section_path: Outside mathematics/Describing physical phenomena
section_name: Describing physical phenomena
paragraph_text: Describing physical phenomena
  
  Although not a physical constant, π appears routinely in equations describing fundamental principles of the universe, often because of π 's relationship to the circle and to spherical coordinate systems. A simple formula from the field of classical mechanics gives the approximate period T of a simple pendulum of length L, swinging with a small amplitude (g is the earth's gravitational acceleration): T ≈ 2 π L g. {\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}.}
  
  One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δ x) and momentum (Δ p) cannot both be arbitrarily small at the same time (where h is the Planck constant): Δ x Δ p ≥ h 4 π. {\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}.}
  
  The fact that π is approximately equal to 3 plays a role in the relatively long lifetime of orthopositronium. The inverse lifetime to lowest order in the fine-structure constant α is 1 τ = 2 π 2 − 9 9 π m e α 6, {\displaystyle {\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m_{\text{e}}\alpha ^{6},} where m is the mass of the electron.
  
  π is present in some structural engineering formulae, such as the buckling formula derived by Euler, which gives the maximum axial load F that a long, slender column of length L, modulus of elasticity E, and area moment of inertia I can carry without buckling: F = π 2 E I L 2. {\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}.}
  
  The field of fluid dynamics contains π in Stokes' law, which approximates the frictional force F exerted on small, spherical objects of radius R, moving with velocity v in a fluid with dynamic viscosity η: F = 6 π η R v. {\displaystyle F=6\pi \eta Rv.}
  
  In electromagnetics, the vacuum permeability constant μ appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation. Before 20 May 2019, it was defined as exactly μ 0 = 4 π × 10 − 7 H/m ≈ 1.2566370614 … × 10 − 6 N/A 2. {\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}\approx 1.2566370614\ldots \times 10^{-6}{\text{ N/A}}^{2}.}