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Balancing Chemical Equations

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The Periodic Table

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Archimedes Principle

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Bernoulli Principle

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Blackbody (Cavity) Radiation and Planck's Hypothesis

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Center of Mass Frame

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Comparison Between Gravitation and Electrostatics

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Coriolis Effect

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Cyclotron Resonance

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Accelerated Reference Frames - Rindler Coordinates

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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics

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Mathematics

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Amplitude, Period and Phase

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Binomial Theorem (Pascal's Triangle)

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Complex Numbers

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Index Notation (Tensors and Matrices)

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Introduction to Conformal Field Theory

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Polynomial Division

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Regular Polygons

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Sets, Groups, Modules, Rings and Vector Spaces

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Similar Matrices and Diagonalization

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Spherical Trigonometry

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Stirling's Approximation

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Sum and Differences of Squares and Cubes

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Symmetric Groups

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Tangent and Normal Line

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Taylor and Maclaurin Series

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The Essential Mathematics of Lie Groups

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The Integers Modulo n Under + and x

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The Limit Definition of the Exponential Function

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Tic-Tac-Toe Factoring

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Trapezoidal Rule

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Vector Calculus

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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops

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Field Dimensions

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Helicity and Chirality

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Regularization and Renormalization

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Scattering - Mandelstam Variables

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Spin 1 Eigenvectors

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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots

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Categorical Data - Crosstabs

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Chebyshev's Theorem

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Chi Squared Goodness of Fit

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Confidence Intervals

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Factor Analysis

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Hypothesis Testing

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Linear Regression

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Monte Carlo Methods

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Non Parametric Tests

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Pearson Correlation

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Permutations and Combinations

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Pooled Variance and Standard Error

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Probability Rules

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Sample Size Determination

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Sampling Distributions

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Programming and Computer Science

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How this site works ...

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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol

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Quantum Field Theory

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Creation and Annihilation Operators

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Field Operators for Bosons and Fermions

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Lagrangians in Quantum Field Theory

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Path Integral Formulation

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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships

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Clebsch-Gordan Coefficients

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Electron Orbital Angular Momentum and Spin

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Entangled States

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Heisenberg Uncertainty Principle

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Ladder Operators

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Multi Electron Wavefunctions

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Pauli Exclusion Principle

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Pauli Spin Matrices

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Photoelectric Effect

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Position and Momentum States

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Probability Current

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Schrodinger Equation for Hydrogen Atom

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Schrodinger Wave Equation

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Schrodinger Wave Equation (continued)

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Spin 1/2 Eigenvectors

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The Differential Operator

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The Essential Mathematics of Quantum Mechanics

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The Observer Effect

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The Schrodinger, Heisenberg and Dirac Pictures

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The WKB Approximation

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Time Dependent Perturbation Theory

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Time Evolution and Symmetry Operations

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Time Independent Perturbation Theory

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Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids

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Fermi-Dirac Statistics

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Intrinsic and Extrinsic Semiconductors

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The P-N Junction

Special Relativity

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Electromagnetic 4 - Potential

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Energy and Momentum, E = mc²

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Lorentz Invariance

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Lorentz Transform

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Lorentz Transformation of the EM Field

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Newton versus Einstein

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Spinors - Part 1

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Spinors - Part 2

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The Lorentz Group

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Velocity Addition

Statistical Mechanics

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Black Body Radiation

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Entropy and the Partition Function

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The Harmonic Oscillator

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String Theory

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Bosonic Strings

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Extra Dimensions

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Introduction to String Theory

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Kaluza-Klein Compactification of Closed Strings

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Strings in Curved Spacetime

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Toroidal Compactification

Superconductivity

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Introduction to Superconductors

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Superconductivity (Lectures 1 - 10)

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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields

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Generators of a Supergroup

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Grassmann Numbers

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Introduction to Supersymmetry

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The Gauge Hierarchy Problem

test

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The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)

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Gauge Theories (Yang-Mills)

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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Last modified: September 19, 2019



Bernoulli Principle
-------------------
The lowering of fluid pressure in regions where the flow velocity is increased.

                                      Δl₂
                                      ||
                                 --------
                               /         A₂, ρ₂, h₂, P₂, v₂
                      Δl₁     /  -------     
                      ||     /  /        
                     --------  /         
                     ----------          
    A₁, ρ₁, h₁, P₁, v₁
    
Continuity Equation: 

A₁Δl₁/t = A₂Δl₂/t
=>  A₁v₁ = A₂v₂ = volumetric flow (m³/s)

Bernoulli Equation:

U = Fd
  = PAd since P = F/A
  = PV
=> P = U/V ... energy per unit volume

          KE per unit volume
         /
P₁ + (1/2)ρv₁² + ρgh₁ = P₂ + (1/2)ρv₂² + ρgh₂

P + (1/2)ρv² + ρgh = constant

It is a statement of the conservation of energy in a form useful for solving problems 
involving fluids. For a non-viscous, incompressible fluid in steady flow, the sum of 
pressure, potential and kinetic energies per unit volume is constant at any point.  


Venturi:

       
^      ||           ||   
| -----  ---        ||     ^
h₁            \     ||     |   h₂
|              -----  ---- |
v       . P₁        .P₂    v
              ----------
             /
  ----------

P₁ = h₁ρg        P₂ = h₂ρg

H = h₁ - h₂ = (1/2)ρ(v₂² - v₁²)

Geyser:
              . .  v₂ = 0
            .      .   
           .   h?     
           .        .  
           .           
     -----  -------   
          ||
          || d
          ||
      ----  ---- v₁ ~ 0, P₁ = dρ_soilg
     |          |
      ----------

 dρ_soilg - P_atmos = (1/2)ρ_water(0 - 0) + (h + d)ρ_waterg

 dρ_soilg - P_atmos = (h + d)ρ_waterg
 
 dρ_soilg - dρ_waterg - P_atmos = hρ_waterg

 {dg(ρ_soil - ρ_water) - P_atmos}/ρ_waterg = h

Torricelli's Law
----------------

Water leaking from a tank.

   P_a
   |
   v
|    |
|----|P₁, v₁, h₁
|    |
|    |
|    |..> P₂, v₂, h₂
|    |  
 ----

P₂ - P₁ = (1/2)ρ(v₂² - v₁²) + ρgh where h = h₁ - h₂

But P₁ = P₂ = P_a

Therefore, 0 = (1/2)ρ(v₂² + ρgh if v₁ ~ 0

=> v₂ =√2gh