[hamdani yusuf (OP)]

In the case where 1 Newton force is applied at 1 meter away from the axis of rotation, and the bolt radius is 10 mm, the thread needs to produce 100 Newton force to cancel out the torque and prevent the bolt from turning. If the friction force by the thread is less than 100 Newton, the bolt will be accelerated rotationally.

So a lubricated bolt can't be tightened to a set torque, everything in my car maintenance manual is wrong, and the wheels will fall off unless I use rusty bolts. I learn something new every day.

No. You just don't understand them correctly. That's why you sound confused.
Let's say the bolt thread got worn out. It can rotate indefinitely. The kinetic friction force it produces is 100 N. To produce a constant torque higher than what caused by the friction you need to keep accelerating the rotation of the bolt.

[hamdani yusuf (OP)]

On the contrary you need to learn some BASIC mechanics.

The most basic mechanics as described by Newton's laws don't involve friction nor elasticity of interacting objects. If you get your basic wrong, it's not surprising that you get confused with slightly more complicated cases.

[paul cotter]

Ok, lets clear this up. What do you think is wrong with my post #795?

[hamdani yusuf (OP)]

Torque can be viewed as tangential force.

but not by anyone who can read the definition of torque.

Which one of these definitions?
a twisting force that tends to cause rotation. (oxford dictionary)

In physics and mechanics, torque is the rotational analogue of linear force. (wikipedia)

Alan said that torque is defined as perpendicular force times radius of rotation.
This is NOT a definition. it's just one of many equations relating torque to other physical quantities.
Some other equations related to torque are:
τ = I.α   = rotational inertia times angular acceleration
τ = dL/dt  = rate of change of angular momentum
τ = W/θ  = work divided by angular displacement
τ = P/ω  = power divided by angular velocity
They are all equally valid relationships. The fact that current standard units of rotational quantities are not always consistent with their involved equations tells us that at least one mistake must have been made in setting the standard.

Perpendicular force times radius of rotation is not the definition of torque. It's just an equation to calculate torque when the perpendicular force and radius of rotation is known. In other cases where they are unknown, we need to use other equations like mentioned above.

For example, a centrifugal pump running at 1500 RPM with 1000 Watt of power. The angular velocity and power consumption can be measured using a tachometer and a power meter. Motor efficiency can be calculated after comparing the power consumption when it's running decoupled and coupled to the pump.
The force by impeller exerted to the liquid is distributed across its surface. The complexity of the impeller shape and pump casing as well as the complexity of fluid dynamics make it hard to calculate the force at every point on the surface of the impeller.
The equation τ = F x R is practically useless here. The equation τ = P/ω  is more applicable in this case.

[hamdani yusuf (OP)]

Ok, lets clear this up. What do you think is wrong with my post #795?

Here's the post.

There is no cancellation of torque when an object fails to turn, there is just insufficient torque to produce rotation. Similarly when one pushes against a wall, the wall does not move but the force is still there.

When the total torque is not zero, the object rotation must be accelerating.
Static friction cancels out acting force up to a limit.
The movement of the wall might be imperceptible to the naked eyes. But it can be measured using appropriate equipment.

[hamdani yusuf (OP)]

He is likely now to deflect the argument by asking "why a computer cannot help".

Reasoning AI models that I asked, Gemini Flash 2.0, ChatGPT-4-turbo, And Grok 3, all of them can identify problems in currently existing standard units of rotational quantities.
So, I asked again to choose from the options provided in the poll of this thread. Grok chooses the option #3.

Yes. They have some known problems, and there are some possible solutions.

Standard units for rotational quantities, like radians for angles and radians per second for angular velocity, work well but have issues. Radians are dimensionless, which can obscure their physical meaning in some contexts, and their use can complicate unit analysis in certain engineering applications. Possible solutions include introducing a distinct angular unit (e.g., a "turn" for 2π radians) or adopting supplementary units for clarity in specific fields, though no single fix addresses all concerns universally.

[hamdani yusuf (OP)]

ChatGPT also chooses option#3.

The best choice is:

3. Yes. They have some known problems, and there are some possible solutions.

Rotational units?like radians being dimensionless, and torque sharing units with energy?have some conceptual and practical issues, especially in teaching and engineering clarity. While these problems aren't catastrophic, they can be improved by redefining or extending the unit system (e.g., treating radians as base units, introducing a distinct unit for torque, etc.). However, no single solution completely solves every issue without trade-offs.

[hamdani yusuf (OP)]

Gemini also chooses option#3.

Yes. They have some known problems, and there are some possible solutions.

It doesn't provide it's consideration, so I had to ask.
What is your consideration?

My consideration is based on the points we discussed earlier about the standard units of rotational quantities. To summarize:
 * Known Problems: As we talked about, the current SI units have some known issues, particularly the dimensional ambiguity of the radian and the resulting overlap in units for physically distinct quantities like torque and energy. Some also argue about inconsistencies with fundamental principles like general covariance.
 * Possible Solutions: We also explored that there are indeed potential solutions proposed, such as the "Nonstandard International units" (NI units). This system suggests assigning a unit to the radian and adjusting related units to achieve better dimensional consistency and differentiation between physical concepts.
Therefore, the statement "Yes. They have some known problems, and there are some possible solutions" accurately reflects the current situation. While the SI units work for most practical purposes, there's room for improvement, and ideas for such improvements exist. The other options are less accurate:
 * Saying they are "already perfect" ignores the acknowledged issues.
 * Claiming "no possible solution" is incorrect, as alternative systems have been proposed.
 * Suggesting "one solution can solve them all" might be an oversimplification, as different proposed solutions might address various issues with varying degrees of success, and the "best" solution might be a matter of ongoing debate and refinement.

[paul cotter]

In connection with #804, I agree there would be imperceptible movement of the wall but that was not the point- the point was that the force is not cancelled. The rest of the post is abject nonsense. If I fail to turn a rusty bolt, a scenario that has often bedevilled me, I can read off the torque that failed to turn the bolt, ie there is NO cancellation of torque and there is no movement.

[hamdani yusuf (OP)]

In connection with #804, I agree there would be imperceptible movement of the wall but that was not the point- the point was that the force is not cancelled. The rest of the post is abject nonsense. If I fail to turn a rusty bolt, a scenario that has often bedevilled me, I can read off the torque that failed to turn the bolt, ie there is NO cancellation of torque and there is no movement.

Imagine someone is stepping on a weight scale.

Her weight exerts a force down on the scale. The scale reacts by exerting a normal force up with equal magnitude. The total force is zero, thus the woman doesn't accelerate anywhere.

In the case of torque wrench that you described, the measuring part (a needle connected to a spring) is located between the acting force (your hand) and reacting force (bolt thread). The deformation of the spring represent the torque applied, which is equal and opposite to the reactionary torque when the acceleration is zero (after a brief transient period).
In the case where there is an obstructing object right below the middle of the wrench, the opposing force is produced between the acting force and the measuring part. Here, the measurement can show much less value, which is affected by the flexibility of the obstructing object as well as the wrench itself.

[hamdani yusuf (OP)]

I also asked DeepSeek R1.
Here's its thought before giving the answer.

Thinking...
>
> Okay, let's dive into this question about rotational quantities and their units. The user is asking if the standard units can be improved, and they've given four options. I need to figure out which one is correct. Hmm.
>
> First, I should recall what the standard rotational quantities are and their units. The main ones are angle, angular velocity, angular acceleration, torque, moment of inertia, etc. The SI units for these are radians (rad) for angle, rad/s for angular velocity, rad/s? for angular acceleration, newton-meters (N?m) for torque, and kg?m? for moment of inertia. Wait, radians are a bit special because they're dimensionless. That might be a point of contention.
>
> I've heard before that some people have issues with radians being dimensionless. For example, when you convert between linear and rotational quantities, like tangential velocity v = ωr. Since ω is in rad/s and r is in meters, the radians kind of disappear, leaving m/s. But radians aren't considered a unit in the same way as meters or seconds. That could lead to confusion, especially in teaching or when dealing with units in equations.
>
> Another thing is torque and energy both having the unit N?m, even though they're different physical quantities. Torque is a vector (cross product of r and F), while energy is a scalar (dot product). But their units are the same, which might cause some confusion. Although, in practice, they are distinguished by context, so maybe that's not a huge problem. But it's a known issue.
>
> Then there's the moment of inertia, which is kg?m?. That's a derived unit, but does that pose any problem? Maybe not inherently, but when teaching rotational dynamics, students might confuse it with other quantities. Not sure if that's a unit problem or just a conceptual one.
>
> Are there proposed solutions to these issues? For the torque vs. energy unit conflict, some have suggested using different names, like the "joule" for energy and keeping N?m for torque, but that's already the case. Wait, actually, joule is specifically for energy, so torque is N?m but not called joules. So maybe that's not as big a problem.
>
> Regarding radians being dimensionless, one idea is to treat radians as a dimensional unit, but that would complicate unit analysis because many equations rely on the angle being dimensionless. For example, in the formula for arc length s = θr, θ has to be in radians, but if radians were a dimension, then the unit for s would be rad?m, which isn't the case. So making radians dimensional might cause more problems.
>
> Another possible issue is the use of revolutions per minute (RPM) in engineering contexts instead of rad/s. While RPM isn't an SI unit, it's widely used, which might lead to unit conversion errors. But that's more of a practicality issue rather than a problem with the units themselves.
>
> I remember reading that the current SI system has some ambiguities, like the unit for angular acceleration being rad/s?, but since rad is dimensionless, it's effectively just 1/s?. However, including radians explicitly helps in clarifying that it's angular. So maybe that's a minor issue but not critical.
>
> So, compiling the known problems: radians being dimensionless leading to potential confusion, torque sharing units with energy, and perhaps the use of non-SI units like RPM. Are there possible solutions? Some have proposed introducing a new unit for angle that isn't dimensionless, but as I thought earlier, that would break existing equations. Alternatively, maybe a different symbol or notation to distinguish torque from energy, but that's more of a nomenclature change rather than a unit change.
>
> Since the problems exist but there's no consensus on a solution that doesn't introduce other issues, maybe the answer is option 3: Yes. They have some known problems, and there are some possible solutions. But it's not like there's one solution that fixes everything (option 4).
>
> Alternatively, if the problems are considered minor and there's no feasible solution, then option 2. But in reality, there are discussions and proposals, even if not universally accepted. So option 3 seems right. The key is whether there are known problems and possible solutions. The existence of possible solutions, even if not perfect or widely adopted, would make option 3 correct.

And here's its official answer.

The standard units for rotational quantities, while functional, have recognized issues. For instance, the radian's dimensionless nature can lead to confusion in unit analysis, and torque sharing the unit N?m with energy might cause ambiguity. Although proposed solutions exist?such as treating radians as dimensional or introducing distinct notations for torque?they often introduce new complexities or lack consensus. Thus, the answer is:

*3. Yes. They have some known problems, and there are some possible solutions.*

This option acknowledges existing issues and the presence of potential fixes without overstating their efficacy or universality.

[hamdani yusuf (OP)]

Gemini 2.5 Pro answers.

The most appropriate answer is: *3. Yes. They have some known problems, and there are some possible solutions.*

It gave long explanation with a lot of references.

*Conclusion:*

The standard units for rotational quantities, particularly the radian, are functional but have recognized conceptual and practical issues. There isn't a consensus that they are "perfect" (Option 1) or that there are "no possible solutions" (Option 2). Multiple solutions have been proposed (Option 3), addressing different aspects of the problems. It's unlikely that one single solution will solve all problems without any trade-offs or changes to existing practices, making Option 4 less likely than Option 3. The ongoing discussions and proposals by bodies like the Consultative Committee for Units (CCU) indicate that the topic is still under active consideration [[1]](https://en.wikipedia.org/wiki/Radian)[[3]](https://indico.global/event/7204/contributions/66892/attachments/32331/59823/Res01background_SI_units.pdf).

The pdf link contains this points.

? Definitions concerning the radian, as a unit of angle, and other
units in the International System of Units (SI) are governed by the
Consultative Committee for Units (CCU), a committee that advises
the International Committee on Weights and Measures (CIPM) of
the International Bureau of Weights and Measures (BIPM).
? Angle units have been problematic since the establishment of the
International System of Units (SI).
o Initially the radian and steradian were classified as
"supplementary units".
o In 1995 the radian was reclassified as a derived unit.
o At the CCU meeting a month ago, it was mentioned that the
earlier terminology might be reestablished.
? The debates about the proper role of the radian have been going
on for many years.
o A committee to study the issue was established many years
ago and subsequently dissolved, because agreement could
not be reached.
o At the CCU meeting a month ago, a subcommittee was
created to again examine the issue.
o IUPAP has a representative on the committee: Eric Shirley,
who is an Associate Commission C2 member and is in line to
be the next US member of C2.
o The GA resolution could have some influence on the
committee.
? One of the problems addressed by the resolution is that both
radians per second and Hz (cycles per second) are called s^-1
(inverse second).  This leads to errors of 2 pi in publications.  The
resolution recommends always specifying which is meant.

[hamdani yusuf (OP)]

Perplexity Sonar answers.

The question of whether the standard units of rotational quantities can be improved involves understanding the nature of these units and their limitations. The standard units, such as radians for angles and angular velocity measured in radians per second, are well-established and effectively serve their purpose in describing rotational motion. However, there might be contexts or applications where alternative representations could offer benefits, such as enhanced clarity or computational efficiency.

Here are some considerations:

1. *Radians as a Unit*: Radians are defined as the ratio of arc length to radius, which provides a natural and intuitive measure for angular displacement. This unit system is consistent with the SI system and aligns well with mathematical formulations, making it robust for most applications[5].

2. *Potential Improvements*: While the current system is robust, there could be scenarios where alternative representations might be beneficial. For instance, in certain engineering or computational contexts, using different units or scales might simplify calculations or improve understanding. However, these would likely be specialized and not universally applicable improvements.

Given these considerations:

- *Option 1* ("No. They are already perfect. Any change will only make them worse.") might be overly optimistic, as it dismisses any potential for improvement.
- *Option 2* ("No. They have some known problems, but there is no possible solution.") is also limiting, as it implies that no solutions exist, which may not be true.
- *Option 3* ("Yes. They have some known problems, and there are some possible solutions.") seems most reasonable, acknowledging that while the current system is effective, there might be specialized contexts where improvements could be beneficial.
- *Option 4* ("Yes. They have some known problems, and one solution can solve them all.") is likely too broad, as it suggests a single solution can address all potential issues, which is unlikely.

Therefore, the most balanced approach is to acknowledge that the current system is effective but could potentially be refined in specific contexts.

*Conclusion*: Option 3 is the most appropriate choice. It acknowledges the robustness of the current system while also allowing for potential improvements in specialized situations.

References
[1] https://openstax.org/books/college-physics-2e/pages/10-2-kinematics-of-rotational-motion
[2] https://www.physicsclassroom.com/calcpad/Rotational-Kinematics
[3] https://phys.libretexts.org/Courses/Prince_Georges_Community_College/PHY_1030:_General_Physics_I/09:_Rotational_Kinematics_Angular_Momentum_and_Energy/9.8:_Problem_Solving
[4]

[5] https://pressbooks.bccampus.ca/douglasphys1107/chapter/9-1-rotation-angle-and-angular-velocity/

[paul cotter]

More gibberish. Just because there is no movement does not mean a force is cancelled. Put a fly under one of the girl's feet and it would be crushed, indicating the force remains unchanged.

[hamdani yusuf (OP)]

Claude 3.5 Haiku answers.

Option 3 is the most accurate response: Yes. The standard units of rotational quantities have some known problems, and there are some possible solutions.

Rotational quantities like angular velocity, angular acceleration, and torque currently use units that have some limitations and potential areas for improvement. Some known issues include:

1. Complexity in dimensional analysis
2. Lack of intuitive representation
3. Potential inconsistencies across different scientific domains
4. Challenges in interdisciplinary communication

Potential solutions might involve:
- Standardizing notation
- Developing more coherent unit systems
- Creating more intuitive scaling
- Improving cross-disciplinary compatibility

However, there's no single universal solution that would resolve all issues simultaneously. Different scientific fields might require slightly different approaches. The key is incremental improvements and collaborative development across disciplines like physics, engineering, and mathematics.

The answer suggests that while current rotational units are functional, there's room for thoughtful refinement and optimization.

[hamdani yusuf (OP)]

ChatGPT o4-mini answers.

I?d go with option **3**:

3. Yes. They have some known problems, and there are some possible solutions.

I needed to ask why.

I picked option 3 because:

1. **Known issues with ?dimensionless? angles** 
   ? In the SI, both the radian and steradian are ?suppressed? to the dimension 1.  That makes angles formally dimensionless, which 
     - hides the fact that torque (N?m) and energy (J = N?m) end up with identical units, even though they?re physically very different quantities, 
     - allows you to write angular velocity ω in s⁻? instead of clearly distinguishing it from ordinary frequency f in Hz (also s⁻?). 

2. **Practical confusions** 
   ? Engineers and scientists routinely mix up degrees, radians, revolutions, cycles, and turns. 
   ? Equations get peppered with factors of 2π or π/180 that wander in and out of unit conversions, making derivations error-prone. 

3. **There are several promising fixes** 
   ? **Introduce an ?angle? base dimension** (e.g. [A]): radians become a true unit A ?rad, steradians A? ?sr, so torque vs energy can be told apart. 
   ? **Treat angular frequency and cyclic frequency consistently**, e.g. always use Hz for cycles/s and rad?s⁻? only when you really need radians. 
   ? **Adopt a ?turn? as a unit** of full revolution (1 turn = 2π rad), so you can write frequencies as turns/s instead of burying the 2π in formulas. 

4. **No single panacea** 
   ? Each proposal has trade-offs: introducing a new base dimension breaks backward compatibility with trillions of textbook equations, 
   ? whereas simply being more disciplined about Hz vs rad s⁻? relies on user rigor, not on the formal SI. 

Because the problems are real but multiple partial remedies exist?none utterly perfect or universally adopted?I landed on ?Yes, known problems exist, and some possible solutions are on the table.?

[hamdani yusuf (OP)]

Llama 4 Maverick answers.

## Step 1: Understand the context of the problem.
The question is asking whether the standard units used to measure rotational quantities can be improved.

## Step 2: Identify the known issues with standard units of rotational quantities.
One of the known problems is the inconsistent treatment of rotational quantities in different systems of units, particularly the distinction between angular quantities and their linear counterparts.

## Step 3: Consider possible improvements or solutions.
Some propose that using radians consistently and appropriately could simplify and improve the measurement and calculation of rotational quantities. Others suggest that distinguishing between different types of rotational quantities (like angular displacement, angular velocity, and angular acceleration) and their units could be beneficial.

## Step 4: Evaluate the given options based on the understanding of the issues and potential solutions.
1. Suggests that the current units are perfect and cannot be improved, which is unlikely given the historical development and ongoing discussions about units in physics.
2. Implies that while there are problems, no solution exists, which contradicts the efforts to improve or simplify units in various areas of physics.
3. Indicates an acknowledgment of problems with the current units and suggests that there are potential solutions, aligning with the idea that there are ongoing discussions and proposals for improvement.
4. Suggests a single, comprehensive solution to all known problems, which might be overly optimistic given the complexity of physical quantities and units.

## Step 5: Analyze the feasibility of each option.
Options 1 and 2 are pessimistic, with the first denying any need for improvement and the second acknowledging problems but denying the possibility of solutions. Option 4 is overly optimistic, suggesting a single solution can address all issues. Option 3 strikes a balance, acknowledging the existence of problems and the potential for solutions.

## Step 6: Choose the most reasonable answer based on the analysis.
Given that there are known issues with the units (such as the treatment of radians and the distinction between different rotational quantities), and there are discussions and proposals for improving these units, the most reasonable answer is that the standard units have known problems and there are possible solutions.

The best answer is 3.

[hamdani yusuf (OP)]

Qwen 3 235B answers.

Thinking...
>
> Okay, so I need to figure out whether the standard units for rotational quantities can be improved. Let's start by recalling what I know about rotational units. There's stuff like angular velocity, torque, angular momentum, radians, etc.
>
> First, I remember that radians are dimensionless units because they're defined as the ratio of arc length to radius. But sometimes in physics, we treat angles as having units even though they're technically dimensionless. That seems a bit inconsistent. Like, if you have a full circle, it's 2π radians, but radians don't have dimensions like meters or seconds. So maybe that's one issue with rotational units?they're treated differently in terms of dimensions compared to other quantities.
>
> Then there's torque, which is measured in newton-meters (N?m), but energy is also in joules, which is equivalent to N?m. However, torque isn't energy, so even though the units are the same, they're different quantities. That might be another problem because the same unit applies to different physical concepts, which could be confusing. Some people suggest using radians in torque units to make it distinct, like N?m/radian or something, but I'm not sure if that's standard.
>
> Angular velocity is in radians per second, but since radians are dimensionless, it's effectively just 1/seconds. But angular acceleration would then be in radians per second squared. However, linear acceleration is meters per second squared. So there's a difference in how rotational and linear quantities are expressed, which might lead to confusion or inconsistency in formulas.
>
> Another point is angular displacement. If you rotate something by 2π radians, it's a full circle, but angular displacement can also be a vector quantity in some contexts (though rotation isn't commutative, so it's not a proper vector). The units here are still radians, but maybe there's a better way to handle that.
>
> I've heard some people argue that using radians consistently in units could help, like explicitly including radians in derived units for rotational quantities. For example, torque could be expressed as N?m/rad to distinguish it from energy, which is N?m. That might make equations more dimensionally consistent when dealing with rotational vs. translational motion.
>
> Also, when converting between rotational and linear motion, you have to multiply by the radius. So angular velocity (rad/s) times radius (meters) gives linear velocity (m/s). But if radians are dimensionless, then technically, you're multiplying 1/s by meters to get m/s. The radian is hidden there, which might be a point of confusion for some. If radians were a base unit, then that conversion would make more sense dimensionally.
>
> There's also the problem with angular acceleration. If you have angular acceleration in rad/s?, when you relate it to linear acceleration (m/s?), you multiply by radius. Again, the radian is there but not explicitly part of the unit, which might make the dimensional analysis less straightforward.
>
> Some proposals I've read about suggest introducing radians as a base unit or modifying the SI system to include angular measures more formally. This could help in distinguishing rotational quantities and making equations more transparent in terms of their dimensional analysis. However, changing established units is a big deal because it affects all of physics and engineering, so there's resistance to such changes unless the benefits are substantial.
>
> Looking at the options given in the problem, the user is asking which conclusion is best. The options are:
>
> 1. Current units are perfect.
> 2. Problems exist but no solution.
> 3. Problems exist and there are possible solutions.
> 4. There's a single comprehensive solution.
>
> From my earlier thoughts, there are definitely issues with how rotational units are handled, especially regarding radians being dimensionless and the overlap in units for different quantities like torque and energy. There are proposals to address these, such as using radians more consistently in units, which would be a possible solution. However, there's no single solution that addresses all problems comprehensively, and the current units aren't perfect. So options 1 and 4 are out. Between 2 and 3, since there are proposed solutions (like changing SI units to include radians as a base unit), option 3 seems correct. The analysis in the original answer supports this, pointing out that option 3 acknowledges both the problems and the potential solutions, which aligns with ongoing discussions in the physics community.
>


The standard units used to measure rotational quantities, such as radians, torque (N?m), and angular velocity (rad/s), face several well-documented challenges. These include the **dimensionless nature of radians**, which can obscure the physical meaning of angular quantities in dimensional analysis, and the **ambiguity in unit usage**?for example, torque and energy both share the unit N?m despite representing fundamentally different physical concepts. Additionally, the **conversion between rotational and linear quantities** (e.g., angular velocity to linear velocity) relies on the radius, but the implicit role of radians in these conversions can lead to confusion. These inconsistencies highlight the need for a more coherent framework for rotational units.

### Key Issues:
1. **Radians as Dimensionless Units**: Radians are defined as the ratio of arc length to radius (dimensionless), yet they are essential for describing angles and rotational motion. This creates ambiguity in dimensional analysis, as angular quantities like angular velocity (rad/s) and angular acceleration (rad/s?) are effectively expressed in terms of inverse seconds (1/s) and inverse seconds squared (1/s?), respectively. This obscures their rotational nature.
   
2. **Unit Overlap**: Torque (N?m) and energy (Joules, equivalent to N?m) share the same unit, even though they represent distinct physical concepts. This overlap can lead to errors in interpreting physical relationships and calculations.

3. **Inconsistent Scaling**: Converting rotational quantities to linear equivalents (e.g., angular velocity ? radius = linear velocity) relies on the radius, but the implicit role of radians in these conversions is not explicitly reflected in the units. This can make dimensional consistency less transparent.

### Proposed Solutions:
- **Explicit Use of Radians in Units**: Incorporating radians into derived units (e.g., torque as N?m/rad) could distinguish rotational quantities from their linear counterparts and clarify their physical meaning. This would align with the principle that units should reflect the nature of the quantity they describe.
 
- **Formalizing Radians as a Base Unit**: Some propose elevating radians to a base unit in the SI system, akin to meters or seconds. This would allow for clearer dimensional analysis and better distinguish rotational quantities from linear ones.

- **Improved Unit Conventions**: Standardizing conventions to explicitly include radians in rotational units (e.g., angular acceleration in rad/s? rather than 1/s?) could reduce ambiguity and improve pedagogical clarity.

### Evaluation of Options:
1. **Option 1 ("Current units are perfect")**: This is demonstrably incorrect, as the issues outlined above are widely recognized in physics and engineering communities. The dimensionless nature of radians and unit overlap are active topics of discussion.

2. **Option 2 ("Problems exist but no solution exists")**: This is overly pessimistic. Numerous proposals (e.g., explicit radians in units, formalizing radians as a base unit) exist to address these challenges, indicating that solutions are both possible and actively pursued.

3. **Option 3 ("Problems exist and there are possible solutions")**: This aligns with the evidence. The ongoing discussions and proposed modifications to unit conventions reflect a recognition of the issues and a commitment to addressing them. For example, the use of radians in torque units (N?m/rad) is already practiced in some contexts to avoid confusion with energy.

4. **Option 4 ("Single comprehensive solution exists")**: This is overly optimistic. While specific solutions (e.g., explicit radians) address particular issues, there is no universally accepted, comprehensive overhaul of rotational units that resolves all challenges. The complexity of physical quantities and their interrelationships makes a single solution unlikely.

### Conclusion:
The most reasonable conclusion is **Option 3**: *The standard units have known problems and there are possible solutions*. This acknowledges the recognized issues (e.g., dimensionless radians, unit overlap) while recognizing the ongoing efforts to refine and improve the measurement of rotational quantities. Proposals such as explicit radians in units and formalizing radians as a base unit represent viable steps toward a more coherent system, even if a single, universal solution remains elusive.

**Final Answer:** 3

[hamdani yusuf (OP)]

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

Thinking...
>
> Okay, so the user is asking about the root cause of the problem with the standard units for rotational quantities. Let me start by recalling the previous discussion where we identified several issues like radians being dimensionless, torque and energy sharing the same unit, and inconsistencies in converting rotational to linear quantities.
>
> Hmm, the root cause... maybe it's the fact that radians are treated as dimensionless. Because if radians were a base unit, then angular quantities would have their own dimensions, making things like torque and energy distinct. But since radians are a ratio of lengths (arc length over radius), they don't have dimensions. That leads to angular quantities like angular velocity (rad/s) effectively being 1/s, which can confuse their rotational nature with linear quantities.
>
> Wait, but why are radians defined as dimensionless in the first place? I think it's because they are derived from the ratio of two lengths (arc length and radius), so mathematically they cancel out. But in practice, angles are a fundamental aspect of rotational motion, so treating them as dimensionless might not capture their physical significance. This creates ambiguity in units for torque, angular momentum, etc., because the angular component isn't explicitly represented in the units.
>
> Another angle (pun intended) is the historical development of units. The SI system was built around translational quantities first, and rotational concepts were added later. Since rotational units were derived from translational ones, they inherited the dimensionless nature of radians. This might have led to the current inconsistencies, like torque sharing units with energy. If rotational units had been given their own base units from the start, maybe this wouldn't be an issue.
>
> Also, when you convert between rotational and linear motion, you use the radius. For example, angular velocity (rad/s) multiplied by radius (m) gives linear velocity (m/s). But since radians are dimensionless, the units just become m/s, hiding the rotational aspect. If radians were a base unit, this conversion would explicitly include radians, making the relationship clearer.
>
> So the root cause seems to be the dimensionless nature of radians in the SI system, which stems from their definition as a ratio of lengths. This leads to inconsistencies in how rotational quantities are represented and conflates different physical concepts (like torque and energy) that share the same unit. Additionally, the historical prioritization of translational over rotational units in the SI system's development has perpetuated these issues.
>
> I should also check if there's a deeper reason beyond the definition of radians. For instance, in dimensional analysis, angles are treated as dimensionless because they are derived from lengths, but physically, angles are a measure of rotation. This disconnect between mathematical definition and physical meaning might be the core problem. If the SI system had recognized angles as a fundamental dimension, it could have avoided these issues.
>
> Therefore, the root cause is the SI system's treatment of radians as dimensionless, which leads to ambiguities and inconsistencies in rotational units. This foundational choice affects how rotational quantities are expressed, conflates distinct physical concepts, and complicates dimensional analysis for rotational motion.
>


The root cause of the issues with standard units for rotational quantities lies in the **SI system's treatment of angles (and thus radians) as dimensionless quantities**. This foundational choice creates inconsistencies and ambiguities in representing rotational motion. Here's a breakdown of the problem:

### 1. **Radians as Dimensionless Units**:
   - Radians are defined as the ratio of arc length to radius (unitless: $ \text{rad} = \frac{\text{m}}{\text{m}} = 1 $). This mathematical definition strips radians of dimensional significance, even though angles physically represent a distinct aspect of rotational motion.
   - **Consequence**: Angular quantities like angular velocity ($ \text{rad/s} $) and angular acceleration ($ \text{rad/s}^2 $) lose their rotational identity in dimensional analysis, effectively reducing to $ 1/\text{s} $ and $ 1/\text{s}^2 $, which are indistinguishable from linear quantities.

### 2. **Ambiguity in Derived Units**:
   - Torque ($ \text{N?m} $) and energy ($ \text{J} = \text{N?m} $) share the same unit despite representing fundamentally different concepts (rotational force vs. energy). Radians are often omitted in torque units, obscuring its rotational nature.
   - **Consequence**: Confusion arises in interpreting physical relationships, as units fail to distinguish between rotational and translational effects.

### 3. **Inconsistent Scaling in Rotational-Linear Conversions**:
   - Converting rotational to linear quantities (e.g., angular velocity $ \omega $ to linear velocity $ v = \omega r $) relies on the radius $ r $, but the implicit role of radians in these conversions is not explicitly reflected in units.
   - **Consequence**: Dimensional analysis becomes less transparent, as the rotational component (radians) is hidden, leading to potential errors in calculations.

### 4. **Historical Development of the SI System**:
   - The SI system was historically built around translational quantities (length, mass, time), with rotational concepts added later. Rotational units were derived from translational ones, perpetuating the dimensionless treatment of angles.
   - **Consequence**: Rotational quantities lack their own base units, forcing them to inherit dimensional inconsistencies from their translational counterparts.

### Why This Matters:
The dimensionless nature of radians creates a disconnect between mathematical definitions (ratios of lengths) and physical meaning (angles as a measure of rotation). If radians were formalized as a base unit (e.g., analogous to meters or seconds), rotational quantities would have distinct dimensional identities, resolving ambiguities in units like torque, angular momentum, and energy.

### Key Insight:
The root cause is **not merely technical but philosophical**: the SI system's choice to treat angles as dimensionless overlooks their physical uniqueness in rotational dynamics. This foundational decision cascades into inconsistencies in unit definitions and dimensional analysis, fueling ongoing debates about revising the SI system to better accommodate rotational quantities.

### Final Answer:
The root cause is the **SI system's treatment of radians as dimensionless**, stemming from their definition as a ratio of lengths. This leads to ambiguities in rotational units, conflates distinct physical concepts (e.g., torque vs. energy), and complicates dimensional analysis, reflecting a deeper disconnect between mathematical definitions and the physical reality of rotational motion.

[alancalverd]

It seems that the function of chatbots is to invent problems where none exist. What a waste of life.