Editing Pi (section)

=== Describing physical phenomena ===
Although not a [[physical constant]], {{pi}} appears routinely in equations describing fundamental principles of the universe, often because of {{pi}}'s relationship to the circle and to [[spherical coordinate system]]s. A simple formula from the field of [[classical mechanics]] gives the approximate period {{math|''T''}} of a simple [[pendulum]] of length {{math|''L''}}, swinging with a small amplitude ({{math|''g''}} is the [[Gravity of Earth|earth's gravitational acceleration]]):<ref>{{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |last3=Walker |first3=Jearl |title=Fundamentals of Physics |edition=5th |publisher=John Wiley & Sons |year=1997 |page=381 |isbn=0-471-14854-7}}</ref>

<math display=block>T \approx 2\pi \sqrt\frac{L}{g}.</math>

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{{math|''x''}}) and [[momentum]] (Δ{{math|''p''}}) cannot both be arbitrarily small at the same time (where {{math|''h''}} is the [[Planck constant]]):<ref>{{cite book |title=College Physics 2e |contribution=29.7 Probability: The Heisenberg Uncertainty Principle |contribution-url=https://openstax.org/books/college-physics-2e/pages/29-7-probability-the-heisenberg-uncertainty-principle |publisher=[[OpenStax]] |first1=Paul Peter |last1=Urone |first2=Roger |last2=Hinrichs |year=2022}}</ref>

<math display=block> \Delta x\, \Delta p \ge \frac{h}{4\pi}.</math>

The fact that {{pi}} 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]] {{math|''α''}} is<ref>{{cite book |last1=Itzykson |first1=C. |author-link1=Claude Itzykson |last2=Zuber |first2=J.-B. |author-link2=Jean-Bernard Zuber |title=Quantum Field Theory |date=1980 |publisher=Dover Publications |location=Mineola, NY |isbn=978-0-486-44568-7 |edition=2005 |url=https://books.google.com/books?id=4MwsAwAAQBAJ |lccn=2005053026 |oclc=61200849}}</ref>

<math display=block>\frac{1}{\tau} = 2\frac{\pi^2 - 9}{9\pi}m_\text{e}\alpha^{6},</math>

where {{math|''m''<sub>e</sub>}} is the mass of the electron.

{{pi}} is present in some structural engineering formulae, such as the [[buckling]] formula derived by Euler, which gives the maximum axial load {{math|''F''}} that a long, slender column of length {{math|''L''}}, [[modulus of elasticity]] {{math|''E''}}, and [[area moment of inertia]] {{math|''I''}} can carry without buckling:<ref>{{cite book |last=Low |first=Peter |title=Classical Theory of Structures Based on the Differential Equation |publisher=Cambridge University Press |year=1971 |pages=116–118 |isbn=978-0-521-08089-7}}</ref>

<math display=block>F =\frac{\pi^2EI}{L^2}.</math>

The field of [[fluid dynamics]] contains {{pi}} in [[Stokes' law]], which approximates the [[drag force|frictional force]] {{math|''F''}} exerted on small, [[sphere|spherical]] objects of radius {{math|''R''}}, moving with velocity {{math|''v''}} in a [[fluid]] with [[dynamic viscosity]] {{math|''η''}}:<ref>{{cite book |last=Batchelor |first=G. K. |title=An Introduction to Fluid Dynamics |publisher=Cambridge University Press |year=1967 |page=233 |isbn=0-521-66396-2}}</ref>

<math display=block>F =6\pi\eta  Rv.</math>

In electromagnetics, the [[vacuum permeability]] constant ''μ''<sub>0</sub> appears in [[Maxwell's equations]], which describe the properties of [[Electric field|electric]] and [[Magnetic field|magnetic]] fields and [[electromagnetic radiation]]. Before 20 May 2019, it was defined as exactly

<math display=block>\mu_0 = 4 \pi \times 10^{-7}\text{ H/m} \approx 1.2566370614 \ldots \times 10 ^{-6} \text{ N/A}^2. </math>