at the poles, so you weigh about 0.5% more at the poles than at the equator.

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So some of the force of gravity (centripetal force) is being used to make you go around in a circle at the Equator (instead of flying off into space) while at the pole this is not needed. The centripetal acceleration at the Equator is given by four times pi squared times the radius of the Earth divided by the period of rotation squared (4×π2×R/T2). Earth's period of rotation is a sidereal day (86164.1 seconds, slightly less than 24 hours), and the equatorial radius of the Earth is about 6378 km. This means that the centripetal acceleration at the Equator is about 0.03 m/s2 (metres per second squared). Compare this to the acceleration due to gravity which is about 9.8 m/s2 and you can see how tiny an effect this is - you would weigh about 0.3% less at the equator than at the poles!

There is an additional effect due to the oblateness of the Earth. The Earth is not exactly spherical but rather is a little bit like a "squashed" sphere (technically, an oblate spheroid), with the radius at the Equator slightly larger than the radius at the poles. (This shape can be explained by the effect of centrifugal acceleration on the material that makes up the Earth, exactly as described above.) This has the effect of slightly increasing your weight at the poles (since you are close to the centre of the Earth and the gravitational force depends on distance) and slightly decreasing it at the equator.

Taking into account both of the above effects, the gravitational acceleration is 9.78 m/s2 at the equator and 9.83 m/s2 at the poles, so you weigh about 0.5% more at the poles than at the equator.

september 2017 by nhaliday

Gimbal lock - Wikipedia

september 2017 by nhaliday

Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space.

The word lock is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes there is no gimbal available to accommodate rotation along one axis.

https://blender.stackexchange.com/questions/469/could-someone-please-explain-gimbal-lock

https://computergraphics.stackexchange.com/questions/4436/how-to-achieve-gimbal-lock-with-euler-angles

Now this is where most people stop thinking about the issue and move on with their life. They just conclude that Euler angles are somehow broken. This is also where a lot of misunderstandings happen so it's worth investigating the matter slightly further than what causes gimbal lock.

It is important to understand that this is only problematic if you interpolate in Euler angles**! In a real physical gimbal this is given - you have no other choice. In computer graphics you have many other choices, from normalized matrix, axis angle or quaternion interpolation. Gimbal lock has a much more dramatic implication to designing control systems than it has to 3d graphics. Which is why a mechanical engineer for example will have a very different take on gimbal locking.

You don't have to give up using Euler angles to get rid of gimbal locking, just stop interpolating values in Euler angles. Of course, this means that you can now no longer drive a rotation by doing direct manipulation of one of the channels. But as long as you key the 3 angles simultaneously you have no problems and you can internally convert your interpolation target to something that has less problems.

Using Euler angles is just simply more intuitive to think in most cases. And indeed Euler never claimed it was good for interpolating but just that it can model all possible space orientations. So Euler angles are just fine for setting orientations like they were meant to do. Also incidentally Euler angles have the benefit of being able to model multi turn rotations which will not happen sanely for the other representations.

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The word lock is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes there is no gimbal available to accommodate rotation along one axis.

https://blender.stackexchange.com/questions/469/could-someone-please-explain-gimbal-lock

https://computergraphics.stackexchange.com/questions/4436/how-to-achieve-gimbal-lock-with-euler-angles

Now this is where most people stop thinking about the issue and move on with their life. They just conclude that Euler angles are somehow broken. This is also where a lot of misunderstandings happen so it's worth investigating the matter slightly further than what causes gimbal lock.

It is important to understand that this is only problematic if you interpolate in Euler angles**! In a real physical gimbal this is given - you have no other choice. In computer graphics you have many other choices, from normalized matrix, axis angle or quaternion interpolation. Gimbal lock has a much more dramatic implication to designing control systems than it has to 3d graphics. Which is why a mechanical engineer for example will have a very different take on gimbal locking.

You don't have to give up using Euler angles to get rid of gimbal locking, just stop interpolating values in Euler angles. Of course, this means that you can now no longer drive a rotation by doing direct manipulation of one of the channels. But as long as you key the 3 angles simultaneously you have no problems and you can internally convert your interpolation target to something that has less problems.

Using Euler angles is just simply more intuitive to think in most cases. And indeed Euler never claimed it was good for interpolating but just that it can model all possible space orientations. So Euler angles are just fine for setting orientations like they were meant to do. Also incidentally Euler angles have the benefit of being able to model multi turn rotations which will not happen sanely for the other representations.

september 2017 by nhaliday

Flows With Friction

september 2017 by nhaliday

To see how the no-slip condition arises, and how the no-slip condition and the fluid viscosity lead to frictional stresses, we can examine the conditions at a solid surface on a molecular scale. When a fluid is stationary, its molecules are in a constant state of motion with a random velocity v. For a gas, v is equal to the speed of sound. When a fluid is in motion, there is superimposed on this random velocity a mean velocity V, sometimes called the bulk velocity, which is the velocity at which fluid from one place to another. At the interface between the fluid and the surface, there exists an attraction between the molecules or atoms that make up the fluid and those that make up the solid. This attractive force is strong enough to reduce the bulk velocity of the fluid to zero. So the bulk velocity of the fluid must change from whatever its value is far away from the wall to a value of zero at the wall (figure 7). This is called the no-slip condition.

http://www.engineeringarchives.com/les_fm_noslip.html

The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity.

https://www.quora.com/What-is-the-physics-behind-no-slip-condition-in-fluid-mechanics

https://www.reddit.com/r/AskEngineers/comments/348b1q/the_noslip_condition/

https://www.researchgate.net/post/Can_someone_explain_what_exactly_no_slip_condition_or_slip_condition_means_in_terms_of_momentum_transfer_of_the_molecules

https://en.wikipedia.org/wiki/Boundary_layer_thickness

http://www.fkm.utm.my/~ummi/SME1313/Chapter%201.pdf

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mechanics
h2o
identity
atoms
constraint-satisfaction
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flux-stasis
chemistry
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reddit
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discussion
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pdf
slides
lectures
qra
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local-global
explanation
http://www.engineeringarchives.com/les_fm_noslip.html

The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity.

https://www.quora.com/What-is-the-physics-behind-no-slip-condition-in-fluid-mechanics

https://www.reddit.com/r/AskEngineers/comments/348b1q/the_noslip_condition/

https://www.researchgate.net/post/Can_someone_explain_what_exactly_no_slip_condition_or_slip_condition_means_in_terms_of_momentum_transfer_of_the_molecules

https://en.wikipedia.org/wiki/Boundary_layer_thickness

http://www.fkm.utm.my/~ummi/SME1313/Chapter%201.pdf

september 2017 by nhaliday

GALILEO'S STUDIES OF PROJECTILE MOTION

august 2017 by nhaliday

During the Renaissance, the focus, especially in the arts, was on representing as accurately as possible the real world whether on a 2 dimensional surface or a solid such as marble or granite. This required two things. The first was new methods for drawing or painting, e.g., perspective. The second, relevant to this topic, was careful observation.

With the spread of cannon in warfare, the study of projectile motion had taken on greater importance, and now, with more careful observation and more accurate representation, came the realization that projectiles did not move the way Aristotle and his followers had said they did: the path of a projectile did not consist of two consecutive straight line components but was instead a smooth curve. [1]

Now someone needed to come up with a method to determine if there was a special curve a projectile followed. But measuring the path of a projectile was not easy.

Using an inclined plane, Galileo had performed experiments on uniformly accelerated motion, and he now used the same apparatus to study projectile motion. He placed an inclined plane on a table and provided it with a curved piece at the bottom which deflected an inked bronze ball into a horizontal direction. The ball thus accelerated rolled over the table-top with uniform motion and then fell off the edge of the table Where it hit the floor, it left a small mark. The mark allowed the horizontal and vertical distances traveled by the ball to be measured. [2]

By varying the ball's horizontal velocity and vertical drop, Galileo was able to determine that the path of a projectile is parabolic.

https://www.scientificamerican.com/author/stillman-drake/

Galileo's Discovery of the Parabolic Trajectory: http://www.jstor.org/stable/24949756

Galileo's Experimental Confirmation of Horizontal Inertia: Unpublished Manuscripts (Galileo

Gleanings XXII): https://sci-hub.tw/https://www.jstor.org/stable/229718

- Drake Stillman

MORE THAN A DECADE HAS ELAPSED since Thomas Settle published a classic paper in which Galileo's well-known statements about his experiments on inclined planes were completely vindicated.' Settle's paper replied to an earlier attempt by Alexandre Koyre to show that Galileo could not have obtained the results he claimed in his Two New Sciences by actual observations using the equipment there described. The practical ineffectiveness of Settle's painstaking repetition of the experiments in altering the opinion of historians of science is only too evident. Koyre's paper was reprinted years later in book form without so much as a note by the editors concerning Settle's refutation of its thesis.2 And the general literature continues to belittle the role of experiment in Galileo's physics.

More recently James MacLachlan has repeated and confirmed a different experiment reported by Galileo-one which has always seemed highly exaggerated and which was also rejected by Koyre with withering sarcasm.3 In this case, however, it was accuracy of observation rather than precision of experimental data that was in question. Until now, nothing has been produced to demonstrate Galileo's skill in the design and the accurate execution of physical experiment in the modern sense.

Pant of a page of Galileo's unpublished manuscript notes, written late in 7608, corroborating his inertial assumption and leading directly to his discovery of the parabolic trajectory. (Folio 1 16v Vol. 72, MSS Galileiani; courtesy of the Biblioteca Nazionale di Firenze.)

...

(The same skeptical historians, however, believe that to show that Galileo could have used the medieval mean-speed theorem suffices to prove that he did use it, though it is found nowhere in his published or unpublished writings.)

...

Now, it happens that among Galileo's manuscript notes on motion there are many pages that were not published by Favaro, since they contained only calculations or diagrams without attendant propositions or explanations. Some pages that were published had first undergone considerable editing, making it difficult if not impossible to discern their full significance from their printed form. This unpublished material includes at least one group of notes which cannot satisfactorily be accounted for except as representing a series of experiments designed to test a fundamental assumption, which led to a new, important discovery. In these documents precise empirical data are given numerically, comparisons are made with calculated values derived from theory, a source of discrepancy from still another expected result is noted, a new experiment is designed to eliminate this, and further empirical data are recorded. The last-named data, although proving to be beyond Galileo's powers of mathematical analysis at the time, when subjected to modern analysis turn out to be remarkably precise. If this does not represent the experimental process in its fully modern sense, it is hard to imagine what standards historians require to be met.

The discovery of these notes confirms the opinion of earlier historians. They read only Galileo's published works, but did so without a preconceived notion of continuity in the history of ideas. The opinion of our more sophisticated colleagues has its sole support in philosophical interpretations that fit with preconceived views of orderly long-term scientific development. To find manuscript evidence that Galileo was at home in the physics laboratory hardly surprises me. I should find it much more astonishing if, by reasoning alone, working only from fourteenth-century theories and conclusions, he had continued along lines so different from those followed by profound philosophers in earlier centuries. It is to be hoped that, warned by these examples, historians will begin to restore the old cautionary clauses in analogous instances in which scholarly opinions are revised without new evidence, simply to fit historical theories.

In what follows, the newly discovered documents are presented in the context of a hypothetical reconstruction of Galileo's thought.

...

As early as 1590, if we are correct in ascribing Galileo's juvenile De motu to that date, it was his belief that an ideal body resting on an ideal horizontal plane could be set in motion by a force smaller than any previously assigned force, however small. By "horizontal plane" he meant a surface concentric with the earth but which for reasonable distances would be indistinguishable from a level plane. Galileo noted at the time that experiment did not confirm this belief that the body could be set in motion by a vanishingly small force, and he attributed the failure to friction, pressure, the imperfection of material surfaces and spheres, and the departure of level planes from concentricity with the earth.5

It followed from this belief that under ideal conditions the motion so induced would also be perpetual and uniform. Galileo did not mention these consequences until much later, and it is impossible to say just when he perceived them. They are, however, so evident that it is safe to assume that he saw them almost from the start. They constitute a trivial case of the proposition he seems to have been teaching before 1607-that a mover is required to start motion, but that absence of resistance is then sufficient to account for its continuation.6

In mid-1604, following some investigations of motions along circular arcs and motions of pendulums, Galileo hit upon the law that in free fall the times elapsed from rest are as the smaller distance is to the mean proportional between two distances fallen.7 This gave him the times-squared law as well as the rule of odd numbers for successive distances and speeds in free fall. During the next few years he worked out a large number of theorems relating to motion along inclined planes, later published in the Two New Sciences. He also arrived at the rule that the speed terminating free fall from rest was double the speed of the fall itself. These theorems survive in manuscript notes of the period 1604-1609. (Work during these years can be identified with virtual certainty by the watermarks in the paper used, as I have explained elsewhere.8)

In the autumn of 1608, after a summer at Florence, Galileo seems to have interested himself in the question whether the actual slowing of a body moving horizontally followed any particular rule. On folio 117i of the manuscripts just mentioned, the numbers 196, 155, 121, 100 are noted along the horizontal line near the middle of the page (see Fig. 1). I believe that this was the first entry on this leaf, for reasons that will appear later, and that Galileo placed his grooved plane in the level position and recorded distances traversed in equal times along it. Using a metronome, and rolling a light wooden ball about 4 3/4 inches in diameter along a plane with a groove 1 3/4 inches wide, I obtained similar relations over a distance of 6 feet. The figures obtained vary greatly for balls of different materials and weights and for greatly different initial speeds.9 But it suffices for my present purposes that Galileo could have obtained the figures noted by observing the actual deceleration of a ball along a level plane. It should be noted that the watermark on this leaf is like that on folio 116, to which we shall come presently, and it will be seen later that the two sheets are closely connected in time in other ways as well.

The relatively rapid deceleration is obviously related to the contact of ball and groove. Were the ball to roll right off the end of the plane, all resistance to horizontal motion would be virtually removed. If, then, there were any way to have a given ball leave the plane at different speeds of which the ratios were known, Galileo's old idea that horizontal motion would continue uniformly in the absence of resistance could be put to test. His law of free fall made this possible. The ratios of speeds could be controlled by allowing the ball to fall vertically through known heights, at the ends of which it would be deflected horizontally. Falls through given heights … [more]

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With the spread of cannon in warfare, the study of projectile motion had taken on greater importance, and now, with more careful observation and more accurate representation, came the realization that projectiles did not move the way Aristotle and his followers had said they did: the path of a projectile did not consist of two consecutive straight line components but was instead a smooth curve. [1]

Now someone needed to come up with a method to determine if there was a special curve a projectile followed. But measuring the path of a projectile was not easy.

Using an inclined plane, Galileo had performed experiments on uniformly accelerated motion, and he now used the same apparatus to study projectile motion. He placed an inclined plane on a table and provided it with a curved piece at the bottom which deflected an inked bronze ball into a horizontal direction. The ball thus accelerated rolled over the table-top with uniform motion and then fell off the edge of the table Where it hit the floor, it left a small mark. The mark allowed the horizontal and vertical distances traveled by the ball to be measured. [2]

By varying the ball's horizontal velocity and vertical drop, Galileo was able to determine that the path of a projectile is parabolic.

https://www.scientificamerican.com/author/stillman-drake/

Galileo's Discovery of the Parabolic Trajectory: http://www.jstor.org/stable/24949756

Galileo's Experimental Confirmation of Horizontal Inertia: Unpublished Manuscripts (Galileo

Gleanings XXII): https://sci-hub.tw/https://www.jstor.org/stable/229718

- Drake Stillman

MORE THAN A DECADE HAS ELAPSED since Thomas Settle published a classic paper in which Galileo's well-known statements about his experiments on inclined planes were completely vindicated.' Settle's paper replied to an earlier attempt by Alexandre Koyre to show that Galileo could not have obtained the results he claimed in his Two New Sciences by actual observations using the equipment there described. The practical ineffectiveness of Settle's painstaking repetition of the experiments in altering the opinion of historians of science is only too evident. Koyre's paper was reprinted years later in book form without so much as a note by the editors concerning Settle's refutation of its thesis.2 And the general literature continues to belittle the role of experiment in Galileo's physics.

More recently James MacLachlan has repeated and confirmed a different experiment reported by Galileo-one which has always seemed highly exaggerated and which was also rejected by Koyre with withering sarcasm.3 In this case, however, it was accuracy of observation rather than precision of experimental data that was in question. Until now, nothing has been produced to demonstrate Galileo's skill in the design and the accurate execution of physical experiment in the modern sense.

Pant of a page of Galileo's unpublished manuscript notes, written late in 7608, corroborating his inertial assumption and leading directly to his discovery of the parabolic trajectory. (Folio 1 16v Vol. 72, MSS Galileiani; courtesy of the Biblioteca Nazionale di Firenze.)

...

(The same skeptical historians, however, believe that to show that Galileo could have used the medieval mean-speed theorem suffices to prove that he did use it, though it is found nowhere in his published or unpublished writings.)

...

Now, it happens that among Galileo's manuscript notes on motion there are many pages that were not published by Favaro, since they contained only calculations or diagrams without attendant propositions or explanations. Some pages that were published had first undergone considerable editing, making it difficult if not impossible to discern their full significance from their printed form. This unpublished material includes at least one group of notes which cannot satisfactorily be accounted for except as representing a series of experiments designed to test a fundamental assumption, which led to a new, important discovery. In these documents precise empirical data are given numerically, comparisons are made with calculated values derived from theory, a source of discrepancy from still another expected result is noted, a new experiment is designed to eliminate this, and further empirical data are recorded. The last-named data, although proving to be beyond Galileo's powers of mathematical analysis at the time, when subjected to modern analysis turn out to be remarkably precise. If this does not represent the experimental process in its fully modern sense, it is hard to imagine what standards historians require to be met.

The discovery of these notes confirms the opinion of earlier historians. They read only Galileo's published works, but did so without a preconceived notion of continuity in the history of ideas. The opinion of our more sophisticated colleagues has its sole support in philosophical interpretations that fit with preconceived views of orderly long-term scientific development. To find manuscript evidence that Galileo was at home in the physics laboratory hardly surprises me. I should find it much more astonishing if, by reasoning alone, working only from fourteenth-century theories and conclusions, he had continued along lines so different from those followed by profound philosophers in earlier centuries. It is to be hoped that, warned by these examples, historians will begin to restore the old cautionary clauses in analogous instances in which scholarly opinions are revised without new evidence, simply to fit historical theories.

In what follows, the newly discovered documents are presented in the context of a hypothetical reconstruction of Galileo's thought.

...

As early as 1590, if we are correct in ascribing Galileo's juvenile De motu to that date, it was his belief that an ideal body resting on an ideal horizontal plane could be set in motion by a force smaller than any previously assigned force, however small. By "horizontal plane" he meant a surface concentric with the earth but which for reasonable distances would be indistinguishable from a level plane. Galileo noted at the time that experiment did not confirm this belief that the body could be set in motion by a vanishingly small force, and he attributed the failure to friction, pressure, the imperfection of material surfaces and spheres, and the departure of level planes from concentricity with the earth.5

It followed from this belief that under ideal conditions the motion so induced would also be perpetual and uniform. Galileo did not mention these consequences until much later, and it is impossible to say just when he perceived them. They are, however, so evident that it is safe to assume that he saw them almost from the start. They constitute a trivial case of the proposition he seems to have been teaching before 1607-that a mover is required to start motion, but that absence of resistance is then sufficient to account for its continuation.6

In mid-1604, following some investigations of motions along circular arcs and motions of pendulums, Galileo hit upon the law that in free fall the times elapsed from rest are as the smaller distance is to the mean proportional between two distances fallen.7 This gave him the times-squared law as well as the rule of odd numbers for successive distances and speeds in free fall. During the next few years he worked out a large number of theorems relating to motion along inclined planes, later published in the Two New Sciences. He also arrived at the rule that the speed terminating free fall from rest was double the speed of the fall itself. These theorems survive in manuscript notes of the period 1604-1609. (Work during these years can be identified with virtual certainty by the watermarks in the paper used, as I have explained elsewhere.8)

In the autumn of 1608, after a summer at Florence, Galileo seems to have interested himself in the question whether the actual slowing of a body moving horizontally followed any particular rule. On folio 117i of the manuscripts just mentioned, the numbers 196, 155, 121, 100 are noted along the horizontal line near the middle of the page (see Fig. 1). I believe that this was the first entry on this leaf, for reasons that will appear later, and that Galileo placed his grooved plane in the level position and recorded distances traversed in equal times along it. Using a metronome, and rolling a light wooden ball about 4 3/4 inches in diameter along a plane with a groove 1 3/4 inches wide, I obtained similar relations over a distance of 6 feet. The figures obtained vary greatly for balls of different materials and weights and for greatly different initial speeds.9 But it suffices for my present purposes that Galileo could have obtained the figures noted by observing the actual deceleration of a ball along a level plane. It should be noted that the watermark on this leaf is like that on folio 116, to which we shall come presently, and it will be seen later that the two sheets are closely connected in time in other ways as well.

The relatively rapid deceleration is obviously related to the contact of ball and groove. Were the ball to roll right off the end of the plane, all resistance to horizontal motion would be virtually removed. If, then, there were any way to have a given ball leave the plane at different speeds of which the ratios were known, Galileo's old idea that horizontal motion would continue uniformly in the absence of resistance could be put to test. His law of free fall made this possible. The ratios of speeds could be controlled by allowing the ball to fall vertically through known heights, at the ends of which it would be deflected horizontally. Falls through given heights … [more]

august 2017 by nhaliday

Isaac Newton: the first physicist.

august 2017 by nhaliday

[...] More fundamentally, Newton's mathematical approach has become so basic to all of physics that he is generally regarded as _the father of the clockwork universe_: the first, and perhaps the greatest, physicist.

The Alchemist

In fact, Newton was deeply opposed to the mechanistic conception of the world. A secretive alchemist [...]. His written work on the subject ran to more than a million words, far more than he ever produced on calculus or mechanics [21]. Obsessively religious, he spent years correlating biblical prophecy with historical events [319ff]. He became deeply convinced that Christian doctrine had been deliberately corrupted by _the false notion of the trinity_, and developed a vicious contempt for conventional (trinitarian) Christianity and for Roman Catholicism in particular [324]. [...] He believed that God mediated the gravitational force [511](353), and opposed any attempt to give a mechanistic explanation of chemistry or gravity, since that would diminish the role of God [646]. He consequently conceived such _a hatred of Descartes_, on whose foundations so many of his achievements were built, that at times _he refused even to write his name_ [399,401].

The Man

Newton was rigorously puritanical: when one of his few friends told him "a loose story about a nun", he ended their friendship (267). [...] He thought of himself as the sole inventor of the calculus, and hence the greatest mathematician since the ancients, and left behind a huge corpus of unpublished work, mostly alchemy and biblical exegesis, that he believed future generations would appreciate more than his own (199,511).

[...] Even though these unattractive qualities caused him to waste huge amounts of time and energy in ruthless vendettas against colleagues who in many cases had helped him (see below), they also drove him to the extraordinary achievements for which he is still remembered. And for all his arrogance, Newton's own summary of his life (574) was beautifully humble:

"I do not know how I may appear to the world, but to myself I seem to have been only like a boy, playing on the sea-shore, and diverting myself, in now and then finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Before Newton

...

1. Calculus. Descartes, in 1637, pioneered the use of coordinates to turn geometric problems into algebraic ones, a method that Newton was never to accept [399]. Descartes, Fermat, and others investigated methods of calculating the tangents to arbitrary curves [28-30]. Kepler, Cavalieri, and others used infinitesimal slices to calculate volumes and areas enclosed by curves [30], but no unified treatment of these problems had yet been found.

2. Mechanics & Planetary motion. The elliptical orbits of the planets having been established by Kepler, Descartes proposed the idea of a purely mechanical heliocentric universe, following deterministic laws, and with no need of any divine agency [15], another anathema to Newton. _No one imagined, however, that a single law might explain both falling bodies and planetary motion_. Galileo invented the concept of inertia, anticipating Newton's first and second laws of motion (293), and Huygens used it to analyze collisions and circular motion [11]. Again, these pieces of progress had not been synthesized into a general method for analyzing forces and motion.

3. Light. Descartes claimed that light was a pressure wave, Gassendi that it was a stream of particles (corpuscles) [13]. As might be guessed, Newton vigorously supported the corpuscular theory. _White light was universally believed to be the pure form_, and colors were some added property bequeathed to it upon reflection from matter (150). Descartes had discovered the sine law of refraction (94), but it was not known that some colors were refracted more than others. The pattern was the familiar one: many pieces of the puzzle were in place, but the overall picture was still unclear.

The Natural Philosopher

Between 1671 and 1690, Newton was to supply definitive treatments of most of these problems. By assiduous experimentation with prisms he established that colored light was actually fundamental, and that it could be recombined to create white light. He did not publish the result for 6 years, by which time it seemed so obvious to him that he found great difficulty in responding patiently to the many misunderstandings and objections with which it met [239ff].

He invented differential and integral calculus in 1665-6, but failed to publish it. Leibniz invented it independently 10 years later, and published it first [718]. This resulted in a priority dispute which degenerated into a feud characterized by extraordinary dishonesty and venom on both sides (542).

In discovering gravitation, Newton was also _barely ahead of the rest of the pack_. Hooke was the first to realize that orbital motion was produced by a centripetal force (268), and in 1679 _he suggested an inverse square law to Newton_ [387]. Halley and Wren came to the same conclusion, and turned to Newton for a proof, which he duly supplied [402]. Newton did not stop there, however. From 1684 to 1687 he worked continuously on a grand synthesis of the whole of mechanics, the "Philosophiae Naturalis Principia Mathematica," in which he developed his three laws of motion and showed in detail that the universal force of gravitation could explain the fall of an apple as well as the precise motions of planets and comets.

The "Principia" crystallized the new conceptions of force and inertia that had gradually been emerging, and marks the beginning of theoretical physics as the mathematical field that we know today. It is not an easy read: Newton had developed the idea that geometry and equations should never be combined [399], and therefore _refused to use simple analytical techniques in his proofs_, requiring classical geometric constructions instead [428]. He even made his Principia _deliberately abstruse in order to discourage amateurs from feeling qualified to criticize it_ [459].

[...] most of the rest of his life was spent in administrative work as Master of the Mint and as President of the Royal Society, _a position he ruthlessly exploited in the pursuit of vendettas_ against Hooke (300ff,500), Leibniz (510ff), and Flamsteed (490,500), among others. He kept secret his disbelief in Christ's divinity right up until his dying moment, at which point he refused the last rites, at last openly defying the church (576). [...]

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britain
history
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stories
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discovery
The Alchemist

In fact, Newton was deeply opposed to the mechanistic conception of the world. A secretive alchemist [...]. His written work on the subject ran to more than a million words, far more than he ever produced on calculus or mechanics [21]. Obsessively religious, he spent years correlating biblical prophecy with historical events [319ff]. He became deeply convinced that Christian doctrine had been deliberately corrupted by _the false notion of the trinity_, and developed a vicious contempt for conventional (trinitarian) Christianity and for Roman Catholicism in particular [324]. [...] He believed that God mediated the gravitational force [511](353), and opposed any attempt to give a mechanistic explanation of chemistry or gravity, since that would diminish the role of God [646]. He consequently conceived such _a hatred of Descartes_, on whose foundations so many of his achievements were built, that at times _he refused even to write his name_ [399,401].

The Man

Newton was rigorously puritanical: when one of his few friends told him "a loose story about a nun", he ended their friendship (267). [...] He thought of himself as the sole inventor of the calculus, and hence the greatest mathematician since the ancients, and left behind a huge corpus of unpublished work, mostly alchemy and biblical exegesis, that he believed future generations would appreciate more than his own (199,511).

[...] Even though these unattractive qualities caused him to waste huge amounts of time and energy in ruthless vendettas against colleagues who in many cases had helped him (see below), they also drove him to the extraordinary achievements for which he is still remembered. And for all his arrogance, Newton's own summary of his life (574) was beautifully humble:

"I do not know how I may appear to the world, but to myself I seem to have been only like a boy, playing on the sea-shore, and diverting myself, in now and then finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Before Newton

...

1. Calculus. Descartes, in 1637, pioneered the use of coordinates to turn geometric problems into algebraic ones, a method that Newton was never to accept [399]. Descartes, Fermat, and others investigated methods of calculating the tangents to arbitrary curves [28-30]. Kepler, Cavalieri, and others used infinitesimal slices to calculate volumes and areas enclosed by curves [30], but no unified treatment of these problems had yet been found.

2. Mechanics & Planetary motion. The elliptical orbits of the planets having been established by Kepler, Descartes proposed the idea of a purely mechanical heliocentric universe, following deterministic laws, and with no need of any divine agency [15], another anathema to Newton. _No one imagined, however, that a single law might explain both falling bodies and planetary motion_. Galileo invented the concept of inertia, anticipating Newton's first and second laws of motion (293), and Huygens used it to analyze collisions and circular motion [11]. Again, these pieces of progress had not been synthesized into a general method for analyzing forces and motion.

3. Light. Descartes claimed that light was a pressure wave, Gassendi that it was a stream of particles (corpuscles) [13]. As might be guessed, Newton vigorously supported the corpuscular theory. _White light was universally believed to be the pure form_, and colors were some added property bequeathed to it upon reflection from matter (150). Descartes had discovered the sine law of refraction (94), but it was not known that some colors were refracted more than others. The pattern was the familiar one: many pieces of the puzzle were in place, but the overall picture was still unclear.

The Natural Philosopher

Between 1671 and 1690, Newton was to supply definitive treatments of most of these problems. By assiduous experimentation with prisms he established that colored light was actually fundamental, and that it could be recombined to create white light. He did not publish the result for 6 years, by which time it seemed so obvious to him that he found great difficulty in responding patiently to the many misunderstandings and objections with which it met [239ff].

He invented differential and integral calculus in 1665-6, but failed to publish it. Leibniz invented it independently 10 years later, and published it first [718]. This resulted in a priority dispute which degenerated into a feud characterized by extraordinary dishonesty and venom on both sides (542).

In discovering gravitation, Newton was also _barely ahead of the rest of the pack_. Hooke was the first to realize that orbital motion was produced by a centripetal force (268), and in 1679 _he suggested an inverse square law to Newton_ [387]. Halley and Wren came to the same conclusion, and turned to Newton for a proof, which he duly supplied [402]. Newton did not stop there, however. From 1684 to 1687 he worked continuously on a grand synthesis of the whole of mechanics, the "Philosophiae Naturalis Principia Mathematica," in which he developed his three laws of motion and showed in detail that the universal force of gravitation could explain the fall of an apple as well as the precise motions of planets and comets.

The "Principia" crystallized the new conceptions of force and inertia that had gradually been emerging, and marks the beginning of theoretical physics as the mathematical field that we know today. It is not an easy read: Newton had developed the idea that geometry and equations should never be combined [399], and therefore _refused to use simple analytical techniques in his proofs_, requiring classical geometric constructions instead [428]. He even made his Principia _deliberately abstruse in order to discourage amateurs from feeling qualified to criticize it_ [459].

[...] most of the rest of his life was spent in administrative work as Master of the Mint and as President of the Royal Society, _a position he ruthlessly exploited in the pursuit of vendettas_ against Hooke (300ff,500), Leibniz (510ff), and Flamsteed (490,500), among others. He kept secret his disbelief in Christ's divinity right up until his dying moment, at which point he refused the last rites, at last openly defying the church (576). [...]

august 2017 by nhaliday

Is it possible to recover Classical Mechanics from Schrödinger's equation? - Physics Stack Exchange

august 2017 by nhaliday

Classical limit of quantum mechanics: https://physics.stackexchange.com/questions/32112/classical-limit-of-quantum-mechanics

https://physics.stackexchange.com/questions/108222/from-quantum-mechanics-to-classical-mechanics

Classical Limit of Quantum Mechanics: https://mathoverflow.net/questions/102313/classical-limit-of-quantum-mechanics

How/when does quantum mechanics become classical mechanics?: https://www.quora.com/How-when-does-quantum-mechanics-become-classical-mechanics

Remarks concerning the status & some ramifications of EHRENFEST’S THEOREM: http://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Ehrenfest's%20Theorem.pdf

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https://physics.stackexchange.com/questions/108222/from-quantum-mechanics-to-classical-mechanics

Classical Limit of Quantum Mechanics: https://mathoverflow.net/questions/102313/classical-limit-of-quantum-mechanics

How/when does quantum mechanics become classical mechanics?: https://www.quora.com/How-when-does-quantum-mechanics-become-classical-mechanics

Remarks concerning the status & some ramifications of EHRENFEST’S THEOREM: http://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Ehrenfest's%20Theorem.pdf

august 2017 by nhaliday

Constitutive equation - Wikipedia

august 2017 by nhaliday

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations.

Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a tensor. Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior.[1] See the article Linear response function.

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Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a tensor. Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior.[1] See the article Linear response function.

august 2017 by nhaliday

diffusion - Surviving under water in air bubble - Physics Stack Exchange

august 2017 by nhaliday

I get d≈400md≈400m.

It's interesting to note that this is independent of pressure: I've neglected pressure dependence of DD and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at STP.

Finally, a bubble this large will probably rapidly break up due to buoyancy and Plateau-Rayleigh instabilities.

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It's interesting to note that this is independent of pressure: I've neglected pressure dependence of DD and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at STP.

Finally, a bubble this large will probably rapidly break up due to buoyancy and Plateau-Rayleigh instabilities.

august 2017 by nhaliday

Note on Water Measurements by Frontinus â€¢ by Ilia Rushkin

org:junk org:edu nibble article history iron-age mediterranean the-classics physics mechanics h2o error critique analysis the-trenches alien-character dirty-hands archaeology stock-flow speed technology science fluid

august 2017 by nhaliday

org:junk org:edu nibble article history iron-age mediterranean the-classics physics mechanics h2o error critique analysis the-trenches alien-character dirty-hands archaeology stock-flow speed technology science fluid

august 2017 by nhaliday

Philosophiæ Naturalis Principia Mathematica - Wikipedia

august 2017 by nhaliday

Newton Papers : Philosophiæ naturalis principia mathematica: https://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/1

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august 2017 by nhaliday

Mainspring - Wikipedia

august 2017 by nhaliday

A mainspring is a spiral torsion spring of metal ribbon—commonly spring steel—used as a power source in mechanical watches, some clocks, and other clockwork mechanisms. Winding the timepiece, by turning a knob or key, stores energy in the mainspring by twisting the spiral tighter. The force of the mainspring then turns the clock's wheels as it unwinds, until the next winding is needed. The adjectives wind-up and spring-powered refer to mechanisms powered by mainsprings, which also include kitchen timers, music boxes, wind-up toys and clockwork radios.

torque basically follows Hooke's Law

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torque basically follows Hooke's Law

august 2017 by nhaliday

Titius–Bode law - Wikipedia

august 2017 by nhaliday

d = .4 + .3*2^n for (n = -inf, 0, 1, 2, ...) in AU

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august 2017 by nhaliday

How & Why Solar Eclipses Happen | Solar Eclipse Across America - August 21, 2017

august 2017 by nhaliday

Cosmic Coincidence

The Sun’s diameter is about 400 times that of the Moon. The Sun is also (on average) about 400 times farther away. As a result, the two bodies appear almost exactly the same angular size in the sky — about ½°, roughly half the width of your pinky finger seen at arm's length. This truly remarkable coincidence is what gives us total solar eclipses. If the Moon were slightly smaller or orbited a little farther away from Earth, it would never completely cover the solar disk. If the Moon were a little larger or orbited a bit closer to Earth, it would block much of the solar corona during totality, and eclipses wouldn’t be nearly as spectacular.

https://blogs.scientificamerican.com/life-unbounded/the-solar-eclipse-coincidence/

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The Sun’s diameter is about 400 times that of the Moon. The Sun is also (on average) about 400 times farther away. As a result, the two bodies appear almost exactly the same angular size in the sky — about ½°, roughly half the width of your pinky finger seen at arm's length. This truly remarkable coincidence is what gives us total solar eclipses. If the Moon were slightly smaller or orbited a little farther away from Earth, it would never completely cover the solar disk. If the Moon were a little larger or orbited a bit closer to Earth, it would block much of the solar corona during totality, and eclipses wouldn’t be nearly as spectacular.

https://blogs.scientificamerican.com/life-unbounded/the-solar-eclipse-coincidence/

august 2017 by nhaliday

rotational dynamics - Why do non-rigid bodies try to increase their moment of inertia? - Physics Stack Exchange

august 2017 by nhaliday

This happens to isolated rotating system that is not a rigid body.

Inside such a body (for example, steel chain in free fall) the parts move relatively to each other and there is internal friction that dissipates kinetic energy of the system, while angular momentum is conserved. The dissipation goes on until the parts stop moving with respect to each other, so body rotates as a rigid body, even if it is not rigid by constitution.

The rotating state of the body that has the lowest kinetic energy for given angular momentum is that in which the body has the greatest moment of inertia (with respect to center of mass). For example, a long chain thrown into free fall will twist and turn until it is all straight and rotating as rigid body.

...

If LL is constant (net torque of external forces acting on the system is zero) and the constitution and initial conditions allow it, the system's dissipation will work to diminish energy until it has the minimum value, which happens for maximum IaIa possible.

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Inside such a body (for example, steel chain in free fall) the parts move relatively to each other and there is internal friction that dissipates kinetic energy of the system, while angular momentum is conserved. The dissipation goes on until the parts stop moving with respect to each other, so body rotates as a rigid body, even if it is not rigid by constitution.

The rotating state of the body that has the lowest kinetic energy for given angular momentum is that in which the body has the greatest moment of inertia (with respect to center of mass). For example, a long chain thrown into free fall will twist and turn until it is all straight and rotating as rigid body.

...

If LL is constant (net torque of external forces acting on the system is zero) and the constitution and initial conditions allow it, the system's dissipation will work to diminish energy until it has the minimum value, which happens for maximum IaIa possible.

august 2017 by nhaliday

gravity - Gravitational collapse and free fall time (spherical, pressure-free) - Physics Stack Exchange

august 2017 by nhaliday

the parenthetical regarding Gauss's law just involves noting a shell of radius r + symmetry (so single parameter determines field along shell)

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august 2017 by nhaliday

Introduction to Scaling Laws

august 2017 by nhaliday

https://betadecay.wordpress.com/2009/10/02/the-physics-of-scaling-laws-and-dimensional-analysis/

http://galileo.phys.virginia.edu/classes/304/scaling.pdf

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf

Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.

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http://galileo.phys.virginia.edu/classes/304/scaling.pdf

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf

Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.

august 2017 by nhaliday

How large is the Sun compared to Earth? | Cool Cosmos

august 2017 by nhaliday

Compared to Earth, the Sun is enormous! It contains 99.86% of all of the mass of the entire Solar System. The Sun is 864,400 miles (1,391,000 kilometers) across. This is about 109 times the diameter of Earth. The Sun weighs about 333,000 times as much as Earth. It is so large that about 1,300,000 planet Earths can fit inside of it. Earth is about the size of an average sunspot!

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august 2017 by nhaliday

Tidal locking - Wikipedia

august 2017 by nhaliday

The Moon's rotation and orbital periods are tidally locked with each other, so no matter when the Moon is observed from Earth the same hemisphere of the Moon is always seen. The far side of the Moon was not seen until 1959, when photographs of most of the far side were transmitted from the Soviet spacecraft Luna 3.[12]

never actually thought about this

nibble
wiki
reference
space
mechanics
gravity
navigation
explanation
flux-stasis
marginal
volo-avolo
spatial
direction
invariance
physics
flexibility
rigidity
time
identity
phase-transition
being-becoming
never actually thought about this

august 2017 by nhaliday

The Earth-Moon system

august 2017 by nhaliday

nice way of expressing Kepler's law (scaled by AU, solar mass, year, etc.) among other things

1. PHYSICAL PROPERTIES OF THE MOON

2. LUNAR PHASES

3. ECLIPSES

4. TIDES

nibble
org:junk
explanation
trivia
data
objektbuch
space
mechanics
spatial
visualization
earth
visual-understanding
navigation
experiment
measure
marginal
gravity
scale
physics
nitty-gritty
tidbits
identity
cycles
time
magnitude
street-fighting
calculation
oceans
pro-rata
rhythm
flux-stasis
1. PHYSICAL PROPERTIES OF THE MOON

2. LUNAR PHASES

3. ECLIPSES

4. TIDES

august 2017 by nhaliday

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