chemistry    eFields    magnetics       Stat!    sciCalcs    'puters    history    people    cset215   


references are at the bottom of this page.
When this file is printed (rarely), I append to the end:

Conventions, special symbols

'==' is used to mean the 2 entities are equivalent, by definition, not equation. HTML doesn't support all the symbols needed for physics and some playful alternatives were used.
1.23¤6 is, for example, 1.23 times 10 to the power 6 or 1230000.
F = T χ V is a vector cross product in this document.
© is the curl operator. etc. The full symbol list is here
aberration of light

The aberration of light (also referred to as astronomical aberration or stellar aberration) is an astronomical phenomenon which produces an apparent motion of celestial objects about their locations dependent on the velocity of the observer. Aberration causes objects to appear to be angled or tilted towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is typically very small, on the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.

Aberration, wikip
action, joule-second
Units: 'Joule sec' (THINK what that must mean! weird)
J*s is energy multiplied by time but it is also momentum * distance, or angular momentum.
Planck's Constant (local).

temp changes but no heat/matter flow. PV changes. energy is transferred to its surroundings only as work.
one of the 2 types of thermodynamic processes in a carnot cycle, the other is isothermal'. See thermodynamic 'processes'.
refs: XAM p52, Adiabatic_process, wikip PT1_Q56 adiabatic expansion. What varies when an ideal gas expands freely in a perfectly rigid and insulated cylinder? This adiabatic process involves no gain/loss of heat. Only the internal energy changes.

aerodynamic drag (parasitic)
N = ½ rho Cd Area_m² v² The force, in Newtons, resulting from 'wind resistance'. Specifically this is about the 'parasitic', not that associated with lift ("induced"). A sphere has a Cd of 0.47
sample problems

no distinguished point that serves as an origin.
no vector can be uniquely associated to a point. (so an AS can't support a mag field (?)).
can't add two points of the space. Subtraction to make displacement vector ok.
wikip, informal description
air density
.symb rho = 1.225 kg/m³. Serway p153.

from Arabic "al-jabr" meaning "restoration". Muhammad ibn Musa al_Khwarizmi 780-850 his book "The Compendious Book on Calculation by Completion and Balancing".

Algebra, wikip
my Algebra book
ampere, 'amp'
1 amp = 1 Coulomb/sec = 1/96485 moles of unitCharges / s = 1 C/sec
_electrons_ per sec? +charges/sec? either?
96485 is called faraday's constant
wikip, Ampere's circutal law
Ampere's Law

"magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) that the loop encloses." His original statement didn't contain mention of what later was called the 'displacement current'.
alt formulation. (line integral of B.dl) = µ0 * I , around any closed path XAM p80
Related: Maxwells_equations, local notes

Amp-hours. (AH)
A big car battery might be labeled "130 AH" suggesting it can store (and supply on demand) 130 amps for an hour or 1 amp for 130 hours. The Peukert Effect shows that the simplest model doesn't work.
In the following:
L is angular momentum, a conserved quantity, resistance to changing rotation speed
    L does not have to be aligned with axis of rotation (full treatment requires a tensor)
    I believe L is a pseudo vector
I is moment of Inertia. It's like mass in the (linear) momentum eqn
    but its role here is to (also) handle the distribution of mass wrt the spin axis.
    Basically the sum of all masses times their (r=distance to axis)².
m is mass, kg but in 'm²' and 'm/s', it's meters
r is radius, m
v is velocity, tangential
w is ω, > 0 for ccw rotation. Its units are really 1/s
L = I ω ; units: kg m²/s, Serway p319, XAM p15
L = m v r, where 'm' is mass. MDHS notes. kg m/s m
See rotation entry

ω has to be perpendicular to the motion

Or for objects which are small relative to the distance from the turning axis (planet), their L can be
L = r χ mv # vector cross product. Serway p315.

Since torque = dL/dt, L can be considered Torque-seconds
T (N•m) = I angAccel; so T/I = angular acceleration

Most angular momentum problems are more involved than ball-on-a-string. The result, L, is still proportional to the spin rate but the computation focusses on I, the moment of inertia which is, in full generality, a weighted integral of the mass as distributed about the spin axis.

see moment of inertia

see radius of gyration

see wikip, Angular Mom for:
     A 'pseudo vector'  when in 3D.
     L = (I=r² m) (ω=(r χ v) / r², angular velocity)
     Angular momentum units can be interpreted as torque-seconds, work-seconds, or energy-seconds.
     L does not have to be aligned w/ turn axis. Can require a tensor.

point particle: Rvec χ Pvec   pseudo vector; torque = dL/dt   SOON: ref! 161115

notes from the old diagram:
    conservation of Angular Momentum
    I.b * ω.b = I.a * ω.a
    (I.b = m Rb²) * om.b = (I.a = m Ra²) * om.a
More work on this topic in pub/radiusOfgyration.html
The Miracle Year (of Einstein's papers)
wikip, Annus_Mirabilis
Anodes, Cathodes
In short:
    - the cathode is usually negative - but some applications switch this.
       - CRT is an example
    - Always: the anode has the opposite polarity of the cathode.
    - in 'normal use', when a device is 'running off' a battery, the cathode
      is positive and the anode is neg. This arrangement is called 'galvanic'
    - when charging a battery, doing electro-plating, or driving several types
      of chemical reactions, the polarities of the anode/cathode are reversed.
      This arrangement is called 'electrolytic'.
   "In chemistry, a cathode is the electrode of an electrochemical cell at which reduction occurs."
 ?? WHOA! "electrochemical" means the solution/cell is being run like a battery charger.
    (that is, electrons are flowing into the cathode (traditional 'current' flowing out...)
    and these just-arrived electrons are being 'accepted' by the solution.)
    Cathode, wikip
Archimedes (and his Principle of Bouyancy)
bouyant force = F = (m = rho * Volume) g
"Upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid."
For any ol' volume of water in water, this force is equal to that of its wt so no motion.
A's principle, wikip His book The Method of Mechanical Theorems, wikip. Find areas of regions and volumes of solids.
SOON: ref11 (plastic phys summary) adds atmos pressure to the above.
arithmetic progression
eg: 4, 15, 26, 37, 48, … (with common difference 11). (wikip)

arctangent function. avail in excel and python. didnt see in typ web search.
python: start Idle34, import math, math.atan2(yVal,xVal) * 57.3 # answer in degrees

101000N/m² = 101¤3 N/m² = 101kPa = ~760mm Hg a unit of pressure.

Thunderstorms charge the ionosphere to ~400kv wrt the surface. the earth is generally positive, ionosphere negative. Clouds will move electrons around so that the bottom of a cloud is negative ('looking' at + ground). Cloud top +. This electric field decreases with altitude. Atmospheric ions created by cosmic rays and natural radioactivity move in the electric field, so a very small current flows, even away from thunderstorms.
Near the surface, the e-field is ~100 Volts/meter.

Note that the horizontal axis represents electron density.

Diagram taken from wikipedia's page on the ionosphere.

wikip, 'Atmospheric Electricity'
wikip, 'Atmosphere of Earth'

atom, components
binding E, SOON

hadron: composite particles, strong force

members: baryon (3 quarks), meson (1 quark, 1 anti)
    baryon  made of 3 quarks. eg proton, neutron
        spins: ½
        neutron: free neutrons decay with a half-life of about 611 seconds
    meson   made fm 1 quark, 1 antiquark. eg: pion, kaon
        spins: 0
        pion: decays in 26ns  ; 'pion' = pi meson.
            members: π0, π+, π-

boson: force carriers, not matter

members:  gluon,photon,Z boson,W boson,Higgs
spins: integers
stats: bose-einstein   (aka 'bosonic' (?) )
gauge bosons are force carriers.
   photon (electromagnetism)
   3 W's and Z of the weak force
   8 gluons of the strong force

fermion: all quarks and leptons. assoc'd w/ matter

members: protons, SOON
spins: half-integers
stats: Fermi-Dirac
Pauli Exclusion P: true
types: b/c of quarks, protons are fermions

lepton: no strong F, beta decay, neutrinos no chg

members: electron,muon,tau; & neutrino versions of previous 3
    muons & taus only seen after high E collisions.
spins:   1/2 integer.
forces, and their relative strengths: gravitation 0, weak 25, E&M 36, strong force 38
antiparticle: true
ref: wikip


members: up,down,charm,strange,top,bottom.

spin theorem
the "integer" of spin is 'hBar'!
chemistry file has blob of info typed in 180301
atom, structural models
1897 Thompson discovers the electron
1904 Thompson promoted his "Plum Pudding model" model.
1909 Rutherford discovered the nucleus  PT1_Q22
     Rutherford Scattering.  Planetary model vs Thomson's Plum Pudding...  PT1_Q22
1911 Rutherford advanced the Planetary model  Rutherford_model, wikip
1913 Bohr (and Rutherford) developed a very good model of the Hydrogen atom, able
     to answer questions of spectra. Bohr_model, wikip
1932 valence shell model. Atomic_orbital, wikip
atomic mass unit, 'u'
# of grams in 1 mole. NOT 'kg'! Carbon is defined to be 12.0000 u's or 12g/mol
Atwood machine
# acceleration: (Serway p 117)
T - m1 g = m1 a # eqn '1'
m2g - T = m2 a # eqn '2';
# add eqn 1 and eqn 2, factor g on left, a on right
g (m2 - m1) = a (m1 + m2)
a = g (m2 - m1) / (m1 + m2) # eqn 'accel', 'a' is acceleration

# line tension calc: --------------- PT1_Q91
# substitute eqn 'accel' above into eqn '1' (or '2'): (ref: wikip 'atwood machine')
T - m1 g = m1 a
m2g - T = m2 a # add these. Not sure I follow the algebra...
T = 2 g m1 m2 / (m1 + m2)
online atwood calculator, free
Avogadro's Number

all gases have same # molecules/mole. 6.02¤23/mole. N subCapA

Wikip: "the field of mechanics that deals with the launch, flight, and behavior of projectiles, ...
and the science or art of designing and accelerating projectiles so as to achieve a desired performance." (edited)
prjt ballistics exists to collect
equation info.

The projectile path is a parabola. For a shot where the target and launch are at the same
elevation, the max height of the path is at x,y=(½ distance, ¼ distance).
refs: TODO add to sciCalcs, update bad links (teach/ gone...)

prjt catapultCA SAFE. bean bags. small, cheap, 30' range. 1802xx
prjt catapult405 AGGRESSIVE. clean up, merge. motivor.rb. PRINTs, 1506xx
prjt teach/robotics has old diagram. think I have better 1602xx
teach/robotics/trajectory.png old diagram. think I have better 1602xx
prjt robotFirst made, gd graphs. 171011
archived prjt 2014for throwing pet treats. powered by tape measure 1409xx
beam (balance)
balance a 'beam', 'see-saw', 'teeter totter' XAM_Q14, p29; XAM 172
ref12, item 26: m1 x1 = m2 x2 (torques cancel).
beam (mechanical)
max deflection w/ normal force pressing beam center, length L.
W = F L³ / 48 E I where E=Young's Modulus.
'I' is the second moment of area of the beam's cross-section. 'I' must be calculated with respect to the axis which passes through the centroid of the cross-section and which is perpendicular to the applied loading. Often, the product EI (known as the flexural rigidity) is a constant. wikip
beat frequency
The beat frequency = difference in frequency of two waves (absolute value).
Becquerel, Bq
"SI derived unit of radioactivity",
1 Bq == 1 nuclear decay per second. Units 'per second' or 1/s
Bernoulli's principle

1738. Bernoulli's Law.
The venturi is frequently the example but the principle applies to wings too..

When speed increases in a venturi, the pressure decreases. (incompressible fluid) principle derived fm Conservation of Energy (KE,PE,Internal).

XAM p7 notes that the 'equation of continuity' v1 A1 = v2 A2 where A's are cross sections in a pipe or venturi.

Biot-Savart Law
This law quantifies gradient dB as a function of distance and angle for a fixed current. The B field is the magnetic field associated w/ electro-magnets. The equation is a line integral over the (wire) path C in which the electric currents flow. The result applies only to a single point, r. Symbol r' is the distance to r from whatever point on the wire is being integrated. ('J', when it appears in some derivations, is 'current density')
  • I is a DC current. r' is the displacement vector, r-l (wrt the wire)
  • 1/r³ only occurs when a full r' is in the numerator (not r hat)
  • The 1 / r² changes in other geometries. See Ampere's Law.
  • The equation's output, B, in SI units, is T m/A.
  • Flipping the order to r' x dl uses a more standard RHR. IMHO

µ0 is a true constant so the expression above doesn't, as written, provide for differing permiability. And the 4π denominator undoes most of µ0. This probably just means that their 1820 efforts predated the standardization efforts which produced units like the Tesla.

References: ref03, XAM p79, Physics Rev Shts, simple, E&M section H. diagram w/ right hand rule
apparently a 2 part vector, as found in cross products. Seen in wikip.
black body
see Stefan-Boltzmann Law (local), and Planck's constant
black hole
an object falling into a black hole appears to slow down as it approaches the event horizon, taking an infinite time to reach it. Black_hole, wikip
block and tackle

image of a double tackle. from wikipedia. Block_and_tackle, wikip

Boltzmann's constant
The Boltzmann constant, 1.38¤-23 J/K, is the constant relating energy over temperature at the particle level, which must necessarily be observed at the collective or bulk level.
It IS the gas constant 'R' divided by Avogadro'sNum (#/mole, often written 'NA')

Written as a lower case 'k' (fgh likes 'kB', wiki uses 'Kb'. b/c there are lots of k's !)

    kB = R / NA = (R=8.314 J/(K•mol) / (NA=6.02¤23/mol) = 1.38¤-23 J/K    

(used in 2 contexts, one "times Avogadro's constant" (to remove the
division in the above paragraph, leaving it as just 'R', the gas constant.
Preferred units seem to be J/K or Joules per degree K (absolute).)

Used in ideal gas law, below
wikip on Boltzmann_constant
Serway p543.
objecto will float if
overall_density of object < density of fluid
ρo < ρf     =>     masso / vol < massf / vol
    Mb = M0 ( 1 - pf/p0 ) where
    M0 is vacuum mass of object, Mb is bouyant mass,
    pf is the density of the fluid, p0 is the density of the object
Boyle's Law
if T const, then P,V inverses see Gas Laws
Brewsters Angle

Brewster's angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection.

When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster (1781–1868).

ref: Phys Eqn. item 49 or so
brownian motion
The random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid.

In 1827, while looking through a microscope at particles trapped in cavities inside pollen grains in water, Robert Brown noted that the particles moved through the water; but he was not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules, making one of Einstein's first big contributions to science.

BTU, 1060 Joules
British Thermal Unit.
1 BTU = 1060 Joules.
1 kwh = 3400 BTUs            =  3604000 J, 3.6¤6
1 gallon of gas = 12500 BTUs = 13250000 J, 1.3¤7
25mpg is 5000 BTUs / mile
Serway p554. Joule is a better choice of units than BTUs
Bulk Modulus

A material's resistance to being squished. It's the inverse of compressibility and used, for example, in calculating the speed of sound in a medium.
speedOfWave = sqrt(B=Bulk Modulus=volumeCompressibility / densityOfMedium) looks wrong. SOON

calculus of variations

ref'd in feynman, book 2, pg 19-3

calorie, 'C'
1 cal= 4.184 J, the energy needed to raise 1 gram water fm 15.5C to 16.5C, Serway p554.
Serway p554 "the mechanical equivalent of heat"

1 Cal = 1 kCal = 1000 cal.  // nutrition calorie. Cal (with the capital C). 1887
10¤7 ergs = 0.239 cal = kwh / 3.6¤6  // cal= kwh/3600000 x 4.184

0.239 cal = kwh/3600000
0.239 cal / 3600000 cal/kwh = 1 kWh
860400 cal = 1 kWh

1 cal = 4.184 * wh/3600   // wh/3600 = watt-second == Joule
      = 4.184 * watt-seconds

marathoner burns 100 kCal per mile.
human body can store 2000 kCal => old runners 'hit wall' at 20 miles.
walking burns 1/8 kwh per mile. whpm=125 !

def cal2boilWater(gallons,tInit_F) : # derivation, code   
    '''To boil 'gallons' of tInit_F (Farenheit) degree water requires'''
    delDegC  = 117.7 - 5/9 * tInit_F
    gramsPerGallon = 3785
    cal =  gramsPerGallon * gallons * delDegC

"put unknown in water, measure T change. deduce specific heat of unknown" SOON_180128: cite or give example, proc wikip

Farads = chargeInCoulombs / Volts. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates.

Another view is: 1 amp * 1 sec (gives 1 Coulomb) generates 1 F (only in a 1 F cap (?). think not...)

The human body has a capacitance of about 150 pF (wikip on Plasma_globe).

Parallel plate capacitance: C = eSubR * e0 * A / d,

    where eSubR is a 0->1 scale for e0, 'd' is the separation distance,
    'A' is the area of the plates.  I assume all SI units.
    e0 units are Farads/meter. // 8.85¤-12 F/m
uF  micro  ¤-6
nF  nano   ¤-9
pF  pico   ¤-12
?F  femto  ¤-15   no such capacitor value (?)
Serway, vol 2, chap 26, page 723 discusses/diagrams parallel plate cap.

PT1_Q82 tips for fast calcs of parallel and series caps.
    C1 || C2 (add like R's in series) = C1 + C2
    C1 && C2 (add like R's in parallel) =  1 / (1/C1 + 1/C2) = C1 * C2 / C1 + C2
  For > 2 parts: // eg 3
      c12  = c1  * c2 / (c1  + c2)  then  c123 = c12 * c3 / (c12 + c3)
dielectric, local notes
wikip covers many related topics, eg 'stray cap'
tuning cap, Legendre!
capacitive reactance

X(subC) = 1 / (ω C) XAM p93


energy stored is ½ C V² XAM p92

ref21, p19: Capacitance, C = e1 A / d where e1 is the 'relative permittivity' (aka 'dielectric constant'), e / e0. In the above, 'A' is area, 'd' is distance between plates.

Also the 'constant' is really freq dependent, ie. not constant.
capacitor, multi-layer
R,L parts are 'wired' in parallel.

Sum the reciprocals of all the caps. Then invert the sum to get final C.

Series of caps make a 'final C' lower than starting values. Consider 4 10uF caps. Cfinal =
= 1 / (1/10 + 1/10 + 1/10 + 1/10)
= 1 / (4/10) = 10/4 = 2.5 uF

carnot, carnot cycle
2 heat reservoirs,2 isothermal, 2 adiabatic 
    French physicist Sadi Carnot, 1824
    XAM p52, Serway p618
    work done is area w/in PV lines.
    1. isotherm Th. absorb heat,   Vol++  P-
    2. adiabatic expand. T,P fall, Vol+   P--
    3. Isotherm Tc. release heat.  Vol--  P+
    4. adiabatic compress. P up,   Vol-   P++
Carnot cycle efficiency = [ (QH - QC)/QH ] * 100 where
    QH is Heat In, QC is Heat Out, and W is Work provided.
    W = (QH - QC)
PVC catapult, seen online. ?instructables?
prjt 'cat45'
see anode, cathode entry.
cathode ray
beam of electrons! aka 'e-beam', PT3_Q54
Coefficient of Drag. principally comes from the shape of an object - but the value doesn't stay constant if the object is scaled up or down. And viscosity affects value too. aero drag, local has skydiver examples, terminal velocity.

α = v²/r ; force = m α
centripetal force on car tires in a turn is m v²/r Serway p140. XAM p17, p142, XAM_Q08


_mass_ center (not just spatial center)

Centroid examples

Centroid, wikip


Center of mass, center of gravity

chain rule manipulation

Serway P174. TODO. don't understand last step.

characteristic impedance


charge on 1 proton or 1 electron is 1.602¤-19 C, where C is Coulomb
Charles Law,"Law of volumes"
if P const, T,V proportional
see Gas Laws

chord(θ) = radius * (2 sin(θ / 2) )

Period, T = 2π  r / v
uniform motion. Serway p85
See centripetal acceleration, α = v²/r   // v is the vel of the tangent line at the edge
cloud chamber
see isopropyl
notes on making CO2
photosynthesis notes. local.
CO2 entry in chemistry page
baking_soda = Sodium bicarbonate, NaHCO3
baking_powder = baking soda + some stuff
Cockcroft-Walton. Voltage multiplier
have built a 3 stage unit from purchased kit parts. needs testing.
Possibly useful for project eFields
coef of linear expansion

wrtn 'α'. ΔLen / Len = α ΔT (kelvin) XAM p44

momentum = mv, always conserved.
elastic    KE also conserved. KE in each dim (x,y,z) conserved.
   "An elastic collision is an encounter between two bodies in which
    the total kinetic energy of the two bodies after the encounter is
    equal to their total kinetic energy before the encounter". wikip
inelastic. KE not conserved. usually heat produced. often bodies coupled as result

A basketball-tennis ball collision was a part of PT1_Q24. So I googled 'collision basketball and tennis ball physics'. lots of pages on this.

basketball hits floor at 10mph, bounces upwards at 10 (ok! pretend!). It meets the little tennis ball. They collide at 20mph. Relative to the basketball, the tennis ball leaves the collision with an energy of 20mph plus its speed. So it rockets up at 30mph.

Ok, someone check all the momenta, KEs ... SOON

from rocket balls

see KE, local
Conduction, Thermal
     k t A deltaT    t=time,secs, A=area,sq cm, deltaT is in degrees C or K
H = --------------
k is the Therm Conduc, a property of the mat'l

XAM p44. See Law of Thermal Conductivity
conformal (map): angle preserving on a small scale.
A book written by Apollonius. got wide circulation.
entities which share some simple but special relationship: examples:
- the other root of a quadratic
- for a complex number a+bi, it's a-bi
- for a point on a sphere, it's the oppositie point, joined by a line thru the center.
conjugate pairs
See uncertainty principle
controlled experiment
See wikip
Coulomb (quantity)
6.24150934¤18 charges per Coulomb;    1/1.602¤-19 = .624¤19 or 6.24¤18 _reciprocals_
1 Amp = Coulomb/sec, 1 Coulomb = 1 Amp-sec

1.602¤-19 C is a _quantum_ of charge.
Coulomb counter, sensor
 TODO: buy!
Sparkfun's LTC4150, $13
Sparkfun LTC4150 pdf
I think the design is clever b/c the sense resistor is independent of the actual
circuit - meaning it should work in a wide range of environments.
Coulomb's Law
Coulomb's Law. Force (repulsive) F = (Ke/Kd) (Q1 Q2 / r²); Q's==charges, result N Newtons
Ke¤9 or, more accurately, 8.988¤9 = 1 / (4π e0) ;
Coulomb's Constant='electrostatic constant',Ke. Kd here is my term for the
dielectric constant of the medium (==1 for vacuum, nearly 1 for air)
chap24, p1, kindle loc 6288, ref09
# alternative view (?): wikip, vac perm
critical angle
The angle of incidence where the exit angle becomes 90°.
TIR, Total Internal Reflection, occurs for angles greater than or equal to the critical angle.
critical angle = arcsin(n2 / n1), where n2 must < n1; 
Example: rays rising into air (n2=1.0) from water(n1=1.333), the critical angle would be
    arcsin(1 / 1.333) = arcsin( 0.75 ) = 48.6 degrees
Angles are measured wrt the 'vertical' (perpendicular to boundary)
Viewed through Snell's Law, n1 sin(θ1) = n2 sin(θ2 == 90°)
PT1_Q19, graphic; PT2_Q46, text, eqns
wikip, critical angle
cross product

r χ F: "r cross F" (the r,F order matters)

The magnitude = r.len * F.len * sin(angle)
( usually written r F sin(θ) )

The vector can be formed w/ this magnitude and the unit vector found w/ right hand rule

length of A χ B, denoted |A χ B|, gives the parallogram area. Halve it to get area of triangle!

A χ B = -(B χ A) # order matters
cubit, about 20"
(wiki) tip of middle finger to elbow.
cuneiform tablets 1 of the writing methods (hieroglyphics the other).
clay tablets. Babylon. 2500 BC (? guesses...)
curl, a vector operator

a vector operator that describes the infinitesimal rotation of a 3D vector field. At every point in the field, the curl of that field is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point.

The curl operator will, for now, be ©. 180310

The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

The alternative terminology rotor or rotational and alternative notations are often used.

Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation Del Cross for the curl.

The name "curl" was first suggested by James Clerk Maxwell in 1871.
curl, wikip

centripetal accel, v²/r = Bqv, implies B = v / (r q) sciCalcs, test notes/problems
dark energy
vacuum energy, wikip
dark matter
Red line is what universal gravitation law would make. Blue is what's observed. Gray are error bars. 'kpcs' is kilo-parsecs.

Milky Way, rotation, wikip

The MOND theory proposed changes to the universal gravitation law. MOdified Newtonian Dynamics

dark matter, wikip
Dawes Limit
No, not approaching headlights on a foggy road. The image was synthesized using equations describing diffraction, subject to the limit defined by Dawes.

Dawes Limit comes down to the following calculation:
R = 11.6 / D where:
D is lens diameter in centimeters,
R is the resolving power, in arcseconds.
yes, these aren't SI units.

decibel, dB
logarithmic measure of sound energy or electrical signal amplitude, referenced
to some standard value. As in wiki article, these units are used for
field   20 log10(F/F0)   Fld=10^(Lf/20db) * Fld0
power   10x log10(ratio)   10dB for 10x power ratio
Being a ratio, dB doesn't behave like true units. In the unit 'dBm', the 'm' means. Such suffixes are not supported by ISO etc. In the unit 'dBV', the 'V' means Volts. The value preceding the dB is 10x the logarithm in question, so 30dB is really a log10 value of 0.3

opposes field. The magnetic dipole of a diamagnetic material being placed on a magnetic field aligns itself to the field of opposed polarity.


material which increases the capacitance of a cap by reducing the internal E field by polarization. That is the dielectric makes a bit of an opposing field thus reducing the base field. The dielectric's field opposes that of the cap's plates. The dielectric is said to be a bit of an insulator too.
"Opposes the field charges on the plate". PT1_Q73

A very short table of dielectric constants. see wikip for more.

vacuum1.0000  _
air1.0006  _
paraffin2.2 rimstar only attempt gave ~1.0
min oil2.3wikip _
pure water   ~ 80. my 'distilled' gave 2.8 !
Note that very few substances get constants larger than 3. Except water!
My homemade caps, to look at dielectrics, were made as a part of project eFields

A dielectric (or dielectric material) is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

wikip was the main ref
vv_permittivity has a note about this.
electric field has related material
diffraction grating
d sin θ = lambda
θ Max(m) = arcsin( lambda / d - sin(incidentAngle)
order 'm' (m 0 not interesting...) , d is distance between grating rulings.

A spot lights up if all the incoming rays constuctively 'interfere' with each other.
That occurs if the distances travelled for all the rays is a multiple of the incoming
dimensional analysis
basics plus more complicated examples
(dimensional analysis is also called 'factor-label' or 'unit-factor'. ref schaum's chem book kindle)
When a conversion is 'affine' like Farenheit-to-Celsius, a function can be provided. see R factor

    mi   mi   3600 s     1   mi
    -- = -- x ---- -- x ---- --
    hr   s     1   hr   1621 m
    note that such units terms start w/ a simple expression (equiv to 1 !)
    1 hr = 3600 sec   and can form 2 units terms.
    1st divide both sides by 3600 ; next divide both by 1
    1 hr                            3600 s
    ------ = 1                      ------ = 1
    3600 s                           1 hr
    AND 'you can alway multiply anything (any term etc) by 1'  justification...
    the use of '-' for a multiplication symbol. supposed to be a flying period.
dimensions: area and volume eqns
0.7854 D² = circle area. 0.7854 L * W = ellipse area. (π/4)
0.5236 D³ = sphere volume (π/6)
area of a sphere = 4 π r²
cross product gives area of parallelogram and/or triangle.
unit "per meter", reciprocal of focal length in meters.
A unit of measurement of the optical power of a lens or curved mirror.
For example, a 3-diopter lens brings parallel rays of light to focus
at 1/3 metre. A flat window has an optical power of zero diopters,
and does not converge or diverge light.
Eye glasses at 1.25 diopters focus at 4/5 meter = 31.5 inches
1.75 diopters focus at 22.5 inches
diode internals
Diodes are made from a single piece of Semiconductor material which has a positive “P-region” at one end and a negative “N-region” at the other, and which has a resistivity value somewhere between that of a conductor and an insulator.

PT2_Q68: -ve N-region,  +ve P-region, conductor, insulator, resistance SOON
A diode does not behave linearly with respect to the applied voltage as it has an exponential I-V relationship and therefore can not be described simply by using Ohm’s law as we do for resistors.

Notice that there is a very small margin between the resistivity of the conductors such as silver and gold, compared to a much larger margin for the resistivity of the insulators between glass and quartz.

Note also that the resistivity of all materials at any one time also depends upon their ambient temperature because metals are also good conductors of heat.

Silicon's outermost shell could contain 8 electrons but neutral Si has 4. This ?promotes its lattice structure and provides electrons under certain circumstances. The P and N doping (1 per 10^7) shifts the diode material's tendencies (+/-).


diode, Skottky Diode
faster than silicon. better reverse leakage at lower freqs. smaller Vf (forward voltage drop) than silicon (eg 1N400x) thus less power Power, Watts = Vf * Iamps

dipole,   dipole moment,  &   dipole field

Some thing or volume has an opposing difference, + vs -, either electric or magnetic. Hence, di - pole. Important to the dipole concept is its "moment", a vector quantity representing the strength and spatial distribution.

The dipole doesn't have to be a magnet or a blob of electric charge. It can be a magnetic area created by a current flowing thru a loop.

Polar compounds like hydrogen fluoride (HF), can have a significant dipole moment.

From an article at
"A dipole field is any field of force that is created by two poles separated in space, such as any electrical or magnetic field. The resultant force from a dipolar field varies as r-3, instead of r-2. A dipolar magnetic field can be generated by a magnet or by an electrical current; it was this realization that lead James Clerk Maxwell to the equations which unified the electrical and magnetic forces into the electromagnetic force."

b² - 4ac
is 0 iff eqn has 1 real root
> 0 iff eqn has 2 real roots
< 0 if eqn has imaginary root
The wiki article mentions polynomials of higher than degree 2 and their discriminants.
quadratic discusses quadratic eqn, factoring
dispersion (of wave shape)
1. separation of a wave into its constituent wavelengths due to interaction with a material (eg thin film). examples: prisms, gratings. XAM p68. Red bends least (_note_)
Flint glass bends more than Crown BK7. Some prisms are plastic.

2. mechanical waves (not light) spread over time. Serway p450, btm.

Dispersion_(optics), wikip
Light's phase_velocity depends on its frequency.
da prism equation, PT2_Q73
Np/No = sin(½ * σ + δ) / sin(½ * σ), Np prism Index of R
I tried to derive this by composing 2 copies of Snell's Law but the angles used here aren't those used by Snell.

(A pretty but not so useful eqn if you're actually trying to find AngleOut as a function of AngleIn // and the indices of refraction...)

SOON. relate to Snell's so I can remember it.

displacement current

an offset current added to Maxwell to Ampere's eqn XAM p81
See Maxwells_equations, very last equation box.

div, divergence

a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink.

Divergence, wikip
Maxwell's Eqns, wikip
doppler shift
f' = f  * ( v+vo  /  (v-vs)  )
where: v = speed of wave through air
vo = observer speed, vs = source speed.  WITH RESPECT TO MEDIUM
v = 1533m/s, vo=9m/s, vs=8m/s. f' = 1415 Hz
rules for choosing signs

examples/ explores speeds, speed needed to shift music notes
XAM p69, XAM p149,XAM_Q35
Serway p496-499,p483

PT1_Q29: 2dufus' ea w/ freq gen. duf1 walks to duf2. what hz heard by duf1 (all such 'peaks') fu/v
        where v is speed of sound; u is walking speed, f is freq of emitted sound
        A hears the sound of his own source, which frequency is f. He also hears the
        sound of the moving source carried by B. The apparent frequency of the source
        with B is f(1+u/v). The beat frequency therefore is f(1+u/v)-f = fu/v.

PT1_Q47: Fr = Fs * c / (c - 2u) where u is meteor's speed. Fs 10Ghz. Fr 12.5  SOON review
dot product
operation on vectors, aka 'scalar product'. 2 methods:
    a º b = Sum (a[i] * b[i])
    a º b = ||a|| ||b|| cos(θ)    vector op
An example of using the dot product to construct vector components can be
found in the class functions in
double slit experiment
The image of a 'peak' is called a 'fringe'. FYI

Tests often ask for the wavelength.

(show: slanted lines are each Integer * wavelength. TODO)

PT1_Q04 test question.
PT1_Q48 ?  Young's experiment (white light(?) in, colored peaks(?))
wave-particle duality.
    wavelength = xd/(nL)   // more readable as "x/n d/L" b/c d/L is slope
    x dist from central pk,
    d dist between slits,
    n order of the fringe.
    L dist, slits to screen

PT1_Q04, PT1_Q50
PT3_Q91 is really about gratings
'e', Euler's number.
e = 2.718281828459

e is the base of the natural log system, 'ln'.

The letter 'e' was chosen b/c it's the 1st letter of Euler !

amazing properties shown at left! oooh !

the Electric Constant, 8.854187¤-12 F/m, Farads/meter.
see more complete story at 'permittivity'
E = mc²
book "Why E=MC²"
Maxwell's eqns used measurements of 'E' and 'B' fields (from Faraday(?))
in producing a wave propagation velocity which happens to be C !
earth: mass = 6¤24 kg, radius = 6.371¤6 m (6 3/8 ¤6),
density 5.5 * water Serway p5.
eccentricity of orbit: 0.017, Serway p368 // Halley's Comet 0.97 76 y period
    problem: what angle/day at perihelion,aphelion at what calendar days.
    (earth: jan 3 is roughly perihelion, july 3-4 aphelion)
earth's mag field 0.25 - 0.65 gauss (wikip). ~0.49 Gauss in WC, 10000Gauss/Tesla
North magnetic pole location is 78.5N, 103.4W. moving towards siberia ~35 miles/year !
The GEOmagnetic N pole is where the large dipole field seems to locate the mag N pole.
    2017: Lat. 80.5°N, 80.5°S; Long. 72.8°W, 107.2°E   wikip. GEO moves v slowly.

ellipse x²/a² + y²/b² = r²,
ecc = sqrt(1 - b²/a²). 0 < ecc < 1.
where a=semi-major axis, b=semi-minor. axes are center to edge
ecc * semi-major axis = linear eccentricity == distance fm foci to center (wikip)
ellipse, wikip
for circle x²+y²=r², e is 0, equiv to ellipse w/ a=b
Other ecc's are better defined by the end-to-end cones (conic sections).
parabola e is 1; hyperbola e > 1.
Eccentricity, wikip
Orbital eccentricities:
    0.0167 Earth
    0.0549 moon
    0.0934 Mars. eccentric enuf to help Kepler.
    0.21   Mercury , the largest of the 8 planets.
    0.25   Pluto. actually a little less than .25
    0.97   Halleys Comet
For the variability of earth's orbit caused by Jupiter, see
eddy current
loops of electrical current induced within conductors by a changing magnetic field in the conductor due to Faraday's law of induction. Eddy currents in conductors of non-zero resistivity generate heat as well as electromagnetic forces. The heat can be used for induction heating. The electromagnetic forces can be used for levitation, creating movement, or to give a strong braking effect. Eddy currents can also have undesirable effects, for instance power loss in transformers. In this application, they are minimized with thin plates...
electricity calcs
I'm too slow... etc
calc power dissipation, DC:  P = I² R
A/C values. look for website which summarizes.  SOON
PT1_Q26 ++.
electric power, generation
V(t) = NAB ω sin(ω t), N = #turns, A=coil area, B= mag field strength,
V == induced voltage in an electric generator. Serway vol2, pg 881

The ω next to the NAB comes from the chain rule while differentiating -cos(ω t)

XAM p85 has similar generator equation. text not as helpful.
electric field
" ~~~ watch ~~~ the ~~~ units, ~~~ Luke". Obiwan...
F = ke q1 q2 / r² # units: N, Newtons, Coulomb's Law
The electric field, E, a vector field

E = ke  SUM ( qi / ri² ); # units: N/C, Newtons/Coulomb
Knowing 'E', the force for a particular charge can be calculated as

F = qE         # units: N, Newtons, again
Note that N/C == V/m == Volts/meter == kg m / (s³ * A)

V, volts is surprisingly different than E above. V is a scalar field with a 1/r calc.

It acts like gravitational PE in that all that seems to matter is distance above a value chosen to be 0 ("ground"). Some material seemed to ignore (even) the sign, treating -v the same as +v.

The 1/r aspect is visible in the units of V: note the 'm; in N•m/C while E's units are just N/C. Nominally V's units are J/C which equals N•m/C.

J/C comes from W=VQ or V=W/Q

Also, it's worthwhile to fold in the dielectric constant of the medium so the larger process is visible. I'm using Kd below to represent a dielectric constant.

Replace ke with ke/Kd

Lastly, the equations can be written in a very readable shorthand
F = ke       q1 q2 / r²    # or
F = ke/Kd q1 q2 / r²    # with a dielectric, Kd

Notes, links to related info:
    'van de graffs work best in low humidity' implies conductivity of air increases w/ humidity.
    electric field project, electric field sciCalcs
    electric field, wikip
    plasma, gsci
    Electromagnetism, wikip
electric field lines
"point in the direction that a positive test charge would accelerate if placed upon the line."
the physics classroom
electric field, HV
Atmospheric_electricity, wikip
St Elmo's Fire requires 1k to 30kv per cm. lower end of V range works w/ pointy ends. N&sub2; glows blue
electric field, making an
ping pong balls circling bowl
Recall that electric fields stronger at pointy pieces. Arc, St Elmo's Fire, corona. Radius the corners.

Youtube 2inch balls are styrofoam?
Aaron Fay, Youtube
"3 ways to charge things
1. friction: rub balloon against shirt. electrons on balloon won't migrate (not conductive).
   e- to the balloon, shirt less e-
2. charge by contact: van de graff. little sparks. or put hand on globe.
   He has a metal globe on a stick, apparently connected to earth ground. 'discharge wand'
3. induction: fine pieces of conductive metal hang inside
   how to make an electroscope in an erlenmyer flask: Youtube
   erlenmyer flask has plate on top. vinyl rub w/ hands charges the vinyl."

Crash Course Phys #25, PBS Digital Studios, Youtube
2 pieces scotchtape ripped off table will repel each other. both are neg
then stick 1 tape to table and the 2nd atop the 1st. then rip off top
off btm tape, then lift btm off table. now they attract
glass rod, rub w/ cloth; cloth becomes neg; glass + // charge by friction
fur against amber. ancient greeks. amber would pull hair and feathers
touching 2 rods equalizes the charge between them. charge by contact.
// hand doesn't count ?
if the 2nd rod doesn't touch the 1st but is close enuf to feel the field,
    its charges will be redistributed such that the end nearer the fully
    charged rod will have an opposite charge.  It is "polarized".
charge by induction charges w/o contact.
'grounding' a charged rod will leak e- and leave rod + (?)
measuring the 'q', charge. unit is -1.602e-19 Coulombs // => 6.24e18 e- per C
F = Ke q1 q2 / r²  ; Ke = 9¤9 N m²/C² for air
electric flux
eFlux. units V * m
Dot product of the electric field and the normal vector to a surface (an area).
electric potential
conserved. Is a scalar thus simpler than dealing with the field all the time.
Serway, vol2, p692
electromagnetic radiation
EM_radiation, wikip
has cool, moving image of changing E and B fields in light.
EMwave3Dfromside.gif, wikip
_my_ electrometer, untested. I plan to add material to eFields.html about this. Its input impedance ~ 1¤12 Ω


"Most good electrometers have input impedance's of 200 teraohms ( = 2¤14 Ω)
The average electronic VOM has 10 megohms ( = 1¤7 Ω)"
the project
mass 9.11¤-31 kg
proton mass is 1.67¤-27
electron volt,'eV'
1 eV =  1.602176620¤-19 J (Joules)
     =  E (Joules) gained/lost by 1 electron moving across e-field of 1 volt.
     =  1 J/C sqrt(2 h a / µ0 e0) where:
        h is planck's constant
        a is the fine structure constant (from spectroscopy?)

eV: Mev (== J = Energy) is used as mass b/c E = M C² so Mev/C² is really the mass.
    It's inaccurate to say a mass is '10 MeV'; it's "10 Mev/C²" (kg).   SOON. check units
    1 eV/c2 is 1.783¤-36 kg so 1 Mev is 1.793¤-30 kg ~= 2 electrons

eV: MeV momentum unit when the true, measured energy, in MeV is divided by C, a speed.
    That changes the scale but the units are those of momentum.

eV is determined experimentally.
electrostatics, materials
"Electrons can be exchanged between materials on contact; materials with weakly bound electrons tend to lose them while materials with sparsely filled outer shells tend to gain them. This is known as the triboelectric effect and results in one material becoming positively charged and the other negatively charged."

"...a human being is represented as a capacitor of 100 picofarads (100e-12) F, charged to a voltage of 4000 to 35000 volts. When touching an object this energy is discharged in less than a microsecond. While the total energy is small, on the order of millijoules..."

wikip, static electricity
see local entry on lightning
electrostatic PE
PE = (q1 q2)/r  Ref "Physics" plastic
"std demo of es is prob 83 in CSET Physics Exam Guide" = plastic comb + dry hair = lg pos chg.
Comb brought close to sml bit of uncharged paper which is initially lifted to the comb. What
happens next?
PT1_Q39: stationary eDipole in a larger eField feels torque, no impetus to move.
PT2_Q83: just wanted calc. 2 protons, 3um apart.
Elements. compendium of geometry. gathered, refined by Euclid. ref26.
Have his book somewhere (tablet(?)). Impressive proof of the distributive property.SOON.ref?
x²/a² + y²/b² = r²
The coefficient names 'a' and 'b' are traditional as is the default assignment of 'a' to 'x' and 'b' to 'y'. More important is which of the coefs is larger as it determines the larger dimension of the ellipse, called the 'major axis'. Since the equation is centered on the Origin, it's common to use 1/2 these axis dimensions as 'semi-major' and 'semi-minor'.

The shape of an ellipse is determined by the ratio of minor to major. The size of an ellipse is controlled by scaling 'a' 'b', letting other params be 'whatever'.

ecc: eccentricity = sqrt(1 - (minor/major)² ).
ecc 0 is a circle while near 1 is a skinny ellipse.

foci, locating:

  • set x to 0; y² = r² * b²; y = sqrt(r² * b²) = r * b (& -b)
  • 'r' was 10/3.85=2.6; 2.6 * 3.85 = 10

The semi-major and semi-minor axis lengths are measured from the Origin, the center of the ellipse.

  • Set x or y to 0 then rearrange what's left of the equation
  • horizontal ellipse: x²/a² = r²; x² = a² r²; max.x = a * r;
  • For a vertical ellipse: max.y = b * r

"linear eccentricity" == Origin to focus distance

As I see it, there are 2 approaches to scaling/defining an ellipse:
  - 'math' considers the major-to-minor axis ratio.
  - 'orbit' starts with eccentricity and derives the ratio of 'a' and 'b'.

The product of 'distance to star' * 'orbital speed' is constant throughout the orbit. R1 * V1 = R2 * V2. See XAM p35, bottom

This ellipse shows an orbit in which the aphelion is 3.85 times the perihelion; the same number but totally different meaning. If Rb, the perihelion, is scaled to 1, the aphelion becomes 3.85. The major axis is now 4.85 . If we knew the minor axis, we'd have the 'a', 'b' coefficients to generate the ellipse (except for 'r', the scale factor).

The second focus will be Rb shy of the top of the figure. The semi-minor axis, 'smi', can be found from the triangle shown in the next section.

To find the dimensions of the minor axis (to eventually find the 'a' and 'b' coefs), the pictured approach is used. Given the vertical orientation, 'b' will be greater than 'a'. 'b' will equal f+Rb and 'a' will be 'smi'.

From the top focus to the bottom of the orbit is 2f + Rb. The bottom focal point contributes another distance Rb to the overall sum, now 2f + 2Rb. The distance from the 2 foci to point 'S', 2h, needs to equal the just calculated length or 2h = 2f + 2Rb.

smi = sqrt(h²) - f²)   # divided by 2, substituted 'smi', cancelled f and -f,...
        = sqrt(Rb²) + 2f * Rb)

graphing calc on windows (example above). See screen shot
click 'Explicit', pull down, choose 'implicit', then click 'add'
    fill in equation F().  eg: x² + y²/14.82
    G() should be 6.75 // the right side of the eqn
    click OK
    click blue dot to edit equation.
    something at the bottom suggests you can save the image as an .svg file
EMF = voltage
ElectroMotive Force
see electric field entry in this doc
energy, 'E'
The sum of the power you've expended to get something
    done (something moved, something lifted etc)

gallon of gas=33433.9 watt-hours  //SOON: these numbers don't all agree...
gallon of gasoline contains the energy of 130 megaJoules, or 36.1 kwh
Energy = Force x Distance  // ref/link: FxDequalsE.html
Enthalpy (of fusion)
amt of heat (to do something, like melt ice) XAM p43
see Hess's law.
Wikip: "enthalpy is a meas of energy in a thermodynamic system.
It's equal to the internal energy + (pressure * volume)".
Internal energy is the energy needed to create the system.
Units are Joules.
Typical symbol 'H'. for total enthapy.

XAM p40-42, units are J/K !

Ptolemy. Epicycles, wikip
SOON. excerpt fm wikip.  Serway p110.
equilibrium constant
solving chem output of a process using the quadratic equation. XAM p117-118
Equivalence principle
(Inertial mass) (Acceleration) = (Intensity of the gravitational field) (Gravitational mass). Equivalence_principle, wikip
An object in free-fall is in actuality inertial, but as it approaches
the planetary object the time scale stretches at an accelerated rate,
giving the appearance that it is accelerating towards the planetary
object when, in fact, the falling body really isn't accelerating at all.

Equivalence_principle, wikip
escape velocity
The V where an object's KE = gravitational PE.
G universal Grav            .667¤-10
M mass of Earth              6e24 kg
r dist fm planet's center.   6.3781¤6 m (wikip)
Earth's value is 11.186 km/s wikip
vis viva equation possibly interesting.
exponential decay math, 170714
heat loss, radioactive decay, RC circuit decay are examples of exponential decay,
'e to the minus x' stuff. Often lower case 't' is time, upper case 'T' is temperature.
heat loss example, text file
of a polynomial
my Algebra 2 book, chapter 5 (?)
quadratic discusses quadratic eqn, other poly topics
The printed copy of gSci includes a page 'Completing the Square'.
Farad, 'F'

One farad is defined as the capacitance across which, when charged with one coulomb, there is a potential difference of one volt. Equally, one farad can be described as the capacitance which stores a one-coulomb charge across a potential difference of one volt.

see capacitance for modifiers like micro, nano etc

'F' = C/V, charge per volt (=J/V² = N•m/V²) (see wikip)

Faraday's Constant
96485 Coulombs / mole. This connects Faraday's work with the chemistry's mole-based measurements.
The electrometer '3B' documentation shows and briefly mentions a silver (alum?) 'Faraday cup' and the documentation tells of physically moving the charge from a metal wand into the cup where it's measured by the 3B-VOM combination.

The online documentation, mostly wikip, describes a 'cup' as a part of an in-vacuum detector system.

Faraday's Law (of Induction)

Volts induced in a loop proportional to d(mFlux)/dt of the loop.

EMF = volts = -N dmFlux / dt where N is number of loops. neg sign due to Lenz, XAM p82 Faraday's Law

ref03: "F's law treats electric fields as product of moving magnets".

see Maxwells_equations, local

I wanna build this! (_metal_ ring needed(?) amplifies signal(?))

see 180221 short vids on Google Pixel2 phone. re inducing V in coil of wire.
FBD, free-body diagrams
See sciCalcs for interesting examples.

FBD, Real World Physics, BEST
XAM p21,22 has some simple but interesting examples

    choose coordinate system, inertial. polar or rectangular.
    May need to reassess mid analysis.
    Issue. 2D? 3D?

    Define a sign convention for your diagrams.

    replace contact locations w/ Force and/or Moment vectors.
    If a magnitude isn't known, assign it a direction ?and a variable name?

    Account for constraints which restrict motion

    replace body forces (eg gravity) with F's thru CM

    simplify. eg: assume constant friction...

    draw FBDs for each body.

    When drawing the forces acting on pairs of bodies in contact with each other, apply Newton's Third Law.

    may need concept of a state machine to handle changes
    eg: model catapult to point object is launched. separate state for decel.

FBD Serway p108,118. // probably influenced by traffic light problem
    draw only those force that act of the objects being isolated.
    Draw ALL forces, including field forces, such as Grav
    The forces are vectors, the (x,y) components will be in the simultaneous eqns.
        (be careful with signs...)
    The X equation should equal 0
    The Y equation should equal the load
    Reduce the X equation to reveal the relation between the 2 tensions.
    substitute into the Y equation.
    pg 109 Serway's "Physics for Scientists and Engineers"
    Neglect friction (usually)..
    If the object is not being accelerated, then all the forces must sum to 0.
    see 'Atwood' as an example
    Physics book authored by Serway explained it well, have exercises.

FBD, Atwood, tensions
FBD traffic light
Free Body Diagram.
What are the tensions T1 and T2, in Newtons, if:
a1 = 30;    # left most angle wrt to horiz. 0-90 degrees
a2 = 45;    # right most angle.
T3 = 90;  # light's weight, Newtons)
T3.x = T2 * cos(a2) + T1 * cos(a1) # horiz force must = 0.
T3.y = T2 * sin(a2) + T1 * sin(a1) # to support light
T2overT1 = cos(a2) / cos(a1)   # ratio of horiz pulls
T2 = T3 / (sin(a2) + T2overT1 * sin(a1))
T1 = T2 * T1overT2
CAUTION: ez to omit the T2overT1 element in a test. 
see Serway, pg 111, bottom and
FBD for friction on a ramp
This entry re-examines the (common) slide-down-a-ramp exercise but focuses on using FBDs. See simpler definition first. Also look for the "ball rolling down a ramp problem"! signif harder.
define what the answer should look like

Variations ask 'at what angle does block start sliding'

FBD for ball on a ramp
At the ramp bottom the total energy: PE=0, KE linear speed, and rotation energy.
XAM p35
FET, Field Effect Transistor
Feynman diagrams
wikip, Feynman_diagram
simple wikip
see also QED, local notes
ficticious forces
centrifugal, corolis. Serway p145.
wikip, Fictitious_force
wiki, Field
flux density
flywheel hoop stress = density * r2 * ω2    //cite? think this is given in I moment list. SOON
1st development. Theophilus Presbyter 1070-1125AD
force, F=M A, F in 'Newtons'
see Newtons_laws
Force = Mass x Acceleration; feynman defines as chg in momentum, vol 1 sect 15+, dMomentum / dT
wikip, Force
Frame dragging

Making the orbit of a nearby test particle precess (as in Mercury's orbit?).
Light, traveling in the direction of rotation of the object, will move past the massive object faster than light moving against the rotation
frame dragging, wikip
stanford spacetime search frame-dragging

F (Newtons) = -Us * NormalForce_Newtons.
F always works against motion so it's never larger than the motive force.
It'll never, on its own, create a motion (opposing the force in question).
CAREFUL; don't blindly take the force and subtract it.

The friction coef between 2 materials depends on the materials themselves, possibly modified by moisture. The speed of the objects also affects the coefficient. Normal treatment simplifies all this by declaring 2 'coefficients' of friction, Us (static, no motion) and Uk (kinetic, moving). These coefficients are mostly determined by the materials. Uk is always smaller than Us.

Fn should be shown as a vector equal & opposite to the vector 'm G cos θ'. The 2 offset each other exactly (there's no acceleration along that direction).

Truly Fn = m G cos(θ) + 'finger force' and the resultant Fn times the appropriate coefficient of friction determines (completely) the friction force shown as 'F'; that is:
F = Fn * Us // in the common, static case.

The down ramp acceleration uses Fd - F.

Serway p109  re names Us Uk etc.
problems, sciCalcs
wikip on Friction. See 'Dry friction'

γ = 1 / sqrt(1 - v²/c²), gamma is a synonym for "Lorentz Factor"
special relativity. Protons from solar flare takes 3 days to reach earth => 1.2 million mph.
    1200000 / 3600 = 333 miles/sec. # == 536000 m/sec
    gamma = 1 / (sqrt(1 - (333/186000)²)
            = 1 / (0.99999839)
            = 1.0000016  => not a signif fraction of C => not "relativistic"
Gas Density

density = nM/V = PM/RT g/mol, M is molecular weight

    PV = nRT   starting place
    P  = nRT/V
    P/RT = n/V ; after this, mult by mole wt, M, of the gas
    PM/RT = nM/V = density
Gas Laws, gas properties

The following 3 laws seem related. Each experimenter held one variable constant and commented on the connections between the other two. To recall which is which, note the T P V letters in the front of the following; they are the items held constant. Use the ideal gas law to figure out how the other 2 parameters varied.

  T Boyle's Law: 1662. P1V1 = P2V2 XAM p46-p47; "if T const, P,V inverses"Serway p542
        above section
  P Charles Law,"Law of volumes": 1801. "if P const, V,T proportional" XAM p47, Serway p542,
        (wk was 1780s). Charles Law, wikip
  V Gay-Lussac's Law,'Pressure Law': 1808. "if V const, P,T proportional"   XAM p47; Serway p542
        Gay-Lussac Law, wikip
    Combined Gaw Law: Volume propor to T/P, often expressed as P1V1/T1 = P2V2/T2  XAM p47
    Ideal Gas Law: PV = nRT  XAM p47, 'R':  see ideal gas constant.
    gas density: d = nM/V = PM/RT  ( g/mol (?)) where M is molecular wt
    equation of state: Serway p542. p564 'state variables'. start at thermal equilibrium.
    thermodynamic variables:  'P' 'V' 'T'
    volume:  for 1 mole any gas, 1 atm, 0C, the volume will be 22.414 'L' (liters).
    average KE of gas molecules: proportional to abs temperature  XAM p45

  Work, W = P deltaV, XAM p54, units L * atm = (Liters=0.001 m³) (atm=N/m²) = N•m = J
     examples integrate V.
Gauss's Law (electric)

Gauss's Law E "eFlux leaving a volume proportional to charge inside". Maxwell, wikip ref03: "electric field lines leave a charge and enter on neg end".
PT1_Q30 (no img): electric eFlux eqn: E A cos(θ)

Gauss's Law (magnetism)
Gauss's Law M "total mFlux thru closed surface is zero". implies no monopoles
Gay-Lussac's Law
'Pressure Law':
  Gas Laws
gear ratio
in / out   52/13 for early Bike Speed problem
gearRatioOverall = rpmWheel(evDesiredMph,diaInches) / motorBestRpm

There are interesting lectures on youtube, by Leonard Susskin (?), a Stanford prof. (171014)


Geometric Algebra
Used by Euclid and the other ancient Greeks. The Euclid book on the Kindle (2016)
has the book in Greek and English, side by side.  Book II "Propositions" (item 2?)
shows the distributive law done in areas (it's clever).
I believe the early stuff looks so bad b/c:
    - their number system was base-60 and not easy to read
    - they had no good way to represent fractions.
    - they really didn't have a clear way of expressing an expression.
wikip, Geometric_algebra
geometric progression
math term for a series of numbers, each made by multiplying the predecessor by a 'common ratio'.
(fgh calls the effect 'exponential').
grad, grad operator

A vector-valued version of derivative function.

The symbol is a 'nabla', an upside down triangle, widest at the top; it is often called 'del'.

see wikip

grating, reflective, diffraction
from wikip.

Edmund's, $90, 1200/mm eg: 656nm H2 line on 1200/mm grating, incident at 90deg
Gravitation, Law of universal
Newton's law of universal gravitation:
                    m1 * m2
    F = G * --------------   where G is 2/3 * 10¤-10 N
Gravitational Constant
Gravitational Constant = 32.2 ft/sec_squared = 9.81 m/sec² The 9.81 value is not accurate at all points on earth. Early english clockmaker discovered that! # derivable fm G²) where 'G' is 2/3¤-10 N

At center of earth ? YahooAnswers says best answer is Zero. EXCEPT the zero answer is ignoring the moon's effect. The barycenter is 1050 miles below the surface and, obviously, it moves.

earthole.html models a 'flight' through the earth, assuming no drag from air (and temperature etc). It ignores that the barycenter of the earth-moon system is 1050 miles below surface. And that barycenter is moving at an approx average of 12.8 degrees per day. Moreover the moon's orbit moves around so much that the actual path of the barycenter moves a lot.

Green's Theorem

Green's Theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter. (planimeter!)

Green's essay, 1828 in pub/doc/

George Green
(over my head. 171004)
layman terms, quora
"conservation of information," or "conservation of phase space volume."
tries to describe (?) all the states (activities) within a certain
config. tries to treat all such with the same approach.
Connects to Langrangian.
lagrange, quora
They (lagrangian and hamiltonian) are mathematically equivalent, in the sense
that the solution to the problem when formulated in one of Hamiltonian or
Lagrangian mechanics is always the same as the solution when formulated in the other.

Mathematically, the two formulations are dual to each other in the sense that
each is the Legendre Transform of the other.  This statement can often be lost
on people in the context of classical physics as the Legendre Transform takes
on a particularly trivial form for the kinds of problems that come up in
classical mechanics, and so it often looks like you aren't really doing
much of anything.
harmonic motion
see SHM,'Simple Harmonic Motion'
harmonic sound

(mostly music...) The harmonics of a note (or frequency appearing in an engineering situation) will be the integer multiples of the fundamental frequency. Don't confuse this with the power of 2 progression of octaves.

procs2study. fundamental freq*
Serway p510 etc.
heat capacity

Energy (heat) req'd to raise a _sample_ 1 degree C Not about energy per unit mass (see specific heat) But the per mole aspect makes this pretty useful, given observations on XAM p43.

The word 'capacity' is interesting b/c it also measures how long it takes some material to transfer its energy enuf that it loses temp. Water's high capacity keeps the bay area warm in winter.

_molar_ heat capacity is just the heat capacity for a mole of a substance. commonly used.
    eg: molar heat capacity = 28J/mol-K,  XAM p42-43
These capacity values depend on phase.
    Solid H Cap isnt same as Molten H Cap
Heat Flow
heat loss example, text file

"energy transfer to cause a chg in Temperature" Serway p554

henry, SI unit of inductance

3 Henrys * 4 amps/sec = 12 Volts => H = v / (amp/s) = v s / amps. The voltage opposes the dI/dt.

In ads, I see units of uH, not H. Haven't seen mH even when value is 1000 uH.

Henry gets around !

PT1_Q09 tests 'unit of self-inductance'. It's just the good ol' Henry.

"The current I makes a mag fld and thus mag flux which opposes change by generating a 'back EMF' that counters the rate of change in I. The ratio of the mag flux to I is called the self-inductance of the circuit. Its SI units are the same as inductance, i.e. Henry."

wikip      inductance, local
inductive reactance, local
about J. Henry
hertzsprung-russell diagram
main-sequence stars
law of constant heat summation.
The total enthalpy change during the complete course of a chemical reaction is
the same whether the reaction is made in one step or in several steps. PT1_Q74
Higgs Field
kindle book "Why does E=MC²...", many refs
kindle book on Higgs...Symmetry.
Higgs particle
Hooke's Law

1678. (F Newtons) = -(k N/m) * (x m)
F is called the 'restoring force'
notes on springs, constants

horsepower, 'hp', 550ft-lb/sec
horsepower: 550lb lifted 1 ft in 1 sec = 550ft-lb/sec = 550 lb, ft/sec, 745.7 W
  "550 lbs 1ft/sec". a 'draft horse' pull is 800N.
  horsepower, wikip

HP(force_lbs,v_mph) = force_lbs x v_mph / 375 //brant pg 152 top (brant==electric car book(?))
// "it takes 1 HP to move 37.5 lbs at 10mph" Brant p152
// i think this really says '1 HP pushes a 37.5 force at 10mph' (?)

HP(torque_ft_lbs,rpm) = torque_ft_lbs x rpm / 5252  
Hubble Constant

ok, ok, so it's not _constant_... "Hubble Parameter"
73.5 km/s/Mpc is the most recent value, published 180222. has changed 50 times(?). Mpc is 'mega parsec' = 3.26 light years.

Huygens principle

"each portion of a wave front acts as a source of new waves"

hydrostatic pressure
applies to atmos too.
hydrostatic pressure, P = densityRho * g * h  XAM p4 
Hyperbola is y²/a² - x²/b² = 1  
    orbit when v greater than escape velocity
    hyperbolic functions: sinh etc
sinh(1) = 1.175; sinh(2) = 3.626              TODO: calculator sequence...
cosh(1) = 1.543; cosh(2) = 3.762
tried to bend the hyperbola eqn into form seen in spacetime interval
y2/a2 - x2/b2 = 1 ; or any constant

            (         a2    )
or  sqrt( a2 -  ---  x2)
           (          b2    )

                (        x2  )
or  a * sqrt( 1 - ---    )
               (        b2  )

hyperbolic functions
Area 'a' is twice the red area. Areas below the x axis are negative.

See animation, wikip. It shows relationship between circular values (sin,cos) and hyperbolic (sinh,cosh)

More on the Hyperbolic_function, wikip

Note equation similarity to spacetime interval metric, Minkowski space.

the hyperbolic cosine function may be used to describe the shape of the curve formed by a high-voltage line suspended between two towers (see catenary).
Hysteresis Curve, aka 'B-H curve'
B is electrically generated magnetism and H is the field in the metal. The image was taken from an article on transformers but ... With new components, the sequence starts at the origin where the electromagnet is off (B == 0) and the metal is not magnetized (H == 0). The electromagnet turns on and the B & H travel up the red path, ending at point 1 which some weird sum of B and H. Removing the B field sends

Ideal Gas Law
note there is a 'molar form' (seen on wikip in text(s) about heat flow (?))
ideal gas constant, 'R' = 8.314 J/(K•mol). ?? IS "molar gas constant" ??TODO  
ideal gas constant aka "Universal Gas Constant", "molar gas constant" 
XAM p47, Serway p543
    0.082 L - atm  # Liter
    8.314 Pa - m³/mol-K

Molar Gas Constant
"as P approaches 0, PV/nT approaches R for all gases". Serway p543 
If you google Ideal gas law. That will tell you that PV=nRT where
P is pressure, V is volume, n is number of moles, R is the gas constant and
T is the temperature. All units in the SI-system.
You can rewrite it to PV=m/M*RT. Where m is mass and M is the mole mass.
Calculate how much a given volume weighs at the ambient condition.
Normal values for P is 1.013 KPa and R=8.314 J/(K*mol).
Next, estimate how hot the air can be (this could be quite tricky) and then
calculate the weight of that air. The difference in mass should tell you
how much lift there is.
Ideal Mechanical Advantage SOON
vector Z = R + Xi, i imaginary. XAM p93

impedance amplitude Z = sqrt(R² + (XL - XC)²) ; phase angle = arctan((XL - XC)/R) # if C only phase angle negative

F deltaT = (F=(m (a=dv/dt)) )*dT = d(m * v) Serway p232. impulse
Since mV is momentum, m * deltaV must be the change in momentum.
Impulse is a vector & is denoted by 'J' (but that isn't Joule!).
// F = m a => 'N' = 'kg m/s2'; F * t => 'N s' = 'kg m/s' = 'm v' = momentum (but J = kg m2/s2 = energy)
Impulse-Momentum Theorem (Serway p232) states that the impulse acting on any system changes
the momentum of the entire system (depends on the scope of 'system').
impulse momentum theorem, wikip
phys classroom. treats conservation of Mom too
L = (µ0 K N² Area) / len_m where:
    L  inductance in henrys
    µ0 is the permeability of free space, 4 π ¤-7 henrys/meter
    K  is the 'Nagaoka coef' which is about 1 when coil length >> diameter.
    N  is the number of turns
    Area  is cross section of coil in square meters
L is the ratio of the voltage to dI/dt, the rate of change of current,
# hmm: NAB in electric power, local,    SOON

No way (I know) to keep the mag field when detaching the inductor from current flow. Spark! (?chg thru R. drain thru R) "decays" (basically) instantly.

where: N = #of turns, D = loop Diameter,
d=wire diameter, µr is relative permiability.
source, all about circuits

The above doesn't depend on length, density...

L has units of henrys (H). Inductors have values that typically range from 1 uH (¤-6 H) to 1 H. Many inductors have a magnetic core made of iron or ferrite inside the coil, which serves to increase the magnetic field and thus the inductance.

Inductance of the air core (above) can be magnified greatly with a ferrite core (permeable core). The core, however, can _saturate_ in which case the field partly 'escapes' and the effective L is reduced.

wikip, Inductor

inductive reactance
XL = ω L = AC angular freq L = 2π * freq * L
XAM p93
energy stored is ½ L I², L is the inductance
#of turns is in embedded in L, inductance (just above)
Test equipment, commercial units: 2 each of { 1uH, 10uH, 100uH, 470uH }
resistance to momentum chg. reqs force. XAM p1
infinite series
as needed for full pendulum swing. its power series, wikip
things so small that there is no way to measure them. The insight with exploiting
infinitesimals was that entities could still retain certain specific properties,
such as angle or slope, even though these entities were quantitatively small.

Archimedes used what eventually came to be known as the method of indivisibles in
his work "The Method of Mechanical Theorems" to find areas of regions and volumes of solids.

Johannes Kepler, area calc of a circle, representing it as an infinite-sided polygon.

1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz.
units="ft² degF hours / BTU"   Serway p574
DOW claims XPS styrofoam has an R value of "five per inch".
The SI equiv might be: (?) //Note dimensional analysis problem w/ F -> C
        ft²  degF  hour             m²        degC(degF)     3600s    BTU
R = ---------------------   =   ------- * ---------------- * ----- * --------
             BTU                      3.3ft²         1                    hour     1055J

There are 2 ways of defining R, SI and 'whatever', (one is 5.68 times the other) see
wikip, R-value
integrals, common
fill in common integrals. need for test SOON.
IndefInt(1/r)(dr)=ln(r)   XAM p208
single-page-integral-table, also printed
wikip, antiderivative
integration by parts
example ref'd in Feynman, bk 2, pg 19-5
internal energy
spin or vibes yes, heat no. Serway p554, SEE XAM p147
motions wrt vol's ctr
the ionized part of Earth's upper atmosphere, from about 60 km (37 mi) to 1,000 km (620 mi) altitude. The ionosphere is ionized by solar radiation. It plays an important role in atmospheric electricity and forms the inner edge of the magnetosphere. It has practical importance because, among other functions, it influences radio propagation to distant places on the Earth.

Feynman, Book 2, has a chapter "Electricity in the Atmosphere".
   10 pA / sq meter, pos charges falling towards negative earth.
   a video talked of a 120' wire driving a "corona motor"
determinant of jacobian matrix
jacobian matrix: partial derivatives

there's an image in pub/doc.
used by small electrometer prjt. think the printed note on that is in the box "library".  2n3819(?)
    shows many. other than getting N channel, dunno what the other attributes are...
Joule, 1 N•m = 1 Watt-second
1 N over 1 m gives 1 J. # Time not involved. ref: Serway p166.
joule: N•m = Kg m²/s² = Kg v² = Pa - meters³ = Watt-sec = cal/4.186;
Joule's Law: P = VI, volts * amps, XAM p87  (=I² R = V²/R)

"(ability to do) work"
(wiki) 'work done. = newton-meter' = 1 watt-second
// Joule(f,dist)
// uses 1 force_lb = 4.448 N  bingo!  1 N = .2248 lbs of force
Joules(force_lbs, miles) =
      = (force_lbs * 4.448) x (miles x 1609 m/mile)
      = (N)                 x (    meters   )    // force times distance.
     == force_lbs x miles   x (1609/4.448=7157)

Joules(nCalories) =  4.185 x nCalories // phys cals. not kCal (nutrition)
Joules * 0.2389   =  nCalories //  1 / 4.185 = 0.2389
-273.15C == 0K, triple point defines it (?!)
Kepler, Johannes
1571-1630 Brahe's observations of Mars, coupled w/ Kepler's calculations, showed an 8 minute error in the beliefs of the time. It led Kepler to prove the orbits were elliptical.
more in "The New Astronomy", Heidelberg 1609
'harmonic law' 1609-1619.
  1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
    • consequence of grav square law
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
    • consequence of conservation of angular momentum. Serway p368, XAM p35
    • for points on major axis, see diagram
  3. period² is proportional to (semi-major axis)³ of its orbit
    • Kepler felt this could be described as the Music of the Spheres, wikip
    • consequence of grav square law
    • see procs "orbits, Kepler's 3rd law" for derivation

The following 3 (local) images should show parts of the proof of Kepler's second law. From

Law 2 derives from the conservation of Angular Momentum.
An interesting, related calculation, from XAM p35, compares speeds at perasis
against the speed of apoasis:
   ωa ra² = ωb rb² (// and ω == v/r)
   (va/ra) ra² = (vb/rb) r²
   va ra = vb rb   # wow! valuable, useful 
   va/vb = rb/ra  or, also valuable, vb = va*ra / rb

Law 3, rearranged algebraically, makes an interesting statement about _all_ the planets (and comets etc) orbiting
one star (or moons around a planet):
   (t1/t2)² = (a1/a2)³
   t1²/t2² = a1³/a2²  expand
   t1²/a1³ = t2²/a2³ rearranged
implication: if ANY 2 planets have the same constant binding their period and orbit area, then all the planets around that star share the same constant, meaning it's a property of the star's mass !
The top equation, from wikip, shows an extension of the 3rd law and the (scale factor).

The bottom equation is rearranged to solve for M, the mass holding the planets in their particular orbits. By measuring 'r' and 'T' for an exoplanet, the mass of the star can be calculated. See sciCalcs.html#ps_orbits

See also orbits and wikip Kepler's laws
kilogram, 'kg', 2.2046 lbs
_geometry_ of motion. not its causes. (wiki) Kinematics begins with a description of the geometry of the system and the initial conditions of known values of the position, velocity and or acceleration of various points that are a part of the system, then from geometrical arguments it can determine the position, the velocity and the acceleration of any part of the system.
XAM p.9 Kinematics, wikip
kinetic energy
An elastic collision conserves both KE and momentum. (ref: wiki momentum)
In cases where an stationary object is about to fall, you can
often calc KE directly from(KE = ½ mv²) = (mgh = PE)

Kinetic_energy, wikip Serway P174. '1/2' fm integration? W = integral of F dx KE rotational= ½ I ω2 See collision entry above.
Kirchoff's Current Law

Kirchoff's 1st Law. currents at any node must sum to 0. XAM p88,91
Kirchoff's Circuit Laws, wikip

Kirchoff's Voltage Law

sum of voltages around a loop = 0
Kirchoff's 2nd Law: XAM p88,91
Kirchoff's Circuit Laws, wikip

relative magnetic permeability. B/µ0 (? was written B/B0. what's B0?)
Kuiper Belt Object,'KBO'
names Varuna, Ixion, Quaoar,
UP313 bigger than pluto

'Kilo-Watt Hour' is electrical energy, equal to 1000 'wh'. A battery pack storing 20 kwh, or 20,000 watt-hours, can ideally deliver 200 watts for 100 hours = 2000 watts for 10 hours etc. Walking burns 1/8 kwh per mile. whpm=125 !

cost of electric energy.
$0.085 per kwh (quote from tech writer on web. US average. 2006/7?)
$0.10  Ed Ring used this value in 2005. might have been a purposefully
    high approximation.
$0.12 / kwh: current cost of electricity in Calif (?)
1 gallon of gasoline contains the energy of 130 megaJoules, or 36.1 kwh.
Lagrangian, wikip
Hamilton, wikip
"The French Newton", 1749-1827
wiki. on the person
Laplacian differential operator
Laplace transform, like fourier transform.
Laplace's_equation, 'potential theory', second-order partial differential equation
Light Ampliflied Stimulated Emission of Radiation
predicted by Einstein. photons clone themselves.
see "Stimulated and spontaneous emission" section of Photon, wikip
latent heat
("hidden heat") energy needed (or released) per J/mass 
plus sign if E released.
latent heat of fusion, synonym 'enthalpy' =(2834.1 - 0.29T - 0.004T2) J/g   Taylor_series
latent heat of vaporization. "condensation'. L.water(T)=2500.8 - 2.36T + 0.0016T2 - 0.00006T3) J/g (wikip)
pls see specific heat, local
python examples/ had
to model the latent heat of vaporization (steam energy)
(as of 180416, examples/ doesnt wk)

Serway p560. and wikip
Law of Thermal Conductivity
P=k A abs(dT/dx), P power 
A area cross section, dx is thickness, dt/dx is gradient.
matl   W/(meter . C)  SOON check units
k thermal conductivity for matl:                    ON THE TEST? REFs!
    glass   0.8
    rubber  0.2
    water   0.6
    wood    0.08
    air     0.0234
pls see Conduction, Thermal
Law of Induction, Lenz'

"induced voltage resists magnetically induced current", 'back EMF'
"the direction of current induced in a conductor by a changing magnetic field due to induction is such that it creates a magnetic field that opposes the change that produced it." PT1_Q34. careful: diagram shows Phi symbol and a Phi(i) (induced(?). phi often mean flux.
There isn't a big algraic expression for his law; the minus sign just shows up in Faraday's Law of Induction.

see XAM p90,p82,quest 45. an inducing current causes a 'counter current' to form - to resist the 1st.
self-excitation, wikip
Law of the Cosines

c²=a² + b² - 2ab cos(oppositeAngle),
any triangle, c is side 'c' in diagram. see XAM p76

from wikip which has a derivation.

Law of the Sines

len(a)/sin(A)=len(b)/sin(B) = len(c)/sin(C) = diam of circumCircle from wikip which has a derivation.

can be stated w/ reciprocals (except for circumscribed circle). XAM p76

LC Circuit

A "tuned circuit" with minimal R generally used in radio receivers. The tiny R keeps it from consuming power. Of course there is alway some R (resistance) so another way of examining the circuit is called LCR.


As frequency increases inductive reactance increases and capacitive reactance decreases. passes only the resonant frequency.


removes the resonant freq from a signal.


Legendre Transform
whew... wikip
less than a full theorem.
Like a procedure offered to speed work/understanding/proving
rimstarorg video how to make. w/ lg pop bottle.
lever arm
a vector. often able to treat it as a scalar. See wiki on torque (and maybe moment of inertia)
light energy
J = h C / λ = h (nu=freq) where: h is Planck's constant, λ in meters, & C is the speed of light. E = h nu which is h f, 'f' freq of the light. "Planck's Energy Eqn"
def freq2joules(hz) :  # yes. hz. 'freq' doesn't wk (here) in radians (ω)
    h = 6.63¤-34 # Planck's constant
    joules = h * hz
    return joules
def lambda2joules(lambda_m) :
    h = 6.63¤-34 # Planck's constant
    c = 3¤8 # speed of light
    joules = h * c / lambda_m
    return joules
lambda2joules(656¤-9)  # H α (Hydrogen Alpha) spectral line
light refraction
Refractive_index and Snell's law of refraction, n1 sin(θ1) = n2 sin(θ2),
   n == index of refraction = speedOfLight / speedInMaterial  = c / v
   PT1_Q07 "refractive index a func of both speed and wavelength. index = C / speed_other_medium
   see critical angle, local
Brewster's Angle == polarization angle.
Dispersion, separation into colors
light wave
Cherenkov radiation results when a charged particle, most commonly an electron,
travels through a dielectric (electrically polarizable) medium with a
speed greater than that at which light propagates in the same medium.
Moreover, the velocity that must be exceeded is the phase velocity of light
rather than the group velocity of light.  SOON: img diagram.
The static charge in air typically breaks down in this way at around 10,000 volts per centimeter (10 kV/cm) depending on humidity.

An average bolt of (negative) lightning carries a current of 30,000 amperes (30 kA), and transfers 15 coulombs of charge and 5¤8 joules. Speed 1¤8 m/s. Can emit gamma rays.

Lightning creates light in the form of black body radiation from the very hot plasma created by the electron flow.

wikip, static electricity, Lightning
wikip, Lightning
light year
5.879¤12 miles, 9.461¤15 meters, ~1¤16 meters
linear programming
SOMEDAY. useful for temp,gas ctl veggies(?). described in Algebra 2 text book. possibly needed for veggie control (RH, temp)
1000 cm³  and  10¤-3 meters³ == volume of 1000 cu cm or,
1 liter = 1/1000 m³. For water 1 liter = 1 kg (at 4 degrees C).
1 liter = 61.02 cu in. so 28.3 liters = 1 cu ft. 1 liter = 0.035 cu ft.
locus,loci  lg class of porisms (!)

Logic, Predicate Calculus, 1st Order Logic
First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.
Lorentz Factor = 'gamma'
130127: 'gamma' derivation in book "Why E=MC²", loc 575

Lorentz Force
"force on charged particle in E and/or M fields"
With E & M fields: F = q(E + V x B)
Just a B field: F = q v B sin θ  # XAM p79 mid and btm and XAM p85.
    if no V, or sin(θ)==0, this reduces to just F = qE
B field measured in Teslas
Maxwell's equations 1861. summary of related equations
relativity calculator
history of the Lorentz Force Law, RHR example
teltron tube, finding charge to mass ratio. Helmholtz coils.

Least Squared Error. See link to leastSquares.html vv_regression. define residuals... TODO

chem light emissions, wikip
magnetics compared
The following shows the magnetic fields for an electromagnet, 'E', and a permanent magnet, 'P'. There are interesting, somewhat subtle differences.
1. The field lines are wide spread while in air and their extensions, in the steel, much denser. The permiability of air is like that of a vacuum, µ0, and the steel obviously has a much large µ value.

2. Note also how the P magnet's field lines emanate close to the upper right corner of the magnet while the electromagnet, E, has field lines emanating from its mid section.

The images, from a kjmagnetics article, are from calculations, not measurements.
magnetic dip angle
How to make a '3D compass', measure the local vector, then calculate field: article. a WOW. excerpt:
"The radial component is given by (eqn below). It varies with the cube of the distance from the Earth (instead of the square) because the magnetic field is a dipole, rather than a monopole, field."
    Br = 2µ0 m cos θ / (4 pi )
The article also says indirectly that the field is distorted near walls or even furniture!
Also found a $50 european article on how to measure dip with a cell phone.
magnetic field,'B'
Good section on history, people: Serway vol2, chap 29
alt names for B   //wikip, 'Magnetic_field'. In a vacuum, B = µ0 H
    mag flux density (wikip) == 'mflux' (fgh term)
    mag flux intensity
    magnetic field

B is measured in Teslas (symbol: T) and Newtons/(meter • ampere) (symbol: N/(m·A)).
B is most commonly defined in terms of the Lorentz Force it exerts on a moving 'test charge'.
The electric current, I amps, is scaled by µ0, the 'magnetic constant', 4π ¤-7 T m/A
    Scaling by 'u' means the action occurs inside a materials with this constant.

Diagram conventions: a circle w/ dot in middle is the field vector coming out of page.
    When the field goes into the page, the symbol is a circle w/ an X in middle.
        (the tail feathers of the arrow).
    See example diagram, below
    drawing a magnet. draw S to N (= - to +) (left to right(?))
    field lines only conceptual.

finding B for common wirings::
About a wire   B = µ0 I   / 2π R  #  2π r == circumference XAM p80
Loops    B = µ0 I   / 2  R  # 
Solenoid    B = µ0 I N / len;   # If  N/len, the 'turns density', gets very small ('len' very   # large), there's very little external field. XAM p80+
B's effect on charges: Magnetic field acting on a charged particle. AP Physics p 438 F = |q| v B sin0 // if v 0 or angle 0, F is 0 example: q= 6¤-6 C, v=4¤5 m/s, B=0.4 T, θ=30 degrees try it (particle2) F = I L © B # L is wire Length XAM p79.4; This works when wire is at an angle to the field mag flux symbol ΦB (Phi sub B). unit Weber, 'Wb' D field. Quora paper said there were 4 fields and modelling all of them obviated the need for µ0, ?e0. 'D' called the Displacement Field. Some materials are diamagnetic, resisting the field. The opposite property is paramagnetic See dipole F = q(E + V x B) // force on charged particle moving thru a B field Lorentz Force Eqn lineIntegral B dA == 0 means #exitingFluxLines = #enteringFluxLines magnetic flux // section below. permiability '2nd rank tensor' (ref19 ?) RHR right hand loop. if fingers bend w/ circle around wire, field points up w/ thumb
magnetic field,'H'
H for permanent magnets. B for electromagnets
H is curl-free ('irrotational') while B is 'solenoidal'
H was figured out by Gilbert using a dipole model as in electrostatics. (model not really appropriate)
'bound current' is a property of atoms and thus applies H field (and spin ?)
magnetic torque (on perm mags), 'tau' = µ0 mDipole H sin(θ) // θ is angle tween H and dipole

The relation between H and B:

Each type, solid or electromagnetic, has a different root cause.

Many sources say "B = µ0 H in a vacuum" but the image at left reveals that the scale factor depends on the material type.

And... for B = u H, note that u can be freq dependent.

The units don't seem to work:
B (Teslas=(Newtons /(Amp meter)) = Newtons/meter per Amp
µ0 Newtons/Amp * H Amps/meter = Newtons/meter

whew! I'm so happy that the image at left says "simplified"...

wikip, Permiability

magnetic field, motor winding
One winding in a field. From online lecture by 'Doc Schuster'.
Right side of winding tries to come out of the page.

torque (tau) = F\ d sin(angle) // angle wrt field, per wire
= (B\ I h) w/2 // add 'N' loops. combine w,h into Area
= (N I A) B sin(angle) // NIA defines 'magnetic moment' TODO where 1/2?

sin(angle) => No torque when hoop seen edge on.

magnetism & 2 wires
magnetic field
straight wire making cylindrical field
Magnet's North to battery +.

RHR (Right Hand Rule). Thumb in direction of current; fingers curl in direction of field.

For the magnitude of a field. use | |
|B\| = (µ0 * I) / (2 π r) = µ0 I / circum T=Teslas

magnetic fld
2 straight wires affecting each other. R the distance between the wires. All units SI.

What force 'F' is applied to wire 2 from wire 1 (Newtons).
'L', a vector aligned with the current, is the Length of the wire affected by the other wire.

F12 is the Force made by wire 1 affecting wire 2. It pulls wire 2 towards 1. For the other case, F21, imagine stepping behind the wire pairs. The equation is basically the same (assuming currents are equal). Thus the 2 wires pull towards each other.

magnetic field
wire loop making a field
Careful! This shows a loop of wire making a field. And it shows that the field produced would be equivalent to a bar magnet mounted as shown.

magnetic flux
(fgh term 'mFlux') XAM p78+ to 80 something. TODO
magnetic levitation
online article told that magnets oriented to concentrate flux in 1 direction. Used on trains etc. google "Electromagnetic Levitation Quadcopter".
magnetic moment
SI unit: Amp m², same unit as 'magnetization'
First source was ref19 (wikip, "magnetic field")

The following is mostly from motor winding:
magnetic moment = N I A      // NIA is #of turns, current, area
The related expression "(N I A) B sin(θ)" calculates a force using the magnetic moment.

see dipole
TODO: dipole related to 'moment' ? Believe 'dipole' is more conceptual; magnetic moment is a calc (vector)
magnetism & relativity
It's said in several places that magnetism is 'just' a feature of special relativity. Here's a quotation from Eli Pasternak, seen on Quora:

"Although this odd attraction was stated initially as an empirical law of nature and the formal calculation was explained as the direction of magnetic fields, as some other answers correctly explain, the root of this apparent oddity was explained by Albert Einstein with special relativity.

When two parallel wires have moving electric charges, let's say negative charges, then they see the non-moving positive charges of the opposite wire more closely spaced since objects in relative motion experience length contraction. This causes opposite electric charges to appear more densely spaced, thus more electrostatic attractive than the two moving charges which are at rest relative to each other. Einstein calculated the forces between these wires and proved that they are identical to the forces one would calculate using magnetic formulas. In essence, "magnetic" forces are just electrostatic forces observed in motion."

magnetism & 2 wires
magnetic north
The + side of an electromagnet points towards earth's north m pole.
gravity 3/8 that of earth
mean density 3.9 g/cm vs 5.5 for earth
see weight for notes on converting fm scale readings to mass, kg. Mass is sometimes calls "gravitational charge" b/c of the similarities between the eqn for Coulomb's law and that for universal gravitation.
mass spectrometer
XAM p85 btm  meas's charge to mass ratio, m/q
Malus' Law
intensity of polarized light passed thru analyzer depends on the angle  TODO
between axis of analyzer and polarizer
Maxwell coil
make a very smooth magnetic field.
Maxwell 3D
magnetFiniteElements copy in ./doc/magnetFiniteElements.pdf refers to " s/w which uses 'mesh' of tetrahedrons.
Maxwell's equations, a set of four equations that could be used to calculate the entirety of electromagnetism.
ref21 pdf, students guide to maxwells ...
XAM p71 suggest setting derivatives to 0 to see simplest case. electroSTATIC.
(wiki on Heaviside says 4 variables: B E J rho.)
the simplified set of eqns:
  • line integral of 'E' div dA = Q/e0 (Gauss' Law Electostatics)
    The 'eFlux', electric flux, leaving a volume is proportional to the charge inside.
  • line integral E dot dS = -dPhi/dt
    The voltage induced in a closed loop is proportional to the rate of change of the mFlux that the loop encloses. 1855 paper "On Faraday's Lines of Force"

XAM p71 shows the eqns for the static cases (nothing changing. simpler).
plastic "Physics" review sheet has abbrev copy of equations.
Gauss' equation for electric fields. ref21

The following diagrams are from Also see grad notes
"Poisson's Equation": The electric flux leaving a volume is proportional to the charge inside.
Gauss' equation for electrostatics:
??? line integral of 'E' div dA = Q/e0

Gauss' Law for Magnetism: The total mag flux through a closed surface is 0 (so no monopoles)
The line integral of 'B' div dA = 0.

Faraday's Law

The Maxwell-Ampere Law: The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) that the loop encloses.
The line integral B dot ds = µ0 I * µ0 * e0 dPhi / dt.
Maxwell added the displacement term.

mechanical advantage
POE Nov '16 prjt/presentation was to calc this # POE 'Principles of Engineering', MDHS class
    IMA  Ideal
    AMA  Actual    'effort force'

ramp  for praxis q 1C6e want MA for each simple machine ?     TODO.   

Mechanical_advantage, wikip
mechanical efficiency
mechanical efficiency = 100 * AMA/IMA
    IMA  Ideal
    AMA  Actual    'effort force'
(IMA) of ramp is Ht/rampLength # not tangent !

Mechanical_efficiency, wikip
see vv_PI re distortion of orbit due to space warping. 160th magnetic strength, polar aligned. trace atmos compared to earth's. stuff from Solar wind. some very cool, some extremely hot 700 C? Dense planet.
meter, 'm', 39.37007874 inches
meter = 3.28084 ft // approx. 1 sq meter = sq_ft * 0.0932

energy used to describe masses. Physicists will often refer to the energy of a particle in MeV, its momentum in MeV, and its mass in MeV. They should say MeV, MeV/c, and MeV/c2, Physicists should say MeV (energy), MeV/c (momentum), and MeV/c2 (mass). quarknet eV entry local

1 millionth of a meter
Milky Way
mass = {0.7 to 2.0} 2e(30+12=42) kg ~= 2e42 kg
    MWay mass as much as "0.7 to 2 trillion solar masses". //trillion = 10e12
    solar mass = 2e30 kg
radius, at sun's position = 26400 ly = 2.64e4 ly = 2.5e20 meters
    ly = 9.5 trillion km = 9.5e12 km = 9.5e15 m
spin rate, at sun. 8.3e-16 radians/sec
    period = 240e6 years * 3.15e7 sec/yr = 7.5e15 sec/rotation = 1.2e15 sec/radian

diameter between 1.5e5 and 200e5 ly
frame dragging
Modulus of Elasticity
aka Young's Modulus, aka Elastic Modulus
Elastic_modulus, wikip
Bending_stiffness, wikip
youTube compares 2 sheets. bending
    saw a note somewhere that steel has a Young's Modulus of 30¤6 psi

Youtube on Mod of E  spring algebra, good. series, parallel
    metal (stretching) is just like a spring. "mexicolibre" ?!
    Force/Area (units of pressure) = Young's Modulus x deltaL / L
    where deltaL is the increased length of rod (spring,whatever) and
    L is the original length of the rod.

    deltaL/L is "strain"
Y Break Rho Vs
steel20 5.2
al 7.0 0.78
nylon0.36 3.2
easier to stretch a rod w/ low value of Y Y's * ¤10 N/m² deltaL = F/A x l/Y or F l = ----- A Y
molar gas constant, 'R'

8.314 J/(K • mol)

molar mass

#of kg for 1 mole of a substance. often (wrongly) called 'molecular weight'


moles per unit volume, m-3

mole, 6.02214¤23

expressed by the Avogadro constant, which has a value of 6.02214e23, the
number of atoms in 12 grams of carbon-12
The mole is one of the base units of the SI, and has the unit symbol 'mol'.

Mole, wikip I somewhat expected that 1 Amp would be a mole / sec. Nope 1 Coulomb/sec and 1 Coulomb has 96485 moles of electrons (or protons) in it. A mole of gas, at STP, requires 22.414 liters of volume. ~6 gallons
molecular weight
ideal gas. M = dRT/P,   
d=density in g or Kg, T kelvin, P atm or Pa
nth moment, Un = distancen Quantity // like a point.
(otherwise this expands into an integral with the above exponent)
'n' is the moment number (as in 'first moment', 'second moment' etc)"

Moment_(mathematics), wikip said "a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points. If the points represent mass, then
  • The zeroth moment is the total mass.
  • The first moment divided by the total mass is the center of mass.
  • The second moment is the rotational inertia.
If the points represent probability density, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third central moment is the skewness, and the fourth central moment (with normalization and shift) is the kurtosis. The mathematical concept is closely related to the concept of moment in physics."

Moment_(physics), wikip said "a moment is an expression involving the product of a distance and a physical quantity. In its most simple and basic form, a moment is the product of the distance to some point, raised to some power, multiplied by some physical quantity such as the force, charge, etc. at that point...

moment of inertia,'I'
'I' is the weighted distribution of mass wrt to a spin axis
Its units, as shown, are 'kg meters² '
sum of masses, each weighted by square of distance from axis
aka 'rotational inertia', 'angular mass'

Radius of Gyration
angular momentum
rotation entry
In its full generality, 'I' is a tensor. demo 'races' sphere and ring down incline: link.

XAM p15
momentum, kg m/s, vector
SI unit kg m/s, or equivalently, N s. Often written as 'p'
Momentum, wikip
mass = 7.34¤22 kg; radius = 1.738¤6 m; grav = 1.622 m/s²
near point
the closest eye can focus: 25cm. XAM p61
eyes focus in front of the retina
NEO = Near Earth Object
celestial body orbiting the Sun with a Apoapis-Periapsis which includes any part
of the earth's orbit (Aphelion-Perihelion). Examples
Bennu   space vehicle to visit 1808xx
Eve     Bennu-bound vehicle getting gravitational boost for trip to Bennu
2018BD  passed earth jan 18. mentioned in
2001AJ129  alt range min ~same as geosync satellites.
Newtonian fluid
teach/doc/physicsSubtest2_*.txt"_1i. Evaluate the statics and dynamics of Newtonian fluids."
Newtonian fluids:  air, water. no fluid perfectly this
non-Newtonian fluids:
_   1i_1 statics   looks like a 'normal' fluid
_   li_2 dynamics  sudden applied force causes fluid to stiffen.
see wikip
Newtons Cradle
PT1_Q76 Not simple! Easy when only 1 ball in motion. For a 5 ball cradle, 2 balls together will cause 2 balls on the other end to move! Moving 3 balls causes the middle ball to always be in motion. The explanation on the multi-ball drops is that the 2 balls impacting aren't actually together. The steel balls compress about 30um. compression time about 200uSec.
see Basic, 'Simple Solution', wikip
More complete solution', wikip
Newton, a unit of Force, kg m/s²
1 N = 1 Kg * 1 m/s²
9.81 N = 1 Kg * 9.81
9.81 N = .2248 lbs * 9.81   # implies 1 force_lb = 4.448 N
Newton, wikip
Newtonian fluid
simplest mathematical models of fluids that account for viscosity.
most general models use tensors
viscous stresses arising from flow are proportional to the rate of deformation over time.
     youtube, complex discussion

Water, air, alcohol, glycerol, and thin motor oil
sand w/ just the right (wrong) amt of water (quicksand)

non-newtonian fluids: what could commonly be called "weird fluids"
It looks like a low viscosity until you do something to it and it
suddenly stiffens up.  There are examples, videos, of people riding bikes
over a trough of such a fluid.

Many salt solutions and molten polymers
ketchup, custard, toothpaste, starch suspensions, maizena, paint, blood, and shampoo
     youtube, cornstarch and water
Newton-meter 'N•m', E=Fd, Joule
'newton-meter', E=Fd energy.
recall that the line integral of force over a path gives total energy   
For straight line motion, E = F * d
Newtons_laws of motion
Law of Inertia. speed doesn't change without an applied force
F=ma. F is the rate of change of momentum. F = d(mv)/dt
F12 = -F21. equal and opposite reaction.
normal distribution (statistics)
Also known as the Gaussian distribution, 'Bell shaped curve', (grading on) The Curve...
Famous, ubiquitous, statistical distribution.

also known as 'bell curve', 'bell-shaped curve'

also known as the 'normal distribution'

wikip article is huge. 31 printed pages

In the std normal, the denominator (sqrt(2 pi)) makes the area under the curve == 1.

binding energy == equiv mass lost as nucleons combine. PE goes neg   
    is the E needed to break apart a nucleus
    max's about Fe or nickel
    = [Z * mass(H) + N mass(Neu) - massOfIsotope] * 931.5 Mev/mass_unit   
carbon12 '12C' used as basis for some constants. mass set at 12.000000

1 electric charge, proton or electron, is 1.6¤-19 Coulombs 

mass unit: Mev/C²
neutrons in nucleus dilute the electric 'Coulomb repulsion'
nuclear density =~ 2.3¤17 kg/m3
nuclear radius, r, = r0 A1/3 where r0 = 1.2 'fm' = 1.2¤-15 m   
radioactive decay: Natoms(t) = Norig * e^(lambda tsecs) where lambda = ln2/(halflife time in secs)
Strong Force.range 2fm = 2¤-15 m  (have seen this limit written as 2¤-14 m)
symbols: Z #protons, N #neutrons, A amu, Z+N
'fm', 'fermi', = 1¤-15m
'u' Serway p1295    (== amu = atomic mass units)
'nodding'. is earth's nutation caused by 'ringing' fm moon collision (w/ "Thera")? wikip gives other reasons. ok?
Oersted's Law
1820. "electric current makes a magnetic field" wikip
Ohm's Law
V=IR. 1827. XAM p86
    German Edu System called it a "web of naked fancies"
oil drop experiment
Millikan. determined charge on the electron.
TODO: add more. like how did he charged the oil drops !
omega, 'angular velocity', symbol ω
ω = v/r, units sec-1, not radians/sec (!)
Serway p270, bottom, says 'radians are dimensionless quantity'. This is fortunate since angular momentum's calc doesn't end up with the proper units if 'radians/sec' were used.
Frequency f = 360°/s or Hz. so ω = 2π freq = radians/s = sec-1
(wikip.hertz says Freq's units are also 'sec-1')
optical resolution

'optical resolution' quantifies the ability of an imaging system to resolve detail. Rayleigh criterion, the theory, often doesn't match reality, pro and con. For telescopes, the atmosphere, eye health, light pollution will all introduce effects.

Rayleigh's theory is based on the 'airy disk', the diffraction pattern made by cylindrical optics.

The sine can be removed because the sin(small angles) is very close to just 'small angle' (in radians). This is a common simplification.

Wikipedia notes that this Rayleigh criterion defines 'diffraction limited' optics, something that's very rare (or very expensive). Many telescope users believe the Dawes Limit is more appropriate. PT3_Q04

This photo shows a real airy disk. If two such 'point sources' are closer than a certain θ, the dark rings of one subtract from the other's brightness.

The slow f-number and long focal length used for this photo amplified the effect(?). A normal telescope doesn't show this.

550e-9m λ (green), 2000mm/25 = 0.08m aperture gives:
1.22 * 550e-9m/0.08m = 8.39e-6 radians. Converting to degrees:
radians * 57.3 = 4.8e-4 degrees. Converting to arc-sec:
degrees * 3600 = 1.73 arc-sec

lens resolution, wikip
airy disk, wikip
a telescope site
optics, real image vs virtual image

It's a 'real image' if you can put film or sensor there and actually intercept light - otherwise it's virtual.
If, in a mirrored setup, the 'image' is behind the mirror, it's virtual.

optics, lenses

Thin Lens equation: 1/S1 + 1/S2 = 1/f so
f = S1 * S2/ (S1 + S2), (like 2 parallel resistors)

'f' will be negative for the example at the left.

Note that the image itself only appears at the 'f' distance WHEN the object is at infinity (to the left). When the object is close to the lens, the image moves to the right of the point 'f'.

Serway, vol 2, pg 1023 or so

optics, mirrors

'Object' on left is the physical object being imaged. The down-faced red arrow, near the diagram center, is the resultant 'real image'. This is 'real' in the sense that a piece of photographic film will produce an image.

The green, yellow, and grey lines are the ray tracings. '1' starts horizontal. '2' heads to the Focal point then bounces to the head of the Real Image. '3' (not always used) goes

The left most red arrow, as above, is being imaged. Note that the resultant image is placed behind (or _inside_) the convex lens. It may not even be possible to place film there. To the observer, however, from the outside near the orginal image, the resultant _virtual_image_ does appear to be on the far side of the mirror.

Note that, though the image is virtual, it's erect.

Serway, vol 2, pg 1023 or so

elliptical usually. angular momentum conserved. XAM p35
can go 'hyperbolic' if speed > escape velocity
A parabolic orbit has eccentricity 1 and its energy is exactly enuf to bring the
   object to a stop at infinity.

vis viva
Orbit, wikip
Parabolic_trajectory, wikip
Elliptic_orbit, wikip
Orbital_state_vectors, wikip
harmonic, SHM, .damped, .driven  Serway p436
simple harmonic motion
how to use, SFE lecture.
Otto Cycle
The Otto Cycle is a 4 cycle version of Carnot Cycle, Serway P623, XAM p52.
'oxidation up' means loss of electrons Redox, wikip, == reduction, loss of + charge, gain -
A paradox is an argument that produces an inconsistency, typically within logic or common sense.
Parallel Axis Theorem
Some components may be not centered on the axis of rotation. Record the position of the original axis of rotation, 'spinAxis', and for this same central part, calculate its moment, I0. For the uncentered part, find its mass 'M' and locate the part's center of mass (Cm). FSind the distance, D, from this new Cm to the 'spinAxis'.
Inew = I0 + M * D², Serway p281
See Radius of Gyration for simple, common shapes.
a form of magnetism whereby certain materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly > 1 (== a small positive magnetic susceptibility) and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect.
wikip,     diamagnetic
paraxial rays
rays making small angle with the optical axis.
3.26 light years, 30¤12 km = 19¤12 mi.
distance at which 1 AU subtends an angle of 1 arc-sec.
see glossary_chemistry
?see? eFields, neg ion generator"
Pascal, Blaise
pascal, unit
Pa= N/m², N per sq meter.
Pascal's Law
1648. P = rho g h # h = fluid column depth XAM p4   
"The pressure in a confined incompressible fluid is the same throughout
the fluid (at that same height)".
deltaP (hydrostatic pressure, Pa, Kg/m3) = (density 'rho') g (9.8m/s²) (heightAboveSensor meters)

Pascal's works are frequently shown with fluid systems having 2 round
openings - but of different diameters. multiplication of forces. hydraulic brakes etc.
Since the pressures at the different pistons are the same, but the
areas of the 2 are different, the forces on the two pistons can differ
from one another (and often 'will').
Pauli Exclusion Principle
"quantum mechanical principle which states that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously." Fermions have 1/2 spin. (1/2,3/2,...): protons,quarks,leptons wikip
The swinging mass is 'bob'. Tension, the other force, does not do work since it always acts in a direction perpendicular to the bob motion.
The parenthetical part, a 'power series', can be omitted if the arc of the swing is < ~ 6 degrees total (?)

physics classroom, KE <=> PE energy transformation. not a video.
pendulum, wikip, general. some math
AP Physics p315 gets into differential equations

pendulum math, wikip

Relationship to Simple Harmonic Motion, XAM_Q13, p171
    F = - (mg/L) s, compare to Hooke's Law (F = - k x)   
   "A simple pendulum may be modeled using the same expression as Hooke's Law for
    displacement 's', but with a spring constant equal to (mg/L)". TODO. substitute. try it

XAM p23-24, p171; Serway p433-435 period Tsec = 2π sqrt(L/g)
    period T = 2π sqrt(I/mgd), I = moment of Inertia. m g D?, TODO   
    physical_pendulum, type of. torsional
    low angle span, similar to harmonic osc
    Acceleration, wikip shows tangential and centripetal accel

The ability of a material to 'accept' more magnetic field (than what it has). That is, when a material reaches magnetic saturation, its permiability is now close to 1.

from table. last ref below:

perm, µ
rel perm

mu-metal 6.3¤-220k-50k used to shield magnetic fields. hmm.
iron 6.3¤-35000 pure but not the extreme alloys
ferrite,Nickle-Zinc8¤-416-640one of several 'ferrites'
carbon steel1.26¤-4100 a typical bolt (?)
neodymium 1.32¤-61.05
water ~ µ00.999where µ0 is vacuum perm.
superconductors0 0 weird. the previous floor was 1

wikip, values


e0, epsilon 0 = 8.854187¤-12 F/m (Farads/meter) The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux (eFlux). In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum (see: vacuum permittivity). Likewise, it is also the ratio of the capacitance of a capacitor using that material as a dielectric, compared to a similar capacitor that has a vacuum as its dielectric.

units same as ? (? m-3 kg-1 s4 A² (electrical) permittivity of free space.    TODO: clean up this area. 171213
is now DEFINED == 1 / (µ0 C²) which nicely guarantees that C² will equal 1/(e0 µ0).
µ0 must be easier to measure accurately...
e0 appears in Coulomb's Law and Maxwell's calculation of the Speed of Light

(from p18 of ref21): "The permittivity of a material determines its response to an applied electric field - in nonconducting materials (called 'insulators' or 'dielectrics'), charges do not move freely, but permittivity in Gauss's law for electric fields is the permittivity of free space (or "vacuum permittivity"), which is why it carries the subscript 0.

(also ref21): most precise units are C/Vm (Coulomb / VoltMeter). Most fundamental units C² s² / (kg m3). C/Vm == Farads/meter

(also ref21): Inside a dielectric, the amplitude of the total E field is generally less than that of the applied field.

Dielectric_permittivity, wikip _relative_
Permittivity_of_free_space, wikip _absolute_. free space.
speed_of_light entry
material 'state': solid, liquid, gas, plasma.
phase velocity
"Phase-velocity represents the vel with which the disturbance is transferred
from one particle to another w/o the actual motion of the particles".
Velocity V, omega angular vel, and k is propagation constant.
V = w/k PT4_Q59

Specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm, corresponding to a photon energy ~3¤-19 J. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5¤-17 J) with a maximum conversion efficiency of 35%.
wikip, Photon_energy

'PV', solar cells.
Solar power
1 kw / sq meter incident energy.
pointing at sun. having sun unobscured...
20% efficient (?)
π 3.1415 926 535 8979 3238 4626 43 3832
error in 15th place for disk held vertically, at earth's surface(?)
3.1415 926 535 8979 3238 4626 43 3832 ... ideal
  1234 567 890 1234 5678 9012 34 5678
         M       C                    C = carbon, portion of 1m circle
carbon atom radius' somewhere around 100pm (averaging VanDerWahls and D orbital)
100pm is 1¤-13m

Considering distortions of what is conceptually a circle (or ellipse)...
Distortion at Mercury's distance fm Sun causes 43 arcsec/100yrs.   
87.969 day orbit. 415.2 orbits/century. *360 = degrees/C = 149472 deg/Cent
* 3600 arcsec/deg = 5.38¤8 arcseconds of orbit/century. has 43 arcsec error
or 8¤-8 ~= 1¤-7

C/D defines Pi. D gets larger so π smaller.
pith (ball)
low density matl fm elderberry plant. holds charge well. Substitute can be styrofoam (ball). ref22, pg 112
Planck's constant,'h'
The "quantum of action", units physical 'action'.
E = h * (nu=frequency), 2/3¤-33 J sec,   

wikip, Atomic_orbital wikip, Planck_constant 1900
Planck's constant, 'hBar'
hBar, the 'reduced Planck Constant', is h/2π and appears in examples of the uncertainty principle
The angular momentum value L must be a multiple of hBar; it's not just a constraint on atoms.
Planck's Law
1900. Max Planck.
Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T.

There are 2 similar renditions of his equation, one using the frequency of the photons, the other using the wavelengths. I am more comfortable with wavelength. Wikip shows both equations.

Planets. and their positions
I updated location to 'walnut creek' and clicked 'planets' on main page. Nasa's 'PlanetaryFactSheet', local copy
Nasa site
wikip, galilean moons

"gaseous mixture of highly charged ions". 4th state of matter. Chemically very active (corrosive).

usually created by heating a gas or by subjecting gas to a strong electromagnetic field. Highly corrosive. Eg: fire, lightning, gas in a florescent or neon lamp. The Sun. Welding arc (?)

St Elmo's fire (plasma), wikip
saltlamp $60 'loud'. ordered it anyhow. 18082x
plasma ball

wikip "clear glass container filled with a mixture of various noble gases with a high-voltage electrode in the center of the container." The choice of noble gas determines the color of the electric arcs. Tesla called it an "inert gas discharge tube".

plasma ball 'tricks' v good.
Platonic figures
Platonic figures: 5 regular solids. conic sections, aka "cosmic figures"
Simeon Poisson. 1781-1840
contributions, wikip, electric fields too.
Some vector equations work in Minkowski space.
most of the attention to polynomials focusses on quadratic
potential_energy, PE=mgh, Joules
PE = mgh (mass grav height) units Joules
lift 100N 1m = 100J  // N•m
U_joules = massKg x gravConstant_m/s² x heightMeters
   =   (183+32)lbs/2.2  x 9.81 m/s2 x 1000'/(32.2/9.81 ft/meter)
   =    215/2.2         x 9.81 m/s2 x 304.8 m
   =     97 Kg          x 9.81 m/s2 x 304.8 m
   = 290000 J
   1 nutritional Calorie is 1 kCal
   290000 J / (4185 J/kCal) = 69 kCals

   kWh = 860.4 kCals so lifting bike & body up 1000' takes
   69/860 = 0.08 kWh

   "walking burns 1/8 kwh per mile", 0.125 kWh
Potential_energy, wikip
power, 'P'
Work/sec = J/sec = 'Watt', energy transfer rate
calc average power {
    energy,work (joules) from (kg,metersLifted)  // assumes grav
        9.8N * kg * meters = joules  // to lift something
        9.8N * ((190+35)lbs / 2.2 lbs/kg) * (1000ft / 3.275 ft/m)
        9.8N x 102 kg * 307 = 307 kJ   // to lift me+bike 1000'
    // wt is probably low; time could be short

    energy (kCal) from (power(watts))
    f_kCal(204 watts)  = ( 204 J/s) / (4185 J/kCal) = kCal/s ; kCal/s * 1500sec  = 73 kCal
    f_kCal(1000 watts) = (1000 J/s) / (4185 J/kCal) = 0.23 kCal/s

# calc time (secs) to boil a given quantity of water
# A calorie of input causes 1g water to rise 1C; 1J increases 4.185 C
# water_g   grams of water, 1000g / liter. same as # of mL
# start_C   starting temp of water
# power_w   input power, watts
def secsTilBoil(water_g,start_C,power_w) :  TODO: add extra heat for phase change   
    tDiff_C = 100 - start_C
    calNeeded = water_g * tDiff_C # of calories needed.
    jNeeded = calNeeded * 4.185;  # 4.185 J / Cal
    tNeeded = jNeeded / power_w # 1 watt-second = 1 J
    return tNeeded

def powerFmHeat_w(water_g, tempStart_c, boilTime_s) :
    return (water_g * (100 - tempStart_c) * 4.185 J/cal) / boilTime_s

the effort a motor system exerts to cause motion.
    This is also the instantaneous rate of energy being consumed.
    When summed over some time interval, this becomes your energy figure.
walking burns 1/8 kwh per mile. whpm=125 ! 1 kw / sq meter = incident solar power. 1 kwh = 3400 BTUs
precession, axis
"precession of the equinoxes". earth orbit period 25772 years 1st pt of Aries, fm zodiac times... 360 degrees / (25772 years). Caused by objects (moon sun(?)) pulling on earth's central bulge (?)
earth, local

If the Earth's pole wanders, then the orbital location where equinox occurs will similarly wander. Yes, recall that the direction the pole is tilted in its 2x23.5 degree circle determines where the equinox is.

precession, orbital
why Earth's orbit axis slowly rotates.
    nBody's pulling
    in mercury's case, 'Sun's oblateness' (central bulge);
         mercury's total is  574.10 secs±0.65 per century
    Gen relativity.
Another way of looking at it is that the circumference of a circular orbit near the Sun is
slightly less than 2π r because of the positive curvature of space-time. – Pete Jackson
general_relativity, wikip
pressure, units 'Pa'=F/A
pascals. Pa, P = F / A (force per unit area), 1 N / sq meter
pressure, wikip
principle of relativity
for now, see da prism equation
projectile motion
PT1_Q52: (in printed sheaf). Deriving 'max height':   
time2apex = Vy / acc
ht = ½ acc time2apex²
ht = ½ Vy²/acc
# and where 'acc' == G
ht = Vy² /(2*9.8 = 19.6)
ht = Vy² * 0.051  # in P51...
# and they made Vy == muzzleSpeed * sin(θ)

PT1_Q54: Target at 400m, height 100m. Cannon pointed 37 degrees above horizon and hits the target. Find muzzle speed, v0

proper time
(?) that measured in rest frame. no. meas'd by a clock passing thru 2 events. depends on clock's path. wiki.
Proper_time, wikip
mass 1.67¤-27 kg 
pseudo vector
eg magnetic field 'B'. in a mirrored image, vector doesnt also reflect,
must reverse sense too.
wikip, the "Fake News" of the math set ?
Test sets. PT's are's Praxis Test #1
(of the 5 I purchased in dec or jan). 1st tried 180109
SS was a short list of questions put out by the agency that makes the test (?)
See prjt re testing, testPhys
Q factor
The quality factor (or Q) of an inductor is the ratio of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, lossless, inductor. High Q inductors are used with capacitors to make resonant circuits in radio transmitters and receivers. The higher the Q is, the narrower the bandwidth of the resonant circuit. Q=(ω L) / R # 'linear increase w/ freq if L & R constant. R increases tho At low freq, Q improved w/ more turns. "L varies as N² while R linear"

ferromagnetic core increase the Q; but have losses that increase with frequency.

In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved.

wiki, Quantum_electrodynamics
wiki, Field
paperback copy in office 171014
170206 important degree 2 stuff in 'ballistics', for roboticsFirst/. see notebook, py
much simpler stuff in solarOven/ (excel...)

ariz edu
derivation of quadratic equation. implies -b/2a is x val for apex and other
part is x distance fm apex to each root.

Completing the square part of factoring
(Khan) For example, solve x˛+6x=-2 by manipulating it into (x+3)˛=7 and then taking the square root.
x + 3 = sqrt(7); x = sqrt(7) - 3 //sqrt 7 is 2.6457
quadratic eqn solver says roots are -5.645 and -0.354    sciCalc
[(sqrt(7)=2.6457) - 3 ] = -0.354
but -2.64 squared is also 7
[ -2.64 - 3 = -5.64 ]

wiki 'quadratic eqn' had this: 2x² + 4x -4 = 0. Ans: x = -1 + sqrt(3), -1 - sqrt(3)// in    

Note: //from vex. FIRST, robotics efforts. projectile trajectory calculation.
The quadratic formula can be dissected into 2 parts
-b/2a and (+- sqrt(b² - 4ac))/2a
The -b/2a locates the apex and the 2nd part defines the distance from the
apex to each of the roots.
2.6457 is -b/2a of the equiv polynomial
+-sqrt(7) are the distances from the apex to each of root
Quantized Oscillator

Serway p1159, vol 2. What is PE of a stretched spring and mass. ½ k x² (?)


A number system that extends the complex numbers. Used by Maxwell in his original papers on electrodynamics. Tossed by Heaviside in favor of the more conventional vector calculus.

A quaternion has 4 components.
Quaternion q = d + aI + bJ + cK where a,b,c,d are reals but I,J,K are imaginary unit-length vectors with the imaginative multiplication (transform!) table shown.

I J = K but J I = -K ! (some similarly to cross products (which is real-valued))

Addition is associative and commutative; multiplication is associative but not commutative.

The complicated rules for multiplication are described in the wikip article mentioned below.

I have books on Q's but this 'Definition' section of the wikip article was better. See 'versor' below, a unit-length quaternion.

There are books on how to use Q's for rotations in computer graphics. Python's 'numpy-quaternion' package is a special download, a WHL file which unpacked with Zip.

Mr Hamilton, wikip
Also Hamilton's last book is on Pinot. 1866. published by his elder son. It interested Feynman.
There's a binder, "Octanions", in the garage or office. 161107 (still around. 2" blue binder 171010)

57.2958 # degrees per radian
radio, DIY
radio tuner, variable caps max out at 360pF.
DIY radio design "modified boyscout radio"; principle mod is "antenna coupler"
youtube author 'squarewave2" "Boyscout Crystal Radio Upgrade"
youtube: "popular crystal radio deisng that's a waste of time - and why"
    it was bogged down w/ antenna's properties.
    this was resolved w/ an "antenna coupler"
antenna coupler: on a toilet paper tube.
    90 turns of #22 wire, 1/8" space, 25 turns #24  the 25T section connects to antenna.
    The antenna coupler was mounted 1" off the board.
Antenna 50' or (better) 100' as high as possible

Can buy a variable capacitance tuner at Antique Electronics ==
Book at Amazon "Radios that work, for free". K E Edwards
radioactive decay
nuclear processes, emissions, binding energies
Natoms(t) = Norig * e^(lambda tsecs) where lambda = ln2/(halflife time in secs)
Radius of Gyration coefficients
Wheeling a stone in a sling of radius 'r' gives an inertial value of  mass * r²

For an irregular object, not a simple stone, the force can be found by F = m * k²
where 'k' acts like the 'r' in the sling example above. 'k',

RofG is principally about calc'g rotational inertia and the result of RofG
considerations is a multiplicative item, k².

F = m r² # for wheeling a sling, rock 'm' kg and r the full ____
Values define where the average mass would be located as a portion of the full radius.

A RADIUS of anything should behave like a radius. Parts of this presentation are not conforming to that notion. And more insight can be had (I hope/think). Look at the expansion of this in ./radiusOfGyration.html
Column 3 'sounds like' a radius. (k2 R2) sounds like a (radius)2 which is then the _correction_factor_, no longer just a radius.
rod/baton, 4th column: L2 or L ?

At left is the real meaning of 'k', the Radius of Gyration. Note that it uses the root(2/5) while some use correction factor, the square, 2/5.

For the sphere example, 'I' can be found with either formula but the 2/5, by itself is not 'k'. Many online aids use the true 'k'.


true Coef
Radius of
Gyration, k
resultant I
m * k²

sphere, ballsqrt(2/5) sqrt(2/5) * R 2/5 R² M * 2/5 R² XAM p15,p35 Serway p278
cigarsqrt(1/3) sqrt(1/3) * L ⅓ L² M * 1/3 L² rotation fm 1 extreme end.
(parallel axis derived, Serway p281)
cylindersqrt(½) sqrt(½) * R ½ R² M * ½ R or disk of const thickness
hoop1 R 1 R² M * R² just like a mass at distance R. duh.
rod/batonsqrt(1/12)sqrt(1/12) * L1/12 L² M * 1/12 L² L==length
rectanglesqrt(1/12) sqrt(1/12) * ? (a²+b²)/12 M*(a²+b²)/12 a,b are length, width
See moment of Inertia, 'I'
See Parallel Axis technique
180207 US railgun aiming to accelerate a 23 pound metal projectile to 4,500 miles an hour,
and the Navy’s goal is an effective range of more than 100 miles. To achieve this feat,
the railgun requires 25 megawatts of power per shot, enough to power 19,000 homes (for
how many milliseconds?).
(source: Popular Science)

A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion. chromatic dispersion, the change in refractive index with optical frequency.

primary rainbow. from 'axis', blue wavelengths bends more than red so the blue-ish colors show up on the inner part of the rainbow's arc. (Blue 40deg, red 42) secondary rainbow( not shown). has the opposite ordering of colors. Blues are on the outside of the arc. Because there are 2 reflections inside each water droplet. (blue 53,red 50) Reds 'face' each other.

light (local note)
rayleigh scattering
scattering of light by particles in a medium, without change in wavelength. It accounts, for example, for the blue color of the sky, since blue light is scattered slightly more efficiently than red.
XAM p74

RC circuit (series)
Time constant T = R*C = 100K * 22uF = 2.2 * 10-5 * 105= 2.2 s
T often written as 'tau', τ
1 time constant, RC, takes the value to 63% of max.
2.2RC to 90%; 5 RC to 1%.   

see impedance and ref31, pg 24, top
RC time constant,'tau'
seconds to rise (or fall) 63%
'tau' = (R.ohms) * (Capacitance.Farads)
charging:    V(t) = V0(1 - e^(-tSecs/tau))   //SOON: python(?)   
discharging: V(t) = V0(    e^(-tSecs/tau))   
When the R & C are to function as a (passive) low pass filter
RC = 1 / (2π cutoffFrequency_Hz) => cutoffFrequency_Hz = 1/(2π RC)
At frequencies > 100khz, it's advised to use RLC, not just RC.
The L (inductor) in RLC is paired w/ the C to make a tuned ckt, the R dampens the response.
For more on RLC, see wikip.
reference frame, inertial
Since objects tend to retain their current velocity due to a property we call inertia, frames that refer to objects with constant speed are known as inertial reference frames.
Introduction_to_special_relativity, wikip 
refractive index  = "C / speed in other medium".  // not func(freq)   
prism PT1_Q93
"Electrodynamics of Moving Bodies", "Annus Mirabilius" _Special_ Relativity, wikip
intro to Special Relativity, wikip
'Now' book (kindle) describes pole-in-barn paradox
relativity calculator
table contents
The errors, the differences between measured data and calculation based on some idealized equation which is supposed to explain the data.
XAM_Q43, XAM_Q46  SOON w/ diagrams
standard resistor values
11 12 13 15 16 18 20 22 24 27 30 33 36 39 43 47 51 56 62 68 75 82 91
11 12    15    18 20 22    27 30 33    39    47 51    62 68    82    fgh's preferred values
eg 3:2. planetary orbit, atomic orbital, or sound
variable resistor used to control current. named by Wheatstone. ref22, pg 21, shows how to build a 'salt water rheostat' for handling very high currents.
richter scale
1928. logarithmic. replaced by moment magnitude scale.
    energy per base (10) goes up 30 per division.
    zero pt 1micron at __... ;
    amplitude and ? wave types.
    kt event about mag 13 or 14.
greek letter, like 'p'. Usually means density. Pronounced 'row'.
Is 'charge density' in electric field descriptions

right hand rule, RHR
RHR is used for vector cross products, spinning objects, and E&M probs. Share basics (?) XAM_Q42

There is a 'left-hand rule' seen in wikip Faraday. The left-hand rule finds the direction of the electromotive force (EMF) directly from Faraday’s law

wikip on magnetic field calls it 'the right hand GRIP rule'. diff ? Curl rh fingers around wire ST thumb points in I's dir, B follows curled fingers.

The same article shows the ? 'right hand palm rule' :-) palm up... Thumb points w/ I, palm up points w/ Force, straight fingers point w/ B field

Align finger with proton's motion,'v'. Align middle finger with the magnetic field. Now your thumb points in the direction of the magnetic force.

For the field around a wire carrying current, align thumb with current dir. fingers wrap in dir to integrate, 'B' field. Serway p845

Similar process for identifying the angular momentum vector of a spinning object. Wrap right hand fingers in the direction of rotation, thumb aligns w/ the vector.

(diagram from

A summary of 2 RHRs was found here

RMS, root mean square
Vrms=Vpeak / sqrt(2)
rotation, in general

ω is the most common unit for spin rate. While it IS radians/sec its units are really 'per second' or sec-1.

Some problems are expressed in full, 360 degree rotations. The spin rate is 'frequency', 'f', and (always) ω = 2π * f. Its units too are really 'per second' or sec-1.

'α' is rotational acceleration, d&ω;/dt, units 'radians/s². ============ TODO !? s-2 ?

To see large copy summarizing rotational and linear, click here. The following image is a subset.

moment of Inertia, 'I', fm object's shape.
angular_momentum 'L' = I * ω kg m²/s

Rotational KE = ½ I ω².
Tangent velocity at disk rim = r * ω
Tangent acceleration at disk rim = r * α
Centripetal Acc, Fc = m v²/r, (r meters)
work torque * angleInRadians = Work, Joules (== N-m) // like Work = F D

example: Earth's rotational KE: 2¤29 Joules, rot KE ============ TODO !? k² ?   
    day: 23.93 hrs, or 86148 secs, or 7.29¤-5 radians/s
    rotational momentum I = 8¤37 kg m²
    => 2¤29 Joules, rot KE   // ref: Serway (?)

rpm, bike wheel
rpmWheel(mph,diaInches) = (mph * 5280 / 60) / (2π * diaInches / 12)
           mph * 5280 * 12          mph * 168
      = ----------------------- = ------------
         2π * diaInches * 60       diaInches
Russell's Paradox, 1901
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.

Zermelo's axioms went well beyond Gottlob Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZFC). Zermelo's solution to the paradox ... turned out to be first-order logic while Russell altered the logical language itself.


sagitta, s = r - sqrt(r² - ell²)

Note that 'l' (ell) goes only ½ way across. It is the 'half chord'. To calculate l, see note on chord.


scientific method
Galileo credited w/ developing it but ~1000AD an Arab did pretty much the same thing.
Since the 17th century, consisting in systematic
Scientific_method, wikip
Serway (the book)
"Physics for Scientists and Engineers", volume 1, 7th edition, 2008
Serway (the man)
Raymond Serway puts out a new physics text every year (?), seemingly changing
    the 2nd author as often. Even tho the editions are different, much of the
    text, photos and diagrams are the same.
SHM, Simple Harmonic Motion
characterized by (eg) y = A sin(kx - ω t) #waveEquation!   
oscillation freq f = sqrt(k/mass) / 2π
'Solve periodic motion problems': XAM p23
Phys By Example, end 'math notes' (red tab). (un)damped harmonic oscillator motion.
    in Angel Flight binder (think this has moved...) TODO_180128
Serway, vol 1, pg 418 notes "force always directed to equilibrium point (x axis in sine wave example).
F = -k x restoring force, sure similar to spring eqn.
accel = d2x/dt2 = - kx/m   
(substitute ω² for k/m SO accel = - ω² * x   

XAM_Q17, pg 175: "What is the max displacement from equilibrium of a 1 kg mass that is attached to a spring with constant k = 100 kg/s² if the mass has a velocity of 3 m/s crossing the equilibrium point? "

KE (½ m v²) at fastest point must equal the spring potential energy (½ k A²)
where 'A' is the amplitude of the wave (largest pull-back).
KE = ½ 1kg (3m/s)² = 0.5kg * 9 m²/s² = 4.5 kg m²/s²
PE = ½ k A² = ½ (100 kg/s²) (A meters)² = 50 A kg m² / s²
KE = ½ 1kg (3m/s)2
KE = 0.5kg * 9 m2/s2
KE = 4.5 kg m2/s2
PE = ½ k A2
PE = ½ (100 kg/s2) (A meters)2
PE = 50 A kg m2 / s2

4.5 kg m2/s2 = 50 (A m2) kg /s2 # (set KE = PE) A2 = 4.5 / 50 = 0.09 # find A squared A = 0.3 m

Simple Harmonic Motion is also covered in AP Physics bk pg 313:
y(t) = A sin(ω T + initialAngle).   comes from
d²y/dt² = -ω² y    #differential eqn in acceleration

F = ma   // on floor (in x)
-kx = m d²x/dt²
-kx/m = d²x/dt²   

SI units
based on MKS, SI_units, wikip
  1. ampere - electric current
  2. candela - luminous intensity
  3. kelvin - temperature
  4. kilogram - mass
  5. meter - length
  6. mole - amount of substance
  7. Pa, Pascal - pressure. 1 N / meter², wikip, unit
  8. second - time
  9. Tesla - A mFlux density of one Wb/m2 = 1 T (tesla)
  10. Wb, Weber = SI unit of magnetic flux ('mFlux')

SI units
unit conversions (not all SI). back cover of Serway
Force  1N = 0.224 lbs // about 2/9
Force  1 lb = 4.448 N // recip .2248 lbs of force
Veloc  1 mi/hr = 1.47 ft/s = .447 m/s = 1.61 km/h
Veloc  1 m/s = 100cm/s = 3.281 ft/s
Veloc  1 mi/min = 60 mi/h = 88 f/s
accel  1 m/s2 = 3.28 ft/s2 = 100 cm/s2
accel  1 ft/s2 = 0.3048 m/s2 = 30.48 cm/s2
presr  1 bar = ¤5 N/m2 = 14.5 lb/in2
presr  1 atm = 760 mm Hg = 76.0 cm Hg
presr  1 atm = 14.7 lb/in2 = 1.013w5 N/m2
presr  1 P = 1 N/m2 = 1.45 ¤-4 lb/in2
time   1 yr = 365.24 days = 3.16¤7 sec
time   1 day = 24h = 1.44¤3 min = 8.64e4 sec
energy 1 J = 0.738 ft lb = N m
energy 1 cal = 4.186 J
energy 1 BTU = 252 cal = 1.054¤3 J
energy 1 eV = 1.602¤-19 J
energy 1 kwh = 3.60¤6 J
power  1 hp = 550 ft-lbs/s = 0.746 kW
power  1 W = 1 J/s = 0.738 ft - lbs/sec
power  1 BTU/h = 0.293 W
sideral (sp)
against stars; vs synodic.  SOON  

a derived unit of ionizing radiation dose in the International System of Units (SI). It is a measure of the health effect of low levels of radiation on the human body. Quantities that are measured in sieverts are intended to represent the stochastic health risk, which for radiation dose assessment is defined as the probability of cancer induction and genetic damage.

Traditionally used for radiation intensities lower than those that make acute and direct damage to a body.

31 emissions/s per gram of natural potassium.
BED==Banana Equivalent Dose = 2.3uSv. 31 emissions/sec/g = 31 Bq/g human flesh.   
BED. 70kg human, 2.5 potassium g/kg(human flesh), * 31 Bq/g of K = 5400 emissions/s   
BED, wikip
gone(?) are: REM (Radiation Equiv Man), RADs
Sievert, wikip
SI unit of conductance, symbol 'G', 1/R, 1/ohms, == 'mho'
3 sigma, 5 sigma rules
About five percent of a Gaussian distribution is more than two standard deviations away from the mean (in either direction). The tradition in particle physics is that the threshold to report “evidence of a particle,” is p < 0.003 (three sigma), and the standard to report a “discovery” is p < 0.0000003 (five sigma, p < 3x10^-7).
Significant Figures

A is only for readability. not signif
B only used to locate the decimal pt. not signif
C signif
D signif. indicates the precision of the measurement
image showing rules
Clocks that are running at different rates lead to the effect of simultaneity being relative.
What causes two objects to be simultaneous? "
Whether two things are simultaneous depends on the _reference frame_

wiki: In physics, the relativity of simultaneity is the concept that distant simultaneity –
whether two spatially separated events occur at the same time – is not absolute,
but depends on the observer's reference frame.

ARE THESE ASSUMED TO BE INERTIAL? (non accelerating)  // seems so
slide rule
log(xy) = log(x) + log(y)

This shows how addition of lengths can represent multiplication (or division). Basic multiplication is supported by the A-B (single decade) and C-D (double decade) scales. more on AB,CD

A,B  2 decade log scales. used for finding squares and roots
C,D  single-decade log scales
K    3 decade logs. used for finding cubes and roots
CF,DF "folded" versions of C-D, start from Pi.
     1. avoid possibility of going off scale w/ a number near 10
     2. multiplying by π is simplified
CI,DI; CIF,DIF  "inverted scales", run fm right to left. simplifies 1/x steps
S    used for finding sines, cosines on C or D scales.
T    find tangents and cotangents on C, CI (or D, DI) scales
ST   sines and tangents of small angles and degree-radian conversion
L    linear scale. used w/ C,D for finding base-10 logs and powers of 10.
LLn  a set of -log-log scales. find logs, exponentials of numbers
Ln   a linear scale, used w/ C,D for finding base e logs and ex

squares B,C
cubes  D,K
TODO. list need then approp scales to use.

see? Used by famous people!
'how to use...'
Slide_rule, wikip
defines other scales
unit of mass (not weight).
"Imperial" unit, about 14.6 kg or 32.1 lbs.
snell's law
n1 * sin(θ1) = n2 * sin(θ2)
θ2 = arcsin(n1 * sin(θ1) / n2)
eg: incident raw, in air, 30deg off vertical, hits horiz water surface (n2 == 1.33).
At what angle is the ray traveling in water?
    θ2 = arcsin(1.0 * sin(30) / (4/3))
              = arcsin( 0.5 * 3/4 ) = arcsin(0.375) = 21.5 degrees

see critical angle, local about total internal reflection.
solar flare
CME. Coronal Mass Ejection; prominence;
video on phone 'Pixel'
Carrington Solar event 1859
solar furnace, cooker, parabola
    prjt solarOven
    "Science Lab", pg 82, using an umbrella
solar power
1 kw / sq meter = incident solar power, at the ground. Higher at top of atmosphere.
solar wind
800 km/s
field 'B' = (µ0 I nTurnsPerMeter), XAM p81   

The magnetic field inside an infinitely long solenoid is homogeneous
and its strength neither depends on the distance from the axis, nor
on the solenoid's cross-sectional area.
Rt Hand Rule (grip), wiki
solid angle
units 'steradians'
sound levels. dB, 130dB max for humans (?), Serway p482
resonant freq
an open tube, L meters, f=intgr * sos / (2 L) sqrt(B=Bulk Modulus=volumeCompressibility / densityOfMedium) speeds of sound: Serway p476 air 331-343 m/s function of pressure ( ~ density ) water 1493 aluminum 6420 sea water 1533 dB sound scale : saturn 5 launch was 220dB, water reduced launch sounds to 142, same as jet

sound wave propagation. PT2_Q60 says "according to Laplace, the prop of sound waves thru a gaseous medium is an adiabatic process, not (Newton's) isothermal process."

1908. Minkowski. negative term non-intuitive, very important
spacetime interval
syn: "Minkowski metric". _invariant_
s² = x² + y² + z² - (ct)² ,  where 's' is distance

See metric used in hyperbolic functions

special relativity
_Special_ Relativity, wikip "Electrodynamics of Moving Bodies"
"Annus Mirabilius"
wikip  intro to Special Relativity
Special_relativity, wikip
specific gravity
The ratio of the density of a substance to the density of a reference substance, usually water. wikip
specific heat,'c'
(better 'specific energy transfer') Serway p556
Specific heat,'c' = heat (Joules) per unit mass (Kg) to raise temp 1 degree C.
(Specific heat sometimes called 'specific heat capacity')
    c = Q / (mass * delta T)  OR, equivalently,  Q (J) = m (Kg) c deltaT

    air's value is 1.00J / (1 Kg * 1 degree)  // degree C or K
At most temps, to raise 1 gram water's temp 1deg C, it takes 1 calorie = 4.186J
water takes 4186J to raise 1kg water 1C;
copper takes 387J to raise 1kg water 1C;   1/10.8 that needed for water!

see sciCalcs  tags, zhe.phase

See latent heat for a heating other examples which goes thru a phase change.
Each emission line & color correspond to the energy released as the electron falls from the higher energy orbital to a lower energy. On the Sun, Hydrogen emits a lot of red light at 656nm and a lesser amount of blue at 486nm. Those wavelengths and colors are from the drop from n=2 to n=1 (656nm) and from n=3 to n=1 (486nm).

Z is the atomic number, 1 for Hydrogen. n is the upper energy level and n'' is the post emission energy level.

R is the Rydberg constant, (1.09737¤7 m-1).

PT3_Q76, question on the formula for the Lyman series.
wikip, H2 spectral series
speed of light
C=670¤6 mph, Wikip

The electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the electric constant, e0 and the magnetic constant, µ0 by the equation C² = 1/(e0 µ0)

spherical coordinates

quantized angular momentum, atomic level

springs, and constant 'k'
Hooke's Law: spring restoring force is -k x where 'x' is displacement and 'k' is the "spring constant".
-k x is a _force_ and thus has units of N, Newtons. k's units are N/m, Newtons/meter.
N/m * ~175+ = lbs/inch. Serway p187, 7.22.
Spring PE = ½ k x² # and is valuable for solving oscillations, harmonic motions.
XAM p31 has the following list of spring properties:
    KE = ½ k A² sin²(ω t + phi)
    PT = ½ k A² cos²(ω t + phi)
    TE = ½ k A²   // total energy
The above diagram is a good match for engineering where 'k' is known but for many applications, like "some stretchy material", the 'k' has to be found by measuring pulling force and displacement.

When you're dealing with a class of similar 'springs, again like rubber bands, the force F should also go up proportionally w/ cross sectional area. And you need to be wary that overstretching a rubber band, or even a spring, can permanently deform it.
Newton entry in gsci
Young's modulus, local
Square root
fast method for doing sqrt's in your head. pretty accurate.
youtube. tecmath "fast square roots".

example 1: sqrt(87)   
81 nearest square below 87.
 9 write this down. draw lines to its right to make a fraction
 6 = 87-81. write this in the numerator of the fraction area.
18 = double the 9; make this the denominator of the fraction
// The above step is from (x+c)² = x² + 2 x * c + c²
9 1/3 is the technique's answer;  calculator says 9.32737...

example 2: sqrt(138)   
121 nearest below. 11² = 121 < 138
 11 write this down. make fraction area (just a line at this point)
 17 = 138-121      ; make this the numerator
 22 = double the 11; make this the denominator
17/22 isn't an easy fraction;
    16/22 = 8/11. 1/11 = 0.090909 so 8/11 would be .7272
    18/22 = 9/11, same apch, = 0.8181
calculator says 11.74
Wikip, Squeeze_mapping
Standard Model (of physics)
img of matrix including the Higgs boson
standing waves
waves of same freq, opposite directions. Serway p505, XAM p66, procs
stationary points are called 'nodes'. points of max motion 'antinodes'.
y = (2 A sin(kx) cos(ω t)  // 'wave function'. left part is envelope, rt part fills w/ signal   
See trig identity needed
tabby's star. weird light curve. amateurs v active. TED talk.
Stefan-Boltzmann Law
P = sigma A e T4 ; SAET4, == Power, S-Tconst, Area, emissivity0-to-1, T thermo Temp
"Total (blackbody) radiant heat E is proportional to 4th power of T_kelvin"
Stefan-Boltzmann Constant, 'sigma', = 5.67¤-8 W/(m² * T4) 5-6-7-8 mnemonic
'e' is 'emissivity' in 0=>1 (mirror 0, black 1) range,
(emissivity is a fudge factor which allows this equation to cover not-so-black bodies.
emissivity is 1 for a true black body and less than one if the object doesn't follow true BB).
Also 'e' has some wavelength dependencies in real life.
    rayleigh-jeans law tried to define BB curve but went infinite as wavelen approached uV. uV catastrophe.
Planck's Law, local
S B constant
stellar aberration
The change in angle is typically very small, on the order of v/c where c is the speed of light and v the velocity of the observer.
On Earth, the maximum amount of aberrational displacement of a star is approximately 20 arcseconds in right ascension or declination.
Stellar_aberration, wikip
steradian, solid angle
On a sphere of radius 'r', a solid angle of '1 sr' defines a cone, centered within the sphere, making an area of r² on the surface of the sphere.

4 π steradian covers a sphere

wikip on steradian
K=appliedForce / deflection; independent of hardness (tempered, rolled etc)
            p    applied force
    K = ---
            w    deflection
Stoke's Law
frictional force – also called drag force – exerted on spherical objects
with very small Reynolds numbers in a viscous fluid. Stokes's law is derived
by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes
(Much different than aero drag for large objects like airplanes)
From wikip
Seen in a wikip discussion on the Millikan oil drop experiment.
Stokes Theorem
TODO(?) ref: XAM p208
aka curl theorem,Wikip
general theorem, Wikip theorem
'Standard Temperature and Pressure': 273.15K and 100kPa(==1 bar)
dimensionless, a ratio of stretch over original length
Wikip, Elastic_modulus
Wikip, Pascal SI  N/m² = 1 Kg/m s²
Force per unit area, measured in pascals,
ref12, item 27
PT2_Q75. good
solar corona. 106K Hbomb, 108K Serway p536
super heating
liquid sodium acetate in hand warmers. super cooling.Serway p562.
when temp drops below a material-dependent threshold, a "phase transition" can occur which changes the state of the symmetry and allows a new set of rules to apply. This type of symmetry breaking was used as a model by Peter Higgs in defining the behavior of the Higgs Field.
kindle book on Higgs...Symmetry.
Wikip, Meissner_effect
1044 joules, Quora aug 28,2017. At 4.e ly, would deliver    how far? must have been a beam?
5¤9 J/m² at earth.
from stars w/ masses 8 to 20 times the sun's mass. T2_Q34
Serway p501. re interference too

refers to a physical condition where a set of predictable outcomes are possible but not yet resolved (or chosen). When a particular change occurs, and some of the previous possible outcomes are no longer possible, "order" has increased but symmetry is said to be "broken". Examples: pencil standing on its point - has possible motions. Once fallen, fewer options.

symmetry, axis of

wrt earth's positions. mercury 116 days, 780 mars

system's E

KE+PE+internalE, from 'Transfer mechanisms' = heating,electricity,... Serway P197

Tachyon, wikip describes some debates which propose odd ways they could exist. "Negative squared" mass...
violation of causality
Taylor series
Tension is a force, units N (Newtons). mass * acceleration etc.
'T' often represents tension in an FBD (Free Body Diagram). Serway p110,p117.
See Atwood machine and the FBD traffic light problem.
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values...
...must be independent of a particular choice of coordinate system
Tensor, wikip
see Feynman, book 2, 26-7
also "Space Time Matter". paperback, book shelf, office
terminal velocity
air resistance
geese in V formation. b/c tip vortex creates small updraft.

1 Weber of mag flux per square meter == B mag fld strength.

Tesla is a measure of magnetic flux DENSITY, equivalent to mag field strength, consequence of the Lorentz Force Law.

1 T equiv units are: N/(A m) (Newton per (Amp meter)) and because Coulomb/sec is an Ampere, 1 T also has units of N/(C m/s) Serway vol2, pg 812

1 N from 1 C at 1 m/s thru 1 T field (B,v perpendicular)

1 T = 10,000 gauss; San Fran's mag field is 0.49 gauss or 0.000049 T or 49 µT
magnetism 1, Khan Academy's, on youtube
magnetism 3, Khan Academy's
“theory” they are not talking about a mere guess but instead "a coherent group of tested general propositions, commonly regarded as correct, that can be used as principles of explanation and prediction for a class of phenomena."

From an argument on Quora
hand closer - more (capacitative) charge - current osc at lower freq
farther - higher freq.
But prettyWoman's video showed pitch higher when hand near vert antenna.
And someone said the freq's were reversed in the electronics.
See entry in electrometer prjt

The instrument's controlling section usually consists of two metal antennas
that sense the relative position of the thereminist's hands and control
oscillators for frequency with one hand, and amplitude (volume) with the
other. The electric signals from the theremin are amplified and sent
to a loudspeaker.
See wikip Operating principles,
IS the study of work, heat, energy in processes. Q(heat) => system[deltaE] => W(work)
The four laws of thermodynamics are:
thermo Law 0: total E const. if sysA equilib sysB AND sysB equilib sysC then sysA equilib sysC
thermo Law 1: Energy i/o chgs system's E per conservation rules.           
thermo Law 2: sum of entropies increases; perpetual machines impossible    
thermo Law 3: Entropy approaches constant as T => 0. S at 0K is about 0.

more detail, from wikip:
Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a
third system, they are in thermal equilibrium with each other. This law helps
define the notion of temperature.

First law of thermodynamics: When energy passes, as work, as heat, or with matter,
into or out from a system, the system's internal energy changes in accord with
the law of conservation of energy. Equivalently, perpetual motion machines of
the first kind (machines that produce work with no energy input) are impossible.

Second law of thermodynamics: In a natural thermodynamic process, the sum of
the entropies of the interacting thermodynamic systems increases. Equivalently,
perpetual motion machines of the second kind (machines that spontaneously
convert thermal energy into mechanical work) are impossible.

Third law of thermodynamics: The entropy of a system approaches a constant
value as the temperature approaches absolute zero. With the exception of
non-crystalline solids (glasses) the entropy of a system at absolute zero
is typically close to zero, and is equal to the natural logarithm of the
product of the quantum ground states.

from ref18, chap 16, "Thermodynamics and Chemical Equilibrium"
    law 1: energy neither created nor destroyed. Or "total E is constant"
           and deltaE = q + w // q is heat input, w is work done on system.
    law 2: entropy of Universe increases during spontaneous reaction.
           (deltaEntropy=deltaS) .ge. q/T // q
           E and H are properties of the system. G is Gibbs free energy
           reactions for which deltaS > Q/T can be spontaneous, otherwise not.
           "all spontaneous processes in nature increase entropy of universe"
    law 3: provides means of calc'g E. Zero E is crystal at 0K

(from Serway (?))
alcohol expands (fastest of common choices(?)) w/ T // max 85C
0 deg water and ice are at equal T's. => no E flow. "ice point of water"
can use pressure chg to reveal T (?presuming volume constant(?))
lines extropolate to absolute zero at -273.16 C 
thermal expansion of joints. p537 ways to handle that.
atoms osc amplitude is 10¤-11 meters. freq 10^13 Hz, spacing 10^-10m
coef of linear exansion α = (deltaL / Li) / deltaT
calcite expands on 1 axis, shrinks on other. CaCO3
thermodynamic processes
TODO. enhance with notes on plastic 'equations' doc
adiabatic   const heat    
isobaric    const pressure
isochoric   const volume  
isothermal  const temperature (not nec heat)
isothermal vs adiabatic  fm Isothermal_process, wikip   
for PT3_Q60

An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir (heat bath), and the change will occur slowly enough to allow the system to continually adjust to the temperature of the reservoir through heat exchange. In contrast, an adiabatic process is where a system exchanges no heat with its surroundings (Q = 0). In other words, in an isothermal process, the value ΔT = 0 and therefore ΔU = 0 (only for an ideal gas) but Q Δ 0, while in an adiabatic process, ΔT 0 but Q = 0. praxis: 0% In isothermal process, no change in internal energy takes place. TODO check. after Delta installed...


model at terminals as open circuit.

Originally it applied only to DC resistive circuits. And each such could be modeled as a voltage source (battery like) and a single resistor. It also works, however, with AC circuits containing both resistive and reactive impedances. Doesn't always get the power consumption right.

All About Circuits adds: Thevenin’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). If we’re dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor components) which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.

wikip shows how
mentioned in "Phys By Example" page 274
time dilation
In the theory of relativity, time dilation is an actual difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses.
Time_dilation, wikip
Gravitational_time_dilation, wikip
dL/dt, units N•m;
Ang Mom  T = r χ F
torqueAtWheel = torqueMotor / overallGearRatio * overall Drivetrain efficiency

"Mathematically, torque is defined as the cross product of the lever-arm distance vector and the force vector, which tends to produce rotation."

The magnitude of torque depends on three quantities: the force applied, the length of the lever arm connecting the axis to the point of force application, and the angle between the force vector and the lever arm. In symbols:

T = r χ F = ||r|| ||F|| sin(θ)

fgh: note that mechanical tracks or other constraints may prevent the force applied to be ideal (perpendicular to the radial line from the center of rotation) - hence the sine etc.

Torque, wikip

The torr (symbol: Torr) is a unit of pressure based on an absolute scale, now defined as exactly 1/760 of a standard atmosphere. Thus one torr is exactly 101325/760 pascals (~133.3 Pa).

Historically, one torr was intended to be the same as one "millimetre of mercury". However, subsequent redefinitions of the two units made them slightly different (by less than 0.000015%). The torr is not part of the International System of Units (SI), but it is often combined with the metric prefix milli to name one millitorr (mTorr) or 0.001 Torr.

The unit was named after Evangelista Torricelli, an Italian physicist and mathematician who discovered the principle of the barometer in 1644 Wikip


Vs/Vp = Ns/Np where Vs is the secondary voltage, Vp is the primary voltage, Ns is the number of secondary windings and Np is the number of primary windings. Divide the number of secondary windings by the number of primary windings, and multiply the source voltage by this ratio. This will give you the output voltage. For example, a voltage source that sends 240 volts through a transformer with 500 primary windings and 100 secondary windings would have an output voltage of 240 * (100/500) = 48 volts.

XAM p82

Trebuchets get their energy from gravity. They (often?) have a sling but that's not the defining characteristic; it's the gravity.

Catapults get energy from tension (usually torsion) and the arm hits a stop near the top of its arc to release the projectile.

"triboelectric" series in the order of most electron attractive
to electron repulsive: (top entries get very negative)
    sulphur  (hmm sulphuric acid good in batteries... Loves more electrons)
    brass     items at the top of this list become more negative.
    silk      vs glass
    cat fur   vs glass.
    ? saran wrap is somewhere on this side of the sequence
(rubber) shoes on carpet; styrofoam
balloon (neg) vs hair (pos) on head
wool, paper or cotton cloth and 4' PVC pipe; AM radio noise;  rod attracts empty can
    I rubbed a half meter piece of schedule 40 pvc w/ old T shirt and it picked up paper scraps.
    Most of the papers only stayed w/ the PVC for a few seconds. Then dropped.

VDG uses such materials to transport charge into the big sphere at the top.

PVC charging a cap:

In one VDG, a rubber belt driven by plastic pulley moves charge up
to VDG's dome (w/ combs).

tinsel ?
saran wrap removes electrons; cat fur adds;
styrofoam PLATES will 'launch' one by one off VDG. rub w/ wash cloth !
vocab TODO. "Wimhurst machine" ?   'electrophorus'

rubbing silk on amber   wikip, Inductance
balloon against hair, wall...
trig identities
sine(a+b)=sin(a)cos(b)+cos(a)sin(b), memorize SOON
Seen in Serway ~p508(?) under Standing Waves
and whatever other identities aren't covered on the plastic review sheets.
triple point
physical conditions in which all 3 phases of a material
are present: solid, liquid, and gas. The 'conditions' are temperature and pressure.

Water's triple point is 0.01C, 4.58mm mercury. Used to define 0 Kelvin as this - 273.16 ref01, Serway p536
Twin_paradox, wikip
my project
Its name: Historically called 'mu naught', written 'µ0'. The µ0 is staying around
but standards boards have been pushing the name "magnetic constant".

Units: wikip shows several equivalent sets of units for it:
    T m/A   : Tesla meters/Amp.  most popular (?)
    H / m   : "henrys" (Inductance) / meter
    N / A² : Newtons per (Amps squared)

1.256¤-6 (= 4π ¤-7) T m/A (or Newtons/Amp) # magneticConstant
Seems this is the reluctance of the vacuum to take on a magnetic field - that
the success in building a field is like compressing a spring. The B fld is PE.
Could be backwards; 'reluctance' doesn't quite feel right.
Maxwell's cross terms suggest the e and m fields play off each other.

// wikip 'Magnetic Field' article implies that the constant linking H and B fields
// is none other than µ0 !  torque = m χ B = µ0 m χ H, in vacuum.

The 4π typically present. But µ0 can be written 1.2566370614¤-6 Newtons/Amp².
This is a 'defined value' and not the result of an experiment.

wikip article Magnetic Fields discusses H fields (b/c perm magnets so avail(?)).
When describing B field, article uses µ0 as above .
When describing H field, they use µ0 as 4π ¤-7 (v*s)/(A•m)
torque 'tau' = m x B = µ0 m x H // "same, up to a multiplicative constant, µ0"

terms being discarded: according to wikip (strong trend towards "magnetic constant")
    mu naught
    magnetic permittivity of vacuum
    permittivity of free space
    vacuum permittivity
    permiability of vacuum

Utility. Where used.
    mid 1800's it was used to calculate the Speed of Light
    Ampere's Law
    wikip, µ0  = 1.256¤-6
ultraviolet light
UVA  315-400  3.10-3.94   long wave, 'black light', not absorbed by ozone layer
UVB  280-315  3.94-4.43   mostly absorbed by ozone layer
UVC  100-280  4.43-12.4   germicidal, completely absorbed by ozone layer and air.
uncertainty principle
heisenberg. 1927.
product of the uncertainties of complementary variables, such as position and momentum,
must be greater than or equal to hBar/2.  hBar is planck's_constant/2π, 'h', divided by 2π.

eg: delta X * delta P >= h/2π SOON units of the multiplication should be J-sec ? examples of conjugate pairs: mass, velocity position and momentum And, far from least important, most/all wave calcs experience precision problems == uncertainty. You tighten down the delta wavelength and have little/no signal... wikip
units, conceptual value of complex units
There are the 'fundamental (SI) units' (eg: kg, meter, sec) and there are 'derived units' formed by combining the fundamental units.

The wikip article, linked below, gives a brief description of the history of these definitions. The evolution began in the 1700's and kg's last tweak rolls out in May, 2019.

Several of these derived units, like Tesla, are composed of several fundamental units. EACH unit, eg N/C * m/s, is a strong hint about how the Tesla (in this case) relates to other concepts and what a user in that area expects from a Tesla (numerator often N, Newtons, FORCE). (N/C)*(m/s) = This conversion factor will output N•m (the numerator) when given C/s (denom). This makes it worthwhile to know the various units, not just one.

    1T = (N/C)*(m/s) // Newton per CoulombOfCharge x meters/sec
    1 Wb/m2 =    // webers per square meter
    1 N/ A • m  // Newtons per Amp•meter
possibly a worthy subject for lectures.
wiki SI base units
wiki SI derived
units, ending on chemicals
-- endings
-ane  alkane, methane
-CH3 methyl group          JUST PLAYING. DUNNO if SIGNIF.
-ene  alkene
    DNA pairs: adenine - thymine; guanine - cytocine
-graphy  topography
-ine  guanine (DNA pieces). fluorine, chlorine,
-line   ethylene
-lipid  phosphlipid
-loid  for chromasome pairing. haploid
-nine   implies 'amino acid'(?)  eg: serine, alanine
-patric Allopatric
-phase   anaphase
-some    allosome, autosome, chromosome
-sis     mitosis
-sphere  photosphere, mesosphere,
-yl      methyl
universe's expansion rate
74.3 plus or minus 2.1 kilometers (46.2 plus or minus 1.3 miles) per second
per megaparsec (a megaparsec is roughly 3 million light-years).
Spitzer Space Telescope. 74 km/s per 3.086¤19 km.
varactor diode
used in "FM radio kit" to tune freqs. see the assembly pg that came with the kit TODO
Van de Graff generator, 'VDG'
youTube van de graaf w/ coke can. SciJoy, 5:02

Safety issues
rubber belt driven by plastic pulley is example of 'triboelectric' material,
metal combs at either end.
    sphere is a capacitor. 200kv - 300kv
    rough values of sphere, 10 pF, and body 1000 ohms, the discharge will be nanoseconds, < msec
   'van de graffs work best in low humidity'.
    body resistance == left to right hand at least 575 ohms
   'discharge wand'. some insist 'necessary!'. sold online. metal ball at one end. grounds the big dome.

    can store 0.5 J ; 'discharge of 1 J hurts everyone severly'.
    advise 20cm or less sphere diameter and voltage < 250kv (so E < 0.35 J)
    keep computers, memory sticks, disks, ?phones? away.
    no if heart condition, pace maker, has embedded medical device of any kind.
    volunteers only. some may not like the small shock.
    after person charges themselves, they are charged. do not grab metal anything (like a pipe)
        touch desk top (?) til charge dissipates.
large mixing bowls.
350kv, good. 'triboelectric transfer', b/w img

Googled 'charging styrofoam balls' 'DIY particle accelerators' ('cyclotron'NotGdResults).
Could cockcroft-walton supply charge things?

VDG/wand, bowl/ball, 5 "things you should never do...", 36min
interesting use of 'discharge wand': she holds it against sphere while near sphere.
Crash Course Phys #32, Indian woman. re Oersted 1820
vacuum pump, expensive, long wine bottle, drilled hole w/ special bit.
    steel 'anode wire' "sputtered" all over inside of bottle; author suggests aluminum next time.
    Subscribed DIY Electron Accelerator - Cathode Ray Tube. "GooFerking Science"
TODO shows variac to 'oil burner ignition transformer'
'homemade crt', zinc sulphide screen, vac pump, hi V 'electrophoresis power supply'.
'homemade crt' gooferking version. erlenmyer vac tube. x-rays possible w/ CRT.
    coated bottom of flask w/ 'beta reactive phosphor'
phosphor can be scraped out of old phosphorescent tubes.
TODO $300 325 kV van de graff,

vector field
cross product local link
dot product local link
divergence local link
grad local link
curl, del χ F local link
a quaternion of length 1.
even r² = -1, Wikip, Versor
vis-viva equation
Leibniz' focus on KE-PE conservation vs Newton-Descarte's Momentum

In astrodynamics, the vis-viva equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight.

Vis viva (Latin for "living force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the aggregate work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done.

For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the vis-viva equation:

  • v is the relative speed of the 2 bodies
  • r is the distance between the 2 bodies
  • a is the semi-major axis
  • G is grav constant
  • M is the mass of the central body
and the product G * M can also be expressed as the standard gravitational parameter using the Greek letter µ.
unit of 'electrical pressure'. Electrons without this 'pressure' behind them will sit still. They'll need that pressure to overcome whatever resistance lies in their path.
volt == W/Q = J/C = Joules/Coulomb = energy/globOfCharges, XAM p77
volts x Q = Work (see units. same as VI=Watts) volt same as 'electromotive force' or EMF
Volt Ohm Meter, aka 'Volt-Ohm-Multimeter', 'multimeter'
# of 'digits', eg: '3 ½ digits' etc.
'impedance, circuit loading'

wikip, properties
H20. in botany/chem file, glossary_chemistry.html
W = J/sec = volts * amps ; W = N•m/sec
a small amount of power. A bathroom light, before LEDs used to take 7 watts.
It was barely warm to the touch.
The human brain, I've heard, takes 60watts at minimum (? after coffee ?)
watt = Joules/sec = volts * amps = Newtons * meters/sec

// pg 3 of 4, physics of auto energy consumption, Koon
watts(force_lbs,v_mph) = force_lbs x v_mph x 746watts/HP / 375
                       = force_lbs x v_mph x 2
watt-hour = 3600 Joules

power, W = V I and V = I R so I = V/R, so W = (V/R * R) * I = I² R  

watt-hours, 'WH'
electrical energy. A 'Watt Hour' is 1 watt for 1 hour, 1/N watts for N hours.
    Usually written "WH" or "Watt-Hr" in referenced material.
watt-hour = 3600 Joules

// derived fm above. works in ev.c too
watt-hours(force_lbs, miles) = Joules(force_lbs,miles)/3600
      = force_lbs x miles x 7157/3600
      = force_lbs x miles x 1.98

testPages/imgs/PT2_Q87_waveTypeFmEqn.png offers 4 named wave types:
y = A sin(wt + kx)
    progressive longitudinal  -kx for progressive
    progressive transverse     if 'x' goes w/ +/-k then perpen to y => transverse
    retrogressive longitudinal +kx for retro
    retrogressive transverse
 my drawing, gsci vv_waveEqn3D, is Y = A sin(x - ct), retro transverse (?)
vv_waveEqn y = A cos(kx - wt)
vv_waves #is where the result of this study should go. It links

Wave Speed. some matl in vv_waves, top

    longitudinal (osc in direction of motion)
        v = sqrt(elasticity / density)
    transverse (oscillations perpendicular to motion). eg guitar string.
        v = sqrt(tension / linearDensity)

A key notion regarding waves is that the matter of the supporting medium is not being moved permanently. It's displaced temporarily then returns to its original position. (physicists have never had surf break on them!)

Light waves are composed of both types of waves. The longitudinal part always obeys the speed of light but the transverse portion reacts to the index of refraction. (sorry. don't really understand this! fgh 171104). Feynman? QED book? soon

standing waves (XAM p66) typ result from interference of 2 waves of same freq
    traveling in opposite directions. There are 'nodes' where no movement.
    max motion at anti-nodes.   dbbl slit Young, local
interference  Serway p501.
    dbbl slit interference given by "n lambda/ d = x / D = sin θ", n == integers   

wave types if rope constrained == transverse, PT1_Q23
wavelengths permitted in certain tube lengths PT1_Q79
XAM p58,
ref11, "Physics", plasticized summary. other types of waves: traveling, standing, quantum mech, harmonic
wave equation
y = A cos(kx - ω t)
will build trial problems in sciCalcs

see Retrogressive...
A wave equation in 3D
As cT increases, the argument to sine 'reaches farther back' into the cT==0 curve. At the red circle, cT == x so the sine argument is 0, mimicing the very start of the wave pattern.

If a time-current 2D image were displayed on a screen, the waves would appear to march towards the right.

'seismic waves', made by earthquakes
P wave  'Primary'. longitudinal. 7-8 km/s  Serway p451
S wave  'Secondary' or 'Surface'. transverse 4 - 5 km/s
XAM p59 defines subtype (Love wave, Rayleigh waves (both Surface waves) ).
weber, Wb
A flux change of 1 Wb/s induces 1 V in a single loop of wire.
The _changing_ flux field _defines_ the Weber: SI unit of magnetic flux.
The directions of the field matter (vs those of the wire).
'A' = B A cos θ, where 'B' is field strength
mag flux DENSITY of 1 Wb/m² = 1 T (Tesla)
(the vv_tesla entry shows T = J/(A m) = Wb/m² implying Wb = J/A)

This was even confusing for standards people. Related discussions started in 1861, got reviewed in 1902, got more focus in 1927, and took a task force 11 years to make solid recommendations. Related standards questions were resolved in 1950. Weber_(unit), wikip

Faraday's Law
Weight is a FORCE, not a mass. So wt = Mass * GravAccel(9.81)
Mass is 'kg' or 'slugs', Wt is Newtons, 'N', or lbs
wt,N = (m,kg) (A=9.8m/s2) so m,kg = (wt,N) / G
wt,N = wt,lbs * 2.2046 (N=kg 9.81)/lb * g.   Note: a pound (mass) is 2.2 kg; to get force (N) fm mass, mult by Grav
mass, local
Wheatstone bridge
Electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component. The primary benefit:
provides extremely accurate measurements. Its operation is similar to a potentiometer.

'Balancing' a Wheatstone bridge means you adjust R2 until Vg is 0.

Invented by Samuel Hunter Christie 1833. Improved by Wheatstone 1843. Wheatstone's initial uses included soils analysis.

Wikip article VERY INTERESTING ! thought-provoking.

Consider using w/ salt water rheostat, ref22, pg 21.

watt hours per mile. written several ways: WH/Mile etc.

This is "energy / mile" and is useful for determining theoretical range of an electric vehicle. A curious aspect of this is that it is E/distance whereas our gas-powered world talks of mpg which is distance/E.

    'Watt-Hour per Mile'. The energy needed to move a car one mile.
    Typical values at 50 mph would be: (figures predate Tesla car, Li batts)
  • 80 whpm motorcycle
  • 130 whpm Impact. induction motor and controller
  • 200 whpm Metro, DC, good aerodynamics
  • 350 whpm truck, running w/ DC, poor aero
// derived fm horsepower(F,Dist). works in ev.c too
whpm(force_lbs) = watt-hours(force_lbs, 1 miles) = Joules(force_lbs,miles)/3600   
      = force_lbs x miles x 7157/3600
      = force_lbs x 1.98
We call this 'energy' and, think about it, F*d IS Newton*Meters.
Fd = work = N•m = J, Force applied over distance. Work
40mJ Energy (liter of gasoline) / 500N = 80km

XAM_Q10, XAM p143. CAREFUL! 10kg item pushed by 5N for 5m is 25J. mass doesn't count!

V = W/Q   XAM p34,p77
Work on 2 charges: W = F r = k (Q1 Q2/ r²) r = k (Q1 Q2 / r)   
and b/c V=W/Q, divide abv by Q2: V = k Q1/r, k is Coulomb's constant (9¤9 N m²/C²) 
work function

context: photoelectric effect (which includes, but isn't limited to, solar panels).

The energy needed to dislodge an electron from a specific metal. Reading on the topic suggests further that the needed energy is also dependent on the local conditions; that is, the electric field being made by the surrounding conditions and materials.

In solid-state physics, the work function (sometimes spelled 'workfunction') is the minimum thermodynamic work (i.e. energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" means that the final electron position is 'far' from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electric fields in the vacuum. The work function is not a characteristic of a bulk material, but rather a property of the surface of the material (depending on crystal face and contamination).

work-energy theorem
"work = Δ E"
work-KE theorem
when only sys chg is speed, net work = chg in KE.
Serway pg175
xenon flash
The main reference, at, contained the following diagrams which show circuit states. Click each to enlarge.

2nd tutorial. Basically shows E = ½ C V²
XNOR gate
XOR w/ an inverter, ie: Not-XOR. This weird-ass gate definition, not seen in decades of dealing in normal circuits, showed up on a practice test for CSET 220. stupid.
wikip "the logical complement of the XOR gate"
yield point
permanently DEFORMS while being bent.
Zeeman effect
atomic spectral lines split, going to 'doublets' or 'triplets'. This caused a huge mystery ~1920 but led to the discovery of additional quantum variables. Great (but v. mathematical) description in ref24.


ref01Serwayphysicspaper,ofcvolume 1,2008. A principle reference
ref02CSET Physics 123,127physicspaper,officemy notes call this 'XAM'. good text,test
ref03CSET Physics Exam Secretsphysicskindle only, tablet'study guide'. no opinion yet
ref04AP Physics C Examphysicspaper,officehuge. good, HARD sample tests.
ref05feynmanphysicspaper,ofc;kindlesvolume 1, online copies, all 3 volumes
ref06feynmanphysicspaper,ofc;kindlesvolume 2
ref07feynmanphysicspaper,ofc;kindlesvolume 3
ref08Weyl,"Space Time Matter"physicspaper,officeDover, 4th edition, 1922; covers tensors.
ref09College Physics
744 Solved Problems
physicsKindle, on phone
and tablet
"Schaum's Outlines", 11th ed.
The ultimate review(?).
ref10 PhysicsByExample.pdf
Physics By Example
physics$pub/doc, and
Angel Flight binder
whew! hard. good.
Only printed a few questions from each chapter
ref11 "Physics" physics plasticized summary, Fry's
ref12 "Physics Equations"physics plasticized summary, Fry's
ref13 Calculus Rev Shts1 math plasticized review shts
ref14 Calculus Rev Shts2 math
ref15 QED paperbackphysicsoffice bookshelf local notes
ref16 Magnetics prjtphysicsp/magnetics/ link to define_ prjt file
ref17 "Electricity and M" physicsKindle unused (?) 171014
ref18 College Chemistry chemistryKindle, tablet Schaum's outline.
ref19 "Magnetic Field"physicsgood on tablet wikip,Magnetic_field
ref20glossary of Physics physicsonlinehuge, new to me 170914
ref21 A student's guide to
Maxwell's Equations
physics Pinot C:pub/doc/
office, binder
250 pages. great reviews. free
book-read-learn project
ref22 Build It Yourself
Science Laboratory
chemistry office bookshelf.
yellow paperback
Really good! hands on
ref23Illus. Guide to Home
Chemistry Experiments
chemistry office bookshelf "All Lab, No Lecture"
ref24 The Story of Spinatomicspaper, ofc difficult reading but well written
ref25 (California) Algebra 2mathofc bookshelf wonderful book. 2008
ref26 Euclid's ElementsmathKindle pre-fractions, pre-algebra!
see his proof of distributive law, ch 2!
ref27 kjmagnetics.comphysicsonline v good. taught me H and M mag fields
Has a calculator
ref28 Real World Physicsphysics online awesome collection of readable articles,
ref29 "Why E = M C²" physics kindle a great book. history, how's and why's
ref30 Higgs...Symmetry.. physics kindle. which device? need to reread it...
ref31 Art of Electronics    duh office book. + lab book. OMG. but dated...1989, 2nd Ed

see Feynman, volume 2, 26-7 (on tensors)
kindle book on Higgs...Symmetry.
Mr Hamilton, wikip. also his last book is on Pinot
George Green's paper is available, file ($doc) or print in office.
Taken from prjt 'magnetics':

Symbol list

These symbol values were found at html codes
'---' means not (yet) assigned (but of possible interest)
use symbol textnotes
multiplication* *multiplication in all computer languages.
10exp ¤ curren'10 to the power', 'raquo' was considered
r accel α alpha rotational acceleration
relativity β beta portion of the speed of light, v/C
cross product χ chi alternative? 'times'='×'
dot product º ordm a º b ; not used as of 180310
curl © copy curl operator
degrees ° deg degree sign, temperature
delta Δ Delta delta, math, differential
- η eta
spin rate ω omega
Ohms Ω Omega
relativity γ gamma relativistic compression
wavelength λ lambda
frequency ν nu
mag flux φ phi ?? both cases represent
mag flux Φ Phi the magnetic flux ??
angle θ theta
u0 µ micro magnetic constant.
units bull
for units multip, esp in denom.
options 'sdot' mult dot or 'middot'
---' apos single '. to get around html string convention
---δ delta could use instead of 'd', derivatives
---÷ divide
---½ frac12 ½
---¼ frac14 1/4
---¾ frac34 3/4
---· middle dot
---π pi
---± plusmn plus-minus
---» raquo ? could be used for '10 to the power __'
--- sdot
---² sup2
---³ sup3
--- uarr ? replace '10 to the..'