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👉Jul 31, 2025 — The value of pi is approximately 3.14, or 22/7. Pi is an irrational number, which means it is not equal to the ratio of any two whole numbers.

Pi | Definition, Symbol, Number, & Facts

Britannica

https://www.britannica.com › Science › Mathematics

Jul 31, 2025 — The value of pi is approximately 3.14, or 22/7. Pi is an irrational number, which means it is not equal to the ratio of any two whole numbers.

modern physics

lorentz factor
γ =  1 √(1 − v2/c2)

time dilation
t =  t0 √(1 − v2/c2)
t = γt0

length contraction
ℓ = ℓ0√(1 − v2/c2)
ℓ =  ℓ0 γ

relative velocity
u′ =  u + v 1 + uv/c2

relativistic energy
E =  mc2 √(1 − v2/c2)
E = γmc2

relativistic momentum
p =  mv √(1 − v2/c2)
p = γmv

energy-momentum
E2 = p2c2 + m2c4

mass-energy
E = mc2

relativistic k.e.
K =  ⎛ ⎜ ⎝ 1  − 1 ⎞ ⎟ ⎠ mc2 √(1 − v2/c2)
K = (γ − 1)mc2

relativistic doppler eff.
λ  =  f0  = √ ⎛ ⎜ ⎝ 1 + v/c ⎞ ⎟ ⎠ λ0 f 1 − v/c

photon energy
E = hf
E = pc

photon momentum
p =  h λ
p =  E c

photoelectric effect
Kmax = E − φ
Kmax = h(f − f0)

schroedinger's	equation
iℏ  ∂  Ψ(r,t) = −  ℏ2  ∇2Ψ(r,t) + U(r)Ψ(r,t) ∂t 2m
Eψ(r) = −  ℏ2  ∇2ψ(r) + U(r)ψ(r) 2m

uncertainty principle
∆px∆x ≥  ℏ  2
∆E∆t ≥  ℏ  2

rydberg equation
1  = −R∞  ⎛ ⎜ ⎝ 1  −  1 ⎞ ⎟ ⎠ λ n2 n02

activity
A =  n t

half life
N = N02−t/T½

absorbed dose
D =  E m

equivalent dose
H = wRD

effective dose
E = wTH

mechanics, thermal physics, waves & optics, electricity & magentism, modern physics

• discussion
• summary
• practice
• problems

Frequently Used Equations

• … symbols
• equations
• constants …

The Physics Hypertextbook
©1998–2025 Glenn Elert
Author, Illustrator, Webmaster

No condition is permanent.

1. mechanics
   1. kinematics
      1. motion
      2. distance and displacement
      3. speed and velocity
      4. acceleration
      5. equations of motion
      6. free fall
      7. graphs of motion
      8. kinematics and calculus
      9. kinematics in two dimensions
      10. projectiles
      11. parametric equations
   2. dynamics i: force
      1. forces
      2. force and mass
      3. action-reaction
      4. weight
      5. dynamics
      6. statics
      7. friction
      8. forces in two dimensions
      9. centripetal force
      10. frames of reference
   3. energy
      1. work
      2. energy
      3. kinetic energy
      4. potential energy
      5. conservation of energy
      6. power
      7. simple machines
   4. dynamics ii: momentum
      1. impulse and momentum
      2. conservation of momentum
      3. momentum and energy
      4. momentum in two dimensions
   5. rotational motion
      1. rotational kinematics
      2. rotational inertia
      3. rotational dynamics
      4. rotational statics
      5. angular momentum
      6. rotational energy
      7. rolling
      8. rotation in two dimensions
      9. coriolis force
   6. planetary motion
      1. geocentrism
      2. heliocentrism
      3. universal gravitation
      4. orbital mechanics i
      5. gravitational potential energy
      6. orbital mechanics ii
      7. gravity of extended bodies
   7. periodic motion
      1. springs
      2. simple harmonic oscillator
      3. pendulums
      4. resonance
      5. elasticity
   8. fluids
   1. density
   2. pressure
   3. buoyancy
   4. fluid flow
   5. viscosity
   6. aerodynamic drag
   7. flow regimes
   9. 
      
2. thermal physics
   1. heat & temperature
      1. temperature
      2. thermal expansion
      3. the atomic nature of matter
      4. gas laws
      5. kinetic-molecular theory
      6. phases
   2. calorimetry
      1. sensible heat
      2. latent heat
      3. chemical potential energy
   3. heat transfer
      1. conduction
      2. convection
      3. radiation
   4. thermodynamics
      1. heat and work
      2. pressure-volume diagrams
      3. engines
      4. refrigerators
      5. energy and entropy
      6. absolute zero
3. waves & optics
   1. wave phenomena
      1. the nature of waves
      2. periodic waves
      3. interference and superposition
      4. interfaces and barriers
   2. sound
      1. the nature of sound
      2. intensity
      3. doppler effect (sound)
      4. shock waves
      5. diffraction and interference (sound)
      6. standing waves
      7. beats
      8. music and noise
   3. physical optics
      1. the nature of light
      2. polarization
      3. doppler effect (light)
      4. cerenkov radiation
      5. diffraction and interference (light)
      6. thin film interference
      7. color
   4. geometric optics
      1. reflection
      2. refraction
      3. spherical mirrors
      4. spherical lenses
      5. aberration
4. electricity & magnetism
   1. electrostatics
      1. electric charge
      2. coulomb's law
      3. electric field
      4. electric potential
      5. gauss's law
      6. conductors
   2. electrostatic applications
      1. capacitors
      2. dielectrics
      3. batteries
   3. electric current
      1. electric current
      2. electric resistance
      3. electric power
   4. dc circuits
      1. resistors in circuits
      2. batteries in circuits
      3. capacitors in circuits
      4. kirchhoff's rules
   5. magnetostatics
      1. magnetism
      2. electromagnetism
      3. ampère's law
      4. electromagnetic force
   6. magnetodynamics
      1. electromagnetic induction
      2. faraday's law
      3. lenz' law
      4. inductance
   7. ac circuits
      1. rc circuits
      2. rl circuits
      3. lc circuits
      4. alternating current
      5. impedance
   8. electromagnetic waves
      1. maxwell's equations
      2. electromagnetic waves
      3. electromagnetic spectrum
5. modern physics
   1. relativity
      1. space-time
      2. mass-energy
      3. general relativity
   2. quanta
      1. blackbody radiation
      2. photoelectric effect
      3. x-rays
      4. antimatter
   3. wave mechanics
      1. matter waves
      2. atomic models
      3. semiconductors
      4. condensed matter
   4. nuclear physics
      1. isotopes
      2. radioactive decay
      3. half life
      4. binding energy
      5. fission
      6. fusion
      7. nucleosynthesis
      8. nuclear weapons
      9. radiobiology
   5. particle physics
      1. the standard model
      2. quantum electrodynamics
      3. quantum chromodynamics
      4. quantum flavordynamics
      5. beyond the standard model
6. foundations
   1. units
      1. international system of units
      2. gaussian system of units
      3. british-american system of units
      4. miscellaneous units
      5. time
      6. unit conversion
   2. measurement
      1. significant digits
      2. order of magnitude
   3. graphs
      1. graphical representation of data
      2. linear regression
      3. curve fitting
      4. calculus
   4. vectors
      1. trigonometry
      2. vector addition and subtraction
      3. vector resolution and components
      4. vector multiplication
   5. reference
      1. special symbols
      2. frequently used equations
      3. physical constants
      4. astronomical data
      5. periodic table of the elements
      6. people in physics
7. back matter
   1. preface
      1. about this book
   2. contact the author
      1. glennelert.us
      2. behance
      3. bluesky
      4. google scholar
      5. instagram
      6. threads
      7. youtube
   3. affiliated websites
      1. hypertextbook.com
      2. midwoodscience.org
   4. stuff for sale
      1. spreadshirt
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