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<p>Giacomo Prando says "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian
in the dimensional analysis of physical equations." For example, a mass hanging by a string from a pulley will rise or drop
by <spanclass="arithmatex">\(y=rθ\)</span> centimeters, where <spanclass="arithmatex">\(r\)</span> is the radius of the pulley in centimeters and <spanclass="arithmatex">\(θ\)</span> is the angle the pulley turns in radians.
When multiplying <spanclass="arithmatex">\(r\)</span> by <spanclass="arithmatex">\(θ\)</span> the unit of radians disappears from the result.
Similarly in the formula for the angular velocity of a rolling wheel, <spanclass="arithmatex">\(ω=v/r\)</span>, radians appear in the units of
<spanclass="arithmatex">\(ω\)</span> but not on the right hand side.
Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical
advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention
and contextual knowledge is "pedagogically unsatisfying".</p>
</div>
<p>At least a dozen scientists have made proposals to treat the radian as a base unit of measure defining its own dimension of "angle",
as early as 1936 and as recently as 2022. This would bring the advantages of a physics-based, consistent, and logically-robust
unit system, with unambiguous units for all physical quantities. At the same time the only notable changes for typical
end-users would be: improved units for the quantities <em>torque</em>, <em>angular momentum</em>, and <em>moment of inertia</em>.</p>
<p>Paul Quincey in his proposal <ahref="../../../appendix/references/#Quincey"><em>"Angles in the SI: a detailed proposal for solving the problem"</em></a> states:</p>
<divclass="admonition quote">
<pclass="admonition-title">Paul Quincey: Angles in the SI: a detailed proposal for solving the problem</p>
<p>The familiar units assigned to some angular quantities are based on equations that have adopted the radian convention,
and so are missing <code>rad</code>s that would be present if the complete equation is used. The physically-correct units are
those with the <code>rad</code>s reinstated. Numerical values would not change, and the physical meanings of all quantities would
also be unaffected.</p>
</div>
<p>He proposes the following changes:</p>
<ul>
<li>The <strong>radian</strong> would become either a new base unit or a 'complementary' unit</li>
<li>The <strong>steradian</strong> would become a derived unit equal to <spanclass="arithmatex">\(1\:rad^2\)</span></li>
<li>
<p>The SI units for</p>
<ul>
<li><em>Torque</em> would change from <spanclass="arithmatex">\(N\:m\)</span> (<spanclass="arithmatex">\(=J\)</span>) to <spanclass="arithmatex">\(J\:rad^{-1}\)</span></li>
<li><em>Angular momentum</em> would change from <spanclass="arithmatex">\(J\:s\)</span> to <spanclass="arithmatex">\(J\:s\:rad^{-1}\)</span> (i.e. <spanclass="arithmatex">\(J/(rad/s)\)</span>)</li>
<li><em>Moment of inertia</em> would change from <spanclass="arithmatex">\(kg\:m^2\)</span> to <spanclass="arithmatex">\(kg\:m^2\:rad^{-2}\)</span> (i.e. <spanclass="arithmatex">\(J/(rad/s)^2\)</span>)</li>
</ul>
</li>
<li>
<p>The option to omit the radian from the SI units for <em>angle</em>, <em>angular velocity</em>, <em>angular frequency</em>,
<em>angular acceleration</em>, and <em>angular wavenumber</em> would be removed, the only correct SI units being
<spanclass="arithmatex">\(rad\)</span>, <spanclass="arithmatex">\(rad/s\)</span>, <spanclass="arithmatex">\(rad/s\)</span>, <spanclass="arithmatex">\(rad/s^2\)</span> and <spanclass="arithmatex">\(rad/m\)</span> respectively.</p>
</li>
</ul>
<p>Paul Quincey summarizes that with the above in action:</p>
<divclass="admonition quote">
<pclass="admonition-title">Paul Quincey: Angles in the SI: a detailed proposal for solving the problem</p>
<p>However, the physical clarity this would build into the SI should be recognised very quickly. The units would tell us that
<spanclass="arithmatex">\(torque \times angle = energy\)</span>, and <spanclass="arithmatex">\(angular\:momentum \times angle = action\)</span>, for example, in the same way that they do for
Dimensional analysis could be used to its full extent. Software involving angular quantities would be rationalised.
Arguments about the correct units for frequency and angular frequency, and the meaning of the unit <spanclass="arithmatex">\(Hz\)</span>, could be left behind.
The explanation of these changes would be considerably easier and more rewarding than explaining how a kilogram-sized mass
can be measured in terms of the Planck constant.</p>
</div>
<h2id="angular-quantities-in-the-si">Angular quantities in the SI<aclass="headerlink"href="#angular-quantities-in-the-si"title="Permanent link">¶</a></h2>
<p>Even though the SI somehow ignores the dimensionality of angle:</p>
<divclass="admonition quote">
<pclass="admonition-title">SI Brochure</p>
<p>Plane and solid angles, when expressed in radians and steradians respectively, are in effect
also treated within the SI as quantities with the unit one. The symbols <spanclass="arithmatex">\(rad\)</span> and <spanclass="arithmatex">\(sr\)</span>
are written explicitly where appropriate, in order to emphasize that, for radians or
steradians, the quantity being considered is, or involves the plane angle or solid angle
respectively. For steradians it emphasizes the distinction between units of flux and intensity
in radiometry and photometry for example. However, it is a long-established practice in
mathematics and across all areas of science to make use of <spanclass="arithmatex">\(rad = 1\)</span> and <spanclass="arithmatex">\(sr = 1\)</span>.
For historical reasons the radian and steradian are treated as derived units.</p>
</div>
<p>It also explicitly states:</p>
<divclass="admonition quote">
<pclass="admonition-title">SI Brochure</p>
<p>The SI unit of frequency is hertz, the SI unit of angular velocity and angular frequency is
radian per second, and the SI unit of activity is becquerel, implying counts per second.
Although it is formally correct to write all three of these units as the reciprocal second, the
use of the different names emphasizes the different nature of the quantities concerned. It is
especially important to carefully distinguish frequencies from angular frequencies, because
by definition their numerical values differ by a factor of <spanclass="arithmatex">\(2\pi\)</span>. Ignoring this fact may cause
an error of <spanclass="arithmatex">\(2\pi\)</span>. Note that in some countries, frequency values are conventionally expressed
using “cycle/s” or “cps” instead of the SI unit <spanclass="arithmatex">\(Hz\)</span>, although “cycle” and “cps” are not units
in the SI. Note also that it is common, although not recommended, to use the term
frequency for quantities expressed in <spanclass="arithmatex">\(rad/s\)</span>. Because of this, it is recommended that
quantities called “frequency”, “angular frequency”, and “angular velocity” always be given
explicit units of <spanclass="arithmatex">\(Hz\)</span> or <spanclass="arithmatex">\(rad/s\)</span> and not <spanclass="arithmatex">\(s^{-1}\)</span>.</p>
</div>
<h2id="strong-angular-extensions-in-the-library">Strong Angular extensions in the library<aclass="headerlink"href="#strong-angular-extensions-in-the-library"title="Permanent link">¶</a></h2>
<p>The mp-units library strives to define physically-correct quantities and their units to provide maximum help
to its users. As treating angle as a dimensional quantity can lead to many "trivial" mistakes in dimensional
analysis and calculation, it was decided to provide additional experimental systems of quantities and units
that follow the above approach and treat <em>angle</em> as a base quantity with a base unit of radian and
<em>solid angle</em> as its derived quantity.</p>
<p>As those (at least for now) are not a part of SI, the <em>plain angle</em> and <em>solid angle</em> definitions can be found
in a dedicated <code>angular</code> system. Those definitions are also used in the <code>isq_angle</code> system of quantities
to make the recipes for angle-based quantities like <em>torque</em> or <em>angular velocity</em> physically correct:</p>
</span><spanid="__span-0-11"><aid="__codelineno-0-11"name="__codelineno-0-11"href="#__codelineno-0-11"></a><spanclass="n">std</span><spanclass="o">::</span><spanclass="n">cout</span><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="s">"Applying a perpendicular force of "</span><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="n">force</span><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="s">" to a "</span><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="n">lever</span><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="s">" long lever results in "</span>
</span><spanid="__span-0-12"><aid="__codelineno-0-12"name="__codelineno-0-12"href="#__codelineno-0-12"></a><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="n">torque</span><spanclass="p">.</span><spanclass="n">in</span><spanclass="p">(</span><spanclass="n">N</span><spanclass="w"></span><spanclass="o">*</span><spanclass="w"></span><spanclass="n">m</span><spanclass="w"></span><spanclass="o">/</span><spanclass="w"></span><spanclass="n">rad</span><spanclass="p">)</span><spanclass="w"></span><spanclass="o"><<</span><spanclass="w"></span><spanclass="s">" of torque.</span><spanclass="se">\n</span><spanclass="s">"</span><spanclass="p">;</span>
</span></code></pre></div>
<p>The above program prints:</p>
<divclass="language-text highlight"><pre><span></span><code><spanid="__span-1-1"><aid="__codelineno-1-1"name="__codelineno-1-1"href="#__codelineno-1-1"></a>Applying a perpendicular force of 500 N to a 20 cm long lever results in 100 N m/rad of torque.
</span></code></pre></div>
<divclass="admonition note">
<pclass="admonition-title">Note</p>
<p><code>cotes_angle</code> is a constant which represents an angle with the value of exactly <code>1 radian</code>.
You can find more information about this constant in <ahref="../../../appendix/references/#Quincey">Quincey</a>.</p>
</div>
<divclass="admonition example">
<pclass="admonition-title"><ahref="https://godbolt.org/z/qfxosv9YM">Try it on Compiler Explorer</a></p>
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