[alancalverd]

Sadly, he has quoted others (including real humans, some of them apparently physics educationalists) in this particular thread, strongly suggesting that he is not alone in his confusion. I think that anyone reading it might end up wiser, or at least calmer, rather than misled and hysterical.

[paul cotter]

Unfortunately there is no shortage of crackpots and wingnuts in academia. I have named some of them previously.

[hamdani yusuf (OP)]

The equation τ = P/ω  is more applicable in this case.

Because it is explicitly derived from τ= F.r.

How do you derive it?

[hamdani yusuf (OP)]

A passive measurement system like Cavendish's torsion balance, the needle doesn't have to move back to where it was. What you said is not necessarily true. Some of your previous posts clearly show your confusions.

" A passive measurement system like Cavendish's torsion balance"
We are not talking about that.
You can tell from the video.

"the needle doesn't have to move back to where it was."
Nobody said it did; in fact I said it might not.
" What you said is not necessarily true."

What part of what I said  (And which I said was necessarily true) is not necessarily true?

It seems like you have a problem with your memory.

And the measurement of torque only happens after the system has moved the needle back to where it was (or, if you want, slightly higher).
Until then there is no measurement.

[alancalverd]

How do you derive it?

By definition  P =F.ds/dt (for any system) = F.r.ω (for a rotating system) = τ.ω

Provided, of course, that you haven't stupidly redefined τ.

[hamdani yusuf (OP)]

How do you derive it?

By definition  P =F.ds/dt (for any system) = F.r.ω (for a rotating system) = τ.ω

Provided, of course, that you haven't stupidly redefined τ.

You can't measure the force on each point of impeller surface.

[hamdani yusuf (OP)]

BC, he will not answer your question. He tries to divert the discourse when questioned after saying something he cannot back up. He has avoided my question, re the "cancellation" of force.

If you are allergic to the word cancellation, and prefer counterbalance instead, be my guess. My point is, if the torque is not completely counterbalanced, there must be an angular acceleration.

[hamdani yusuf (OP)]

BC, he will not answer your question. He tries to divert the discourse when questioned after saying something he cannot back up. He has avoided my question, re the "cancellation" of force.

I know. And that's why I can't understand why the mods don't ban him.

Perhaps because they already understand my point. You are wondering why because you haven't yet.

[hamdani yusuf (OP)]

I don't see him as quite harmless, he is the source of so much error that a third party could easily be misled. He does not debate, he just states his erroneous views and then doubles down on them, refusing to answer pertinent questions or tries to divert the discussion. Infuriating. I suggested previously that all his material be put in one thread and designated "an essay in confusion, too long to read".

You should be able to understand my points by reading my comparison tables I've marked as the best answer in this thread. So far no one has pointed out any objection on them. You can try to be the first.

[alancalverd]

You can't measure the force on each point of impeller surface.

In fact you can (it's standard practice when designing propellors, wings or turbines) but it's not worth doing once you have designed, built and installed your impeller because you can simply measure the torque applied by the driving axle. Or, of course, the power input to the motor.

[alancalverd]

my comparison tables

All you have done there is compare apples with oranges, complain that they aren't the same, invent a new fruit with a shiny skin and segmented pulp, and insist that it is a better orange. You then go on to ignore the fact that your new fruit is a useless absurdity. 

[hamdani yusuf (OP)]

You can't measure the force on each point of impeller surface.

In fact you can (it's standard practice when designing propellors, wings or turbines) but it's not worth doing once you have designed, built and installed your impeller because you can simply measure the torque applied by the driving axle. Or, of course, the power input to the motor.

Generally, you derive a complex concept from one or more simpler concepts. Those simpler concepts must be easier to define or easier to measure.

[hamdani yusuf (OP)]

my comparison tables

All you have done there is compare apples with oranges, complain that they aren't the same, invent a new fruit with a shiny skin and segmented pulp, and insist that it is a better orange. You then go on to ignore the fact that your new fruit is a useless absurdity. 

Which table are you referring to? Can you specify which table, row, and column that contains the most fatal error? What should it contain instead?

[hamdani yusuf (OP)]

I also asked Qwen a follow up question about the root cause of the problem.
...

Let's see what other thinking AI models have to say about it. Starting with ChatGPT o4 mini

The single deepest source of all the downstream headaches is the SI?s choice to treat angles (and solid angles) as ?dimensionless.?  In practice:

1.  A plane angle θ is defined as the ratio of arc-length to radius 
       θ = s / r 
    both in metres , θ carries no net dimension, so ?radian? is logged away as just ?1.?

2.  Likewise a solid angle Ω = area / r^2 is dimensionless (steradian = 1).

By suppressing the dimension of angle, the SI:

- makes torque (N?m) and energy (J = N?m) formally identical in units,
- collapses ω [rad/s] and f [cycles/s, i.e. Hz] both into 1/s,
- forces us to juggle 2π and π/180 ?by hand? whenever we switch between revolutions, degrees, turns and radians,
- breaks the power of dimensional analysis to catch mix-ups involving rotational quantities.

In short, the root cause is the **decision** to define radian and steradian as pure numbers rather than as genuine units carrying their own dimension.

Furthermore, by defining the ratio as dimensionless, we don't know if it should be multiplied or divided instead to calculate without additional information.

[hamdani yusuf (OP)]

Gemini 2.5 pro answers.

*Thinking...*

The root cause of the problems with standard units of rotational quantities, particularly the radian, stems from its definition within the International System of Units (SI) as a **dimensionless derived unit**. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[2]](https://www.quora.com/Why-is-a-radian-dimensionless)

Here's a breakdown of why this is the core issue:

1.  **Definition as a Ratio of Lengths:** The radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[2]](https://www.quora.com/Why-is-a-radian-dimensionless) Mathematically, θ (in radians) = arc length (s) / radius (r). Since both arc length and radius are measured in units of length (e.g., meters), these units cancel out (m/m = 1). [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[2]](https://www.quora.com/Why-is-a-radian-dimensionless) This makes the radian, by its very definition in the SI system, a dimensionless quantity ? a pure number. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[2]](https://www.quora.com/Why-is-a-radian-dimensionless)

2.  **"Dimensionless" Implies No Fundamental Unit:** Because it's dimensionless, the radian is not considered a base unit in the SI system, unlike meters for length, kilograms for mass, or seconds for time. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[3]](https://www.quora.com/Why-are-the-radian-and-steradian-not-considered-SI-base-units) It's treated as a derived unit equal to 1. [[4]](https://en.wikipedia.org/wiki/Radian) This means that, from a dimensional analysis perspective, angles don't have their own fundamental "angle" dimension. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)

3.  **Consequences of Dimensionlessness:**
    *   **"Disappearing" Units:** In many physics equations involving rotational quantities, the "radian" unit seems to appear or disappear, which can be confusing and lead to errors if not handled carefully. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[5]](https://www.reddit.com/r/PhysicsStudents/comments/a8r44d/why_do_radians_just_disappear_during_solving_for/) For example, in `tangential velocity (v) = angular velocity (ω) * radius (r)`, if `ω` is in rad/s and `r` is in meters, `v` comes out in m/s. The "rad" effectively vanishes. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)
    *   **Ambiguity with Other Quantities:**
        *   **Torque vs. Energy:** Torque (force x distance, N?m) and energy (e.g., work done, Joules, which is also N?m) end up having the same fundamental SI dimensions (M L? T⁻?). While the SI recommends using N?m for torque and J for energy to distinguish them, the underlying dimensional similarity can be problematic. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless) Some argue torque should be J/rad to explicitly include the angular nature. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)
        *   **Frequency vs. Angular Frequency:** Frequency (Hertz, Hz, which is s⁻?) and angular frequency (radians per second, rad/s) both reduce to the dimension of s⁻? in the SI system. This can lead to confusion and potential errors involving factors of 2π. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)
    *   **Mathematical Functions:** Trigonometric functions (sin, cos, tan) and series expansions (like Taylor series for these functions) are mathematically cleaner when angles are expressed in radians, precisely because the radian is treated as a pure number. [[6]](https://math.stackexchange.com/questions/2605865/why-are-radians-dimensionless)[[7]](https://www.themathdoctors.org/radians-why-and-when-they-are-better/) If the radian had a dimension, these formulas would require a conversion factor to make the arguments of the functions dimensionless. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[6]](https://math.stackexchange.com/questions/2605865/why-are-radians-dimensionless)

4.  **Convention vs. Physical Reality:** While the dimensionless nature of the radian is a convention based on its definition, some argue that "angle" is a distinct physical quantity that should have its own dimension. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)[[6]](https://math.stackexchange.com/questions/2605865/why-are-radians-dimensionless) They believe that treating it as dimensionless is a source of inconsistency when applying units to real-world physical phenomena. [[1]](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)

In essence, the decision to define the radian as a ratio of two lengths, making it inherently dimensionless in the SI system, is the fundamental reason for the subsequent issues of ambiguity, disappearing units in equations, and the debate over whether angle should be considered a fundamental dimension. While this definition offers mathematical conveniences in some areas (like calculus), it creates conceptual and practical challenges in others, particularly in dimensional analysis and the clear representation of physical quantities involving rotation.

---
Learn more:
1. [unit of measure - Why radian is dimensionless? - Mathematics Stack Exchange](https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless)
2. [Why is a radian dimensionless? - Quora](https://www.quora.com/Why-is-a-radian-dimensionless)
3. [Why are the radian and steradian not considered SI base units? - Quora](https://www.quora.com/Why-are-the-radian-and-steradian-not-considered-SI-base-units)
4. [Radian - Wikipedia](https://en.wikipedia.org/wiki/Radian)
5. [Why do radians just "disappear" during solving for units in an equation? - Reddit](https://www.reddit.com/r/PhysicsStudents/comments/a8r44d/why_do_radians_just_disappear_during_solving_for/)
6. [Why are radians dimensionless? \[duplicate\] - Math Stack Exchange](https://math.stackexchange.com/questions/2605865/why-are-radians-dimensionless)
7. [Radians: Why, and When, They Are Better - The Math Doctors](https://www.themathdoctors.org/radians-why-and-when-they-are-better/)

[paul cotter]

Counterbalanced forces indicates equal and opposite forces while cancellation of force indicates it's removal- not the same thing at all. The girl on the scales maintains her downward force due to gravity, similarly a torque which fails to produce rotation is still a torque.

[Bored chemist]

BC, he will not answer your question. He tries to divert the discourse when questioned after saying something he cannot back up. He has avoided my question, re the "cancellation" of force.

I know. And that's why I can't understand why the mods don't ban him.

Perhaps because they already understand my point. You are wondering why because you haven't yet.

You.
Did.
Not.
Answer.

[hamdani yusuf (OP)]

Let's see one of the link mentioned by Gemini.

https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless

Angles are dimensionless quantities (e.g. m/m for rad and m?/m? for sr). They have no base units in SI, meaning angles have no fundamental existence (contrary to a length or a time), they are derived from something else. This actually creates problems and since a long time proposals have been made to give angle a true dimension. Such change has many implications on other quantities.

The status of angle units has never been clear for the BIPM, the bureau in charge of the SI system, but as making angle units true base units creates more problems than it solves, there is a status quo.

An angle is a ratio, but there are different ratios

If you look at the definition of (plane) angle measurement units, they are all the ratio of a length to a length, so have dimension of m/m = 1, i.e. they are dimensionless.

1 rad: The angle subtended by an arc of a circle that has the same length as the circle's radius. Ratio 1/1.

1 turn: The angle subtended by the circumference of a circle at its center. Ratio 2π
/1.

1?: 1 turn / 360. Ratio 2π
/360.

1 gon (grad, gradient): 1 turn / 400. Ratio 2π
/400.

The angle unit indicates which ratio is used

Note angle comes from latin angulus, apex/corner, and is the corner made by the intersection of two lines/planes. Angle in science is actually a shortcut for angle measure.

And as a matter of fact drawing an angle is easy, but measuring an angle requires some specific construction, e.g. a circle (the ratio of the arc to the radius is the measure), a square (the angle made by the diagonals is 1/4 of a turn), etc. All computations ultimately lead to the ratio of a length to a length. So an angle measure, regardless of the unit used, has no dimension.

However the unit indicates which ratio was used, an angle of n?
arc/radius is not an angle of n?
1/360. So we need to explain which reference was used, this is the definition of a unit.

The assumption is when we use radians, we can omit the unit, and the implicit radian unit in math is due to the simplification it allows, e.g. in Euler's formula linking angles, Euler number and complex numbers.

Still angle units, plane (rad = 1m/m) and solid (sr = 1m?/m?), are of a special kind. From a SI standpoint, they have been supplementary units, separate from base units and derived units. This special class of units was removed in 1995:

The Comit? International des Poids et Mesures, in 1980, having observed that the ambiguous status of the supplementary units compromises the internal coherence of the SI, has in its Recommendation 1 (CI-1980) interpreted the supplementary units, in the SI, as dimensionless derived units.

Dimensionless radian is also a problem

Angle units are now derived units. However this classification and the fact angles are considered dimensionless is strongly challenged as it creates inconsistencies when applied to the real world.

E.g. torque is a force resulting from rotation. It is currently measured in N m, which is a Joule and doesn't reflect an angular quantity. It would be more meaningful to use J rad−1
, but this requires radian to be a base unit with a dimension, this would be the 8th base unit of the SI.

[alancalverd]

One thing that has become clear is that chatbots do not understand the concept or use of dimensions, and in the case of Gemini, is even capable of generating lies: 

torque is a force resulting from rotation

indeed!

Nor, it seems, are chatbots or ignorant educationalists aware of the difference between a scalar (energy) and a vector (torque). I'll admit to possibly having added to the confusion by occasionally writing F . r when the conventional representation should be F x r. No excuse, but in mitigation the number we use is of course the magnitude of the product.

So my advice is to ignore anything but your own common sense, experience, understanding of basic physics and mathematics, and the standard nomenclature that everyone else uses.

[alancalverd]

Which table are you referring to?

All of them. Apples are not oranges.

Consider a simple case. Momentum is a vector in the direction of motion. Angular momentum is a vector, but nothing is moving in the direction of the vector.