From 8cf2951c2d545f9957efae35658ff02dd4991561 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Carlos=20A=2E=20Michel=C3=A9n=20Str=C3=B6fer?= Date: Sun, 3 Dec 2023 21:01:34 -0700 Subject: [PATCH] improve documentation --- docs/src/proposed.md | 30 ++++++++++++++++++++++-------- 1 file changed, 22 insertions(+), 8 deletions(-) diff --git a/docs/src/proposed.md b/docs/src/proposed.md index 8fc248f..a3037c2 100644 --- a/docs/src/proposed.md +++ b/docs/src/proposed.md @@ -10,26 +10,40 @@ different, but most (i) include angle as a base dimension, (ii) use a "fundament constant, typically equal to one radian, to modify the equations of physics, and (iii) define solid angle as a derived dimension equal to angle squared. The different proposals, however, take different approaches to which equations or -quantity units to modify. +quantity units to modify. +One exception not considered here is a class of proposals that define a new dimension for *radius length* rather than for *angle*. Some examples of how common equations would be modified in such systems, using ``θ₀=1rad``: - - Trigonometric functions: + - Trigonometric & exponential functions: + ``cos(θ) → cos(θ/θ₀)`` + ``cos(ωt+φ) → cos([ωt+φ]/θ₀)`` + + ``Aℯⁱᶿ → Aℯ^[iθ/θ₀]`` - - Angular frequency and frequency: + - Angular frequency and angular wave number: + ``ω=2πf → ω=2πfθ₀`` - - Arc length ``s`` and other kinematic equations: + + ``k=2π/λ → k=2πθ₀/λ`` + + - Arc length ``s`` and sector area: + ``s=rθ → s=rθ/θ₀`` - - For dynamic equations there's more variability between the different proposals. - As an example, [Quincey_2021](@cite) would modify torque (and its unit) while leaving the units of work intact, as + + ``A=½r²θ → A=½r²θ/θ₀`` + + - kinematic equations: + + + ``v=rω → v=rω/θ₀`` + + ``a=rω² → a=rω²/θ₀²`` + + - For dynamic equations, there's more variability between the different proposals. + As an example, [Quincey_2021](@cite) would modify the units of torque and moment of inertia while leaving the units of work and energy intact. - + ``T=(𝐫×𝐅) → T=(𝐫×𝐅)/θ₀`` - + ``W=Tθ`` + + ``T=(𝐫×𝐅) → T=(𝐫×𝐅)/θ₀`` (torque, new units: ``J/s``) + + ``I=Σ(mᵢ⋅rᵢ)/θ₀²`` (moment of inertia, new units: ``kg⋅m²/rad²``) + + ``L=Iω=(𝐫x𝐩)/θ₀`` (angular momentum, new units: ``J/(rad/s)``) + + ``W=Tθ`` (work) + + ``E=½Iω²`` (kinetic energy) ## *DimensionfulAngles.jl*'s relation to these proposals