[alancalverd]

Let's say the object is a long thin stick 1 meter long, 1 kg mass, floating in free space. 1 N force is applied to one of its end, perpendicular to the length. What is the torque?

  0.5 Nm, assuming the stick is homogeneous.

[alancalverd]

What does Iα equal to, in your opinion?

Depends on a whole lot of information you haven't given me. Could be zero.

But if you use your definition of "torque", it could be infinity, which is nonsense.

[alancalverd]

Consider tightening a 12 mm bolt onto a flange - say the wheel bolt of a small car.

The initial torque required to spin the bolt into the thread is very small - you can do it with your fingers, and to a first approximation α = τ/I. Let's use a pneumatic driver.

Now the bolt head makes contact with the flange. The torque required to turn it increases rapidly, and the rate of rotation decreases, until eventually we are applying 110 nm, the air motor is stalled,  and the bolt isn't turning at all.

θ = 100 radian, or thereabouts, since the thread first engaged.

α = 0

I  is unchanged

but τ = 110

So τ ≠ I α

[hamdani yusuf (OP)]

If the system does rotate, the distance from the action point of the force to the center of rotation is both the lever arm and the rotation radius.

Not necessarily. See my example with a work bench.

Can you do it without any change in position of the lever?

In principle, yes. In practice, any real lever will bend a bit (indeed some torque wrenches use the bending to measure the tporque), but the applied torque is independent of the elasticity of the lever: whether you use a rigid bar or a flexible one, torque is just the product of force x distance.

Here's a thought experiment in a workshop. We wanted to release a bolt from a corroded valve. It's clamped on a bench using a vise. A large wrench was used in an attempt to turn the bolt to release it. After a force was applied, the bolt didn't turn. It moved the whole bench instead.
It reminds you the definition of torque in terms of cross product between force and radius of rotation. It's not the radius of the object.

Here's the diagram for simplified version of the case.

The length of the wrench only represents an expected value for the radius of rotation. But the actual rotation in this case is around the bottom of left leg of the bench. The radius that should be used to calculate torque is the radius of actual rotation.
Let me remind you that unexpected results come from false assumptions.

Now, in other case where there is no actual rotation, even so slightly, what is the radius of rotation that you will use to calculate torque?

Here's another example to show that expected rotational radius is not always the same as the real rotational radius.
It's similar to previous case, but this time a solid object is obstructing the rotation.
The question is, what's the torque produced by the force?

[hamdani yusuf (OP)]

Let's say the object is a long thin stick 1 meter long, 1 kg mass, floating in free space. 1 N force is applied to one of its end, perpendicular to the length. What is the torque?

  0.5 Nm, assuming the stick is homogeneous.

What if it isn't?

[hamdani yusuf (OP)]

What does Iα equal to, in your opinion?

Depends on a whole lot of information you haven't given me. Could be zero.

But if you use your definition of "torque", it could be infinity, which is nonsense.

What other information do you need?
In what case does it become infinity?

[hamdani yusuf (OP)]

Consider tightening a 12 mm bolt onto a flange - say the wheel bolt of a small car.

The initial torque required to spin the bolt into the thread is very small - you can do it with your fingers, and to a first approximation α = τ/I. Let's use a pneumatic driver.

Now the bolt head makes contact with the flange. The torque required to turn it increases rapidly, and the rate of rotation decreases, until eventually we are applying 110 nm, the air motor is stalled,  and the bolt isn't turning at all.

θ = 100 radian, or thereabouts, since the thread first engaged.

α = 0

I  is unchanged

but τ = 110

So τ ≠ I α

The difference is caused by a reactionary torque, which makes the net torque zero.
The thread on the bolt can also be worn out, which makes the bolt continuously rotating with a constant angular velocity. It means that the net torque is also zero. The reactionary torque by friction balances the torque produced by the driver.

[hamdani yusuf (OP)]

This is becoming quite tiresome. Two highly qualified individuals, namely AC and BC and myself have repeatedly outlined your errors but when cornered you just digress and obfuscate and I personally see this as bad manners.

You should try to think for yourself sometimes. Try to construct your own reasoning.
Don't always rely on argument from authority.

[hamdani yusuf (OP)]

I've changed the title from
What's the unit of torque
to
Can we improve the standard units of rotational quantities?

Torque is just one of some problematic standard units of rotational quantities, as shown in my tables.

[paul cotter]

There is nothing problematic with the standard definition of torque, the problem lies in your lack of understanding.

[hamdani yusuf (OP)]

There is nothing problematic with the standard definition of torque, the problem lies in your lack of understanding.

What's your definition of torque?
Do you agree with the Oxford dictionary?

[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.

[hamdani yusuf (OP)]

I asked some AI chatbot, Can we improve the standard units of rotational quantities?
Meta's AI answered
 

The standard units for rotational quantities are well-established in the International System of Units (SI). However, there are ongoing discussions and proposals to potentially redefine or improve units in various areas of physics.

If you have specific ideas or suggestions for improving units related to rotational quantities, I'd love to hear them!

I asked further about the ongoing discussions and proposals. It gave some links.

https://www.scirp.org/journal/paperinformation?paperid=131623

Improved Units of Measure in Rotational Mechanics
Richard James Petti
Independent Researcher, Arlington, USA.

Abstract

The SI system of units in rotational mechanics yields correct numerical results, but it produces physically incorrect units of measure in many cases. SI units also violate the principle of general covariance?the general rule for defining continuous coordinates and units in mathematics and mathematical physics. After 30+ years of wrestling with these problems, the ultimate authority on units of measure has declared that Newton?meter and Joule are not equivalent in rotational mechanics, as they are in the rest of physics. This article proposes a simple modification to SI units called ?Nonstandard International units? (?NI units?) until a better name is agreed upon. NI units yield correct numerical results and physically correct units of measure, and they satisfy the principle of general covariance. The main obstacle to the adoption of NI units is the consensus among users that the radius of rotation should have the unit meter because the radius can be measured with a ruler. NI units assigned to radius should have units meter/radian because the radius is a conversion factor between angular size and circumferential length, as in arclength = rθ. To manage the social consensus behind SI units, the author recommends retaining SI units as they are, and informing users who want correct units that NI units solve the technical problems of SI units.

1. Introduction

Rotational mechanics is a centuries-old field of classical physics. Isaac Newton wrote about rotational mechanics in Principia Mathematica. Newton derived Kepler?s second law from his mechanics and law of gravitation [1] . In 1746 Daniel Bernoulli and Leonard Euler proved the conservation of angular momentum [2] .

The SI system of units for rotational mechanics always provides correct numerical results [3] , but it assigns physically incorrect units of measure to key variables. This problem has been known for decades. The Finite Element Method is now the accepted approach to solving continuum and some discrete component problems in all areas of engineering [4] . However, it is particularly challenging to assign appropriate units of measure in rotational models.

These issues have led Bureau International des Poids et Mesures (BIPM) to declare that Newton-meter and Joule are not interchangeable in rotational mechanics, though they are interchangeable in the rest of physics.

This work compares SI units with a modified system of units called ?Nonstandard International units? (?NI units?) until a better name is agreed upon. NI units satisfy all requirements of the principle of general covariance, which is the basic rule for defining differentiable coordinates and associated units of measure in differential geometry and mathematical physics. SI units violate this principle in all cases where SI units differ from NI units.

The conversions between SI and NI units are very simple:

1) From NI units to SI units: remove unit rad everywhere it appears in NI units.

2) From SI units to NI units: assign to angular position θ unit rad, and apply the rules of general covariance to assign units to other variables; for example,
angular velocity ω=lim_Δt→0 Δθ/Δt
 must have units rad/s.

It appears that the user community accepted SI units for rotational mechanics for three reasons.

1) Most users are convinced that the radius of rotation ?r? must be assigned the unit meter because it can be measured with a ruler. Insistence on this condition is the root of all the problems. NI units assign to r the units m/rad.

2) SI units yield correct numerical results, as do NI units.

3) Most users valued conditions (a) and (b) more than the consistency of physical units and the principle of general covariance.


4. The Principle of General Covariance

In the twentieth century, differential geometry and general relativity came together to solve the most difficult challenges in coordinate systems in mathematical physics. General relativity generally has no orthonormal coordinates and no path-independent parallel translation; the geometrical length of a line segment generally is not the difference in coordinates, and the shape of space time itself changes continuously in dynamic situations.

The solution to these challenges is the principle of general covariance [10] [11] which requires that the coordinates, equations, tensor fields, and units of measure in mathematical physics are invariant under arbitrary differentiable coordinate transformations. A coordinate transformation is essentially the Jacobian of a differentiable map of one coordinate system to a second coordinate system. The transformation usually consists of multiplication by a conversion factor for each pair of similar units (or at most an inhomogeneous linear transformation, as for Kelvins and degrees Celsius). This principle meets the needs of the rest of physics, including Newtonian mechanics, general relativity, and quantum field theory.

Here are example applications of general covariance.

? If coordinate ?x? has unit meter, and coordinate ?t? has unit second, then dx/dt must have units m/s.

? If a coordinate system requires that θ be measured in radians, then the variable θ must have the unit radian.

? If the definition of a derived variable includes θ, then units for the derived variable must include the unit for θ.

Assignments of SI units to variables angle, torque, and angular momentum violate the principle of general covariance. NI units observe the principle of general covariance everywhere.

The article blames this violation on the presence of curvature in the base manifold. The author has never seen or imagined a case where violation of the principle of general covariance is not cause by incorrect in incomplete application of the principle.

5. Summary

SI units yield correct numerical results in rotational mechanics. However, these units do not yield physically correct units of measure. Three problems remain:

? Requiring angular position θ to be measured in radians but assigning no units to θ violates the principle of general covariance.

? Users cannot change the angular unit in rotational mechanics, because all angular units have been removed from the variables and equations. The variables and equations do not indicate where to put the conversion factors.

? Assigning to torque τ units of energy, either N-m or J, contradicts the practical result that ?τ? has physical meaning Δenergy/Δangle, or J/rad. To avoid equating torque with ?ordinary energy,? BIPM asserts that J and N-m are interchangeable everywhere in physics, except in rotational mechanics. This assignment is a glaring violation of the principle of general covariance.

SI units violate the principle of general covariance in numerous places.

NI units solve all the problems with units of measure in rotational mechanics. The key changes are:

? Angular position θ has unit rad, and θ should be measured in radians. These units enable the equations of rotational mechanics to provide the information needed to convert any angular unit to any other angular unit.

? Radius ?r? should have units m/rad. Many metrologists resist this assignment because the radius can be measured with a ruler, so it appears that ?r? must have the unit meter. NI units treat radius r as a conversion factor between circumference measured in meters and circumference measured in radians. Circumference can be measured with a ruler without violating the principle of general covariance.

One might ask why the engineering and physics communities settled for units with these problems. The author believes this occurred because:

? A strong consensus among users preferred that radius has the unit meter. This condition is the root of the problems.

? SI units deliver correct numerical results.

? Most of the community cares more about the two conditions above than about consistency among units of measure or the principle of general covariance.

The SI system of units is probably the most reasonable system that assigns to radius the unit meter.

6. Recommendations

The main factor favoring SI units for rotational mechanics is that SI units are more familiar to users. In particular, users are more familiar with radius as a basic variable with unit meters.

The main factors favoring NI units for rotational mechanics are:

? NI units observe the principle of general covariance, SI units do not.

? The NI units for angle, torque and other variables match the physical meanings of these variables. SI units do not.

? NI units retain the equivalence of J and N-m. According to BIPM, SI units must invalidate the equivalence of J and N-m.

? Users can change the unit assigned to the angle because locations of angular units in the equations indicate where to include conversion factors for different angular units.

The author?s recommendations are:

? Users who want physically correct units should use NI units. This solves the technical problems with SI units.

? Users who don?t need physical units?or cannot accept that radius has units m/rad?can continue to use SI units.

It also gave another link, but it contains the same information as previous link. https://www.researchgate.net/publication/357717817_Improved_Units_of_Measure_in_Rotational_Mechanics

There are many similarities between my idea and the article above. The main difference is in the proposed change to be made. As I mentioned earlier, the problem can be solved simply by distinguishing rotational radius from geometric radius, and give them different standard units. While geometric radius can retain its current unit of meter (which refers to distance between center and circumference), the unit for rotational radius should be changed to meter (which refers to the arclength displacement) per radian (which refers to change in direction). This change would cause all standard units for rotational quantities consistent with all of equations that they are involved in without having to live with two different standards.

[Bored chemist]

I asked some AI chatbot, ...

Why?
Here's a picture of it getting things stupidly wrong today.

Stawberry.jpg (28.18 kB . 1341x530 - viewed 226 times)

[hamdani yusuf (OP)]

I asked the same question to ChatGPT-4-turbo.

That's a very interesting question.

The short answer is: yes, theoretically, we can improve the standard units for rotational quantities, but it would involve rethinking deeply embedded conventions in physics and engineering.

Right now, rotational quantities (like angular displacement, angular velocity, angular acceleration) are usually measured using:

radians (rad) for angles

radians per second (rad/s) for angular velocity

radians per second squared (rad/s?) for angular acceleration

moment of inertia in kg?m?

torque in newton-meters (N?m)


Potential improvements could focus on:

1. Making units dimensionally consistent.

Radians are technically "dimensionless" because they are the ratio of arc length to radius (both measured in meters). But in practice, we treat radians like real units.

Some physicists suggest that we should formally treat radians as actual units, just like meters or kilograms, to avoid confusion.



2. Connecting better to linear quantities.

Linear and rotational motion are very analogous (displacement vs angular displacement, velocity vs angular velocity, etc.), but the way we write units makes them seem more different than they are.

A consistent "rotational" version of Newton?s laws (F = ma becomes τ = Iα) could be better highlighted if units were framed more systematically.



3. Introducing a unit for torque separate from energy.

Torque (N?m) and energy (joules = N?m) have the same units, but they are conceptually different. Some propose that torque should have its own unit name to avoid confusion.



4. Simplifying base units for rotation-heavy fields.

In mechanical or aerospace engineering, special rotational base units could make calculations faster or more intuitive, e.g., revolutions per minute (RPM) made a base unit instead of always needing to convert to rad/s.





---

An example of a more systematic system might look like:

Define rad as a base unit.

Define torque with a special unit (say, "torq" = N?m) distinct from energy.

Possibly define a "rotational mass" unit for moments of inertia (e.g., kg?m? becomes a new unit like "spinners" or something).



---

Would you like me to sketch a proposal for a full improved system?
(It could be a fun exercise ? and surprisingly useful if you're designing something new, like a new teaching curriculum, or a sci-fi world.)

[hamdani yusuf (OP)]

I asked some AI chatbot, ...

Why?

It can give the link to related article and new research as shown above.
While Meta's AI isn't smart enough to give thoughtful answer, some smarter models can give better answers, as shown in my next posts.

[hamdani yusuf (OP)]

And here's the answer from Gemini 2.0 Flash.

While the standard SI units for rotational quantities work well for calculations and are widely accepted, some physicists and metrologists argue that they have certain inconsistencies and could be improved for better physical interpretation and adherence to fundamental principles.
Here are some points of discussion regarding potential improvements:
Issues with Current SI Units
 * Dimensional Ambiguity: The radian, the SI unit for angle, is considered dimensionless. This can lead to situations where units don't clearly reflect the physical nature of the quantities. For instance, torque (force ? distance) has the same SI unit (Newton-meter, N⋅m) as energy (force ? distance), even though they represent different physical concepts. In rotational contexts, torque is also related to angular displacement, further complicating dimensional analysis.
 * Violation of General Covariance: Some argue that the current SI system in rotational mechanics violates the principle of general covariance, a crucial aspect of defining continuous coordinates and units in advanced physics and mathematics.
 * Radius Unit: The radius of rotation is typically assigned the unit meter (m), even though it acts as a conversion factor between angular size and arc length (\text{arc length} = r \theta). This contrasts with the idea that a conversion factor should have units that facilitate the conversion (e.g., if \theta is in radians, r could conceptually have units of m/rad to yield arc length in meters).
Proposed Improvements and Alternative Units
One proposed modification introduces "Nonstandard International units" (NI units). Key aspects of this system include:
 * Unit for Angle: Assigning the unit "radian" (rad) to angular position (\theta).
 * Radius Unit: Assigning the unit meter per radian (m/rad) to the radius (r). This makes the relationship \text{arc length} = r \theta dimensionally consistent: (\text{m/rad}) \times (\text{rad}) = \text{m}.
 * Consistent Units: Deriving units for other rotational quantities based on the principle of general covariance. For example, angular velocity (\omega = d\theta/dt) would have units of rad/s.
 * Torque and Energy: NI units would differentiate between torque and energy units more clearly.
Other alternative units or perspectives include:
 * Revolutions per minute (rpm): While not an SI unit, rpm is commonly used in engineering and practical applications to describe rotational speed.
 * Degrees: Although widely used, degrees are not part of the SI system and require conversion to radians for most physics formulas.
 * Hertz (Hz): For rotational frequency, Hertz (cycles per second or s⁻?) is sometimes used, although radians per second (rad/s) is the SI unit for angular velocity.
Advantages of Potential Improvements
 * Physical Correctness: Improved units could better reflect the underlying physical nature of rotational quantities and their relationships.
 * Dimensional Consistency: A more consistent system could simplify dimensional analysis and reduce the risk of errors.
 * Adherence to Fundamental Principles: Aligning the unit system with principles like general covariance could be beneficial for advanced physics.
 * Clarity in Applications: Distinguishing between quantities like torque and energy through their units could enhance clarity in various fields.
Challenges and Considerations
 * Widespread Adoption: The current SI system is deeply ingrained in scientific and engineering practices. Any significant change would require a massive effort in education, standardization, and updating existing literature and equipment.
 * User Acceptance: Many users prioritize correct numerical results and familiarity over theoretical consistency in units. A shift might face resistance if it doesn't offer immediate practical advantages for most users.
 * Complexity: Introducing new base or derived units could potentially add complexity to the unit system, at least initially.
Conclusion
While the current SI units for rotational quantities are functional for most purposes, there are valid arguments for improvement based on physical interpretation and fundamental principles. Proposed alternatives like NI units aim to address these issues by introducing a unit for angle and adjusting the units of related quantities for better consistency. However, the significant challenges associated with widespread adoption mean that any major shift in the standard units for rotational quantities is unlikely in the near future unless compelling practical advantages and a strong consensus emerge within the scientific and engineering communities.

BTW, it can correctly count the number of Rs in strawberry.

[hamdani yusuf (OP)]

The main problems those AI can identify in current standard are essentially about Dimensional Ambiguity and Violation of General Covariance.
Unfortunately, they cannot come up with satisfactory solutions yet, just like those in charge with setting, reviewing, teaching, and revising the standard.
But by their ability to identify those problems, they are no longer classified in level 0 in their awareness scale of current problems in rotational quantities.

[Bored chemist]

Meta's AI isn't smart enough

Then why the F*&^*^ did you use it?

That's what I keep asking.
Why use AI which can not be relied on to answer simple questions?
It hardly matters that it can't count the Rs in a word. We know the right answer to that, and we can all laugh at it.

But if you ask it a question to which you do not know the answer, you don't know if it gets it right or not.
(I hesitate to say this  but it's like asking the Bible for moral advice then, when it tells you slavery is a good thing, you dismiss that as metaphorical. You can only do that because you already know slavery is wrong).

If you need to know the answer in advance in order to check if the AI can answer it, then there's no point asking teh AI, is there?

[alancalverd]

Let's say the object is a long thin stick 1 meter long, 1 kg mass, floating in free space. 1 N force is applied to one of its end, perpendicular to the length. What is the torque?

  0.5 Nm, assuming the stick is homogeneous.

What if it isn't?

The stick will rotate about its center of mass, so if you know its mass/length distribution you can calculate the effective torque and subsequent motion. But only if you use the correct definition of torque.