# Strong Angular System¶

## Some background information¶

As per today's SI, both radian and steradian are dimensionless. This forces the convention to set the angle 1 radian equal to the number 1 within equations (similar to what natural units system does for c constant).

Following Wikipedia:

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 $$y=rθ$$ centimeters, where $$r$$ is the radius of the pulley in centimeters and $$θ$$ is the angle the pulley turns in radians. When multiplying $$r$$ by $$θ$$ the unit of radians disappears from the result. Similarly in the formula for the angular velocity of a rolling wheel, $$ω=v/r$$, radians appear in the units of $$ω$$ 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".

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 torque, angular momentum, and moment of inertia.

Paul Quincey in his proposal "Angles in the SI: a detailed proposal for solving the problem" states:

Paul Quincey: Angles in the SI: a detailed proposal for solving the problem

The familiar units assigned to some angular quantities are based on equations that have adopted the radian convention, and so are missing rads that would be present if the complete equation is used. The physically-correct units are those with the rads reinstated. Numerical values would not change, and the physical meanings of all quantities would also be unaffected.

He proposes the following changes:

• The radian would become either a new base unit or a 'complementary' unit
• The steradian would become a derived unit equal to $$1\:rad^2$$
• The SI units for

• Torque would change from $$N\:m$$ ($$=J$$) to $$J\:rad^{-1}$$
• Angular momentum would change from $$J\:s$$ to $$J\:s\:rad^{-1}$$ (i.e. $$J/(rad/s)$$)
• Moment of inertia would change from $$kg\:m^2$$ to $$kg\:m^2\:rad^{-2}$$ (i.e. $$J/(rad/s)^2$$)
• The option to omit the radian from the SI units for angle, angular velocity, angular frequency, angular acceleration, and angular wavenumber would be removed, the only correct SI units being $$rad$$, $$rad/s$$, $$rad/s$$, $$rad/s^2$$ and $$rad/m$$ respectively.

Paul Quincey summarizes that with the above in action:

Paul Quincey: Angles in the SI: a detailed proposal for solving the problem

However, the physical clarity this would build into the SI should be recognised very quickly. The units would tell us that $$torque \times angle = energy$$, and $$angular\:momentum \times angle = action$$, for example, in the same way that they do for $$force \times distance = energy$$, $$linear\:momentum \times distance = action$$, and $$radiant\:intensity \times solid\:angle = radiant\:flux$$. 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 $$Hz$$, 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.

## Angular quantities in the SI¶

Even though the SI somehow ignores the dimensionality of angle:

SI Brochure

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 $$rad$$ and $$sr$$ 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 $$rad = 1$$ and $$sr = 1$$. For historical reasons the radian and steradian are treated as derived units.

It also explicitly states:

SI Brochure

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 $$2\pi$$. Ignoring this fact may cause an error of $$2\pi$$. Note that in some countries, frequency values are conventionally expressed using “cycle/s” or “cps” instead of the SI unit $$Hz$$, 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 $$rad/s$$. Because of this, it is recommended that quantities called “frequency”, “angular frequency”, and “angular velocity” always be given explicit units of $$Hz$$ or $$rad/s$$ and not $$s^{-1}$$.

## Strong Angular extensions in the library¶

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 angle as a base quantity with a base unit of radian and solid angle as its derived quantity.

As those (at least for now) are not a part of SI, the plain angle and solid angle definitions can be found in a dedicated angular system. Those definitions are also used in the isq_angle system of quantities to make the recipes for angle-based quantities like torque or angular velocity physically correct:

using namespace mp_units;
using namespace mp_units::si::unit_symbols;
using mp_units::angular::unit_symbols::deg;

const quantity lever = isq_angle::position_vector(20 * cm);
const quantity force = isq_angle::force(500 * N);
const quantity angle = isq_angle::angular_measure(90. * deg);
const quantity torque = isq_angle::torque(lever * force * angular::sin(angle) / (1 * isq_angle::cotes_angle));

std::cout << "Applying a perpendicular force of " << force << " to a " << lever << " long lever results in "
<< torque.in(N * m / rad) << " of torque.\n";


The above program prints:

Applying a perpendicular force of 500 N to a 20 cm long lever results in 100 N m/rad of torque.


Note

cotes_angle is a constant which represents an angle with the value of exactly 1 radian. You can find more information about this constant in Quincey.