11 ms·

I always frowned upon the unit "radian". Radian is defined in terms of the ratio of two quantities both having length dimensions. So by this definition angle sh

by leni536 11y ago

I always frowned upon the unit "radian". Radian is defined in terms of the ratio of two quantities both having length dimensions. So by this definition angle should be dimensionless. I really don't get radian.

JadeNB 11y ago
Angle, like any other quantity that is fed to a pure mathematical function (like the sine or cosine function), must be dimensionless [0]; and indeed it is. 'Degree' (as in 30 degrees) is not really a unit, but the number π/180, representing π units of arc along a circle of radius 180 units. 'Radian' is also not a unit, but rather the number 1, representing 1 unit of arc along a circle of radius 1 unit. EDIT: [0] For example, one way of defining sin(x) is as the sum of the power series `x - x^3/6 + x^5/120 - …`; this cannot be dimensionally consistent unless x is dimensionless. Another is to define y(x) = sin(x) to be the solution of `d^2y/dx^2 = -y` satisfying certain initial conditions; but the units of `y` can only be equal to those of its second derivative if the argument of `y` is dimensionless. EDIT 2: One easy way to fix the potential dimensional inconsistency is to replace `d^2y/dx^2 = -y` by `d^2y/dx^2 = -k^2 y`, where `k` is just a conversion factor for dimensional consistency (so that `[k] = [x]^{-1}`). Then a solution is `y = sin(k x)`; and, once again, the argument of the sine function is dimensionless.
cyphar 11y ago
I've always ribbed against my physics professors using the notation of rad/s. To be fair, they'd make it clear that the reason they used it was to remind the students that the quantities were angular (ie \omega not velocity).
- jason_s 11y ago
  If you work in the field of motor control, you need this. (Not only that, but you need to distinguish electrical rad/s and mechanical rad/s; motors with N pairs of magnetic poles undergo N electrical cycles for each mechanical cycle.)
jason_s 11y ago
Angle is technically dimensionless. Not sure how to explain the "zen" of radians; it's kind of the angular equivalent of "e" in natural logarithms. People who first learn logarithms say WTF, what is this "e" thing? why not just log10 everywhere? But when you get into calculus you find that "e" is the only log base that doesn't introduce extra baggage; if you take the derivative of a^x the result is ln a * a^x, and if you take the derivative of log_a x = ln x / ln a, you get 1/(x ln a). In both cases the "ln a" term is 1 only when a = e. For radians, the functions sin wt and cos wt (w = "omega") have derivatives of w * cos wt and -w * sin wt when wt is measured in radians. Otherwise there is an additional constant of proportionality that depends on the angular "unit". (the term "unit" in quotes because it's still dimensionless but just related to radians by a dimensionless scaling factor, like 180/pi for radians <-> degrees or 1/(2*pi) for radians <-> revolutions.

by leni536 11y ago

JadeNB 11y ago

Angle, like any other quantity that is fed to a pure mathematical function (like the sine or cosine function), must be dimensionless [0]; and indeed it is. 'Degree' (as in 30 degrees) is not really a unit, but the number π/180, representing π units of arc along a circle of radius 180 units. 'Radian' is also not a unit, but rather the number 1, representing 1 unit of arc along a circle of radius 1 unit. EDIT: [0] For example, one way of defining sin(x) is as the sum of the power series `x - x^3/6 + x^5/120 - …`; this cannot be dimensionally consistent unless x is dimensionless. Another is to define y(x) = sin(x) to be the solution of `d^2y/dx^2 = -y` satisfying certain initial conditions; but the units of `y` can only be equal to those of its second derivative if the argument of `y` is dimensionless. EDIT 2: One easy way to fix the potential dimensional inconsistency is to replace `d^2y/dx^2 = -y` by `d^2y/dx^2 = -k^2 y`, where `k` is just a conversion factor for dimensional consistency (so that `[k] = [x]^{-1}`). Then a solution is `y = sin(k x)`; and, once again, the argument of the sine function is dimensionless.

cyphar 11y ago

I've always ribbed against my physics professors using the notation of rad/s. To be fair, they'd make it clear that the reason they used it was to remind the students that the quantities were angular (ie \omega not velocity).

jason_s 11y ago
If you work in the field of motor control, you need this. (Not only that, but you need to distinguish electrical rad/s and mechanical rad/s; motors with N pairs of magnetic poles undergo N electrical cycles for each mechanical cycle.)

jason_s 11y ago

Angle is technically dimensionless. Not sure how to explain the "zen" of radians; it's kind of the angular equivalent of "e" in natural logarithms. People who first learn logarithms say WTF, what is this "e" thing? why not just log10 everywhere? But when you get into calculus you find that "e" is the only log base that doesn't introduce extra baggage; if you take the derivative of a^x the result is ln a * a^x, and if you take the derivative of log_a x = ln x / ln a, you get 1/(x ln a). In both cases the "ln a" term is 1 only when a = e. For radians, the functions sin wt and cos wt (w = "omega") have derivatives of w * cos wt and -w * sin wt when wt is measured in radians. Otherwise there is an additional constant of proportionality that depends on the angular "unit". (the term "unit" in quotes because it's still dimensionless but just related to radians by a dimensionless scaling factor, like 180/pi for radians <-> degrees or 1/(2*pi) for radians <-> revolutions.