[hamdani yusuf (OP)]

This is a spin-off of my other thread. I thought the discussion would be short and would quickly come into conclusion, but it seems I was not the case. Let's discuss into more details here.

I asked Gemini for other Physical Measures with the Same Units

Excellent question! There are several other physical quantities that share the same units, even though they represent different concepts. Here are a few examples:
...
2. Torque and Energy
 * Both measured in newton-meters (Nm) or joules (J).
 * Torque is a rotational force, while energy is the capacity to do work.
...

While the other points are basically pairs of the same measures, the second point tells a different story. There is something missing here.

I told Gemini that energy is torque times the angle of rotation.

You're absolutely right!
Energy = Torque x Angle of Rotation
This equation is a fundamental relationship in rotational mechanics. Here's a breakdown:
 * Torque: A measure of the force that can cause an object to rotate around an axis. It's analogous to force in linear motion.
 * Angle of rotation: The amount of rotation an object undergoes around an axis.
When you multiply torque by the angle of rotation, you get the work done, which is a form of energy.
Key points:
 * This equation holds true for constant torque. If the torque varies, integration is required.
 * The units of torque (Newton-meter) multiplied by the unit of angle (radians) give the unit of energy (joules).

It means that the proper unit for torque is Joule/radian, or Newton meter/radian.
I found many people being confused about measurement units involving radian, which is often omitted for being a dimensionless quantity.
The unit radian for measuring angle of rotation is convenient since the traveling distance of a point due to 1 radian rotation equals the radius of rotation.

https://en.wikipedia.org/wiki/Radian

An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.

[hamdani yusuf (OP)]

No. Newton.meter is a unit of energy (dimension ML²T^-2) or torque - your standard torque wrench is calibrated by hanging a weight on the end!

Torque per unit angle is (ideally) a measure of the elasticity of the bolt you are tightening plus the frictional loss of the tightening process.

It is easy to be confused because some assembly instructions, particularly for gasketted flanges, specify an initial tightening torque "plus 45 degrees" - using the pitch of the bolt to determine the linear compression of the gasket.

Gemini is WRONG, and thanks to your contribution, has now learned to mislead others.

Let's say I have a standard torque wrench with 1 meter length. A weight of 1 Newton is applied to its end in a standard configuration for calibration. When the wrench doesn't rotate at all, no work has been done, thus 0 Joule, because the angle of rotation is 0.
If it rotates by 1 milli radian, then the energy of 1 milli Joule has been transferred.

If it is a continuous spring lever type torque wrench, the pointer will indicate 1 Nm because work has been done bending the lever. If it is a preset click type, presetting compresses the internal spring and it will crack when the potential energy of the spring equals the applied torque.

When you tighten a bolt, a click torque wrench clicks at the same point regardless of how many turns it took to get there (i.e. the length of the bolt and the pitch of the thread) so angle is irrelevant.

Limitations of the particular measuring device that you use should not affect the definition of the quantity being measured.
Let's say a sensitive sensor is installed for the torque measuring device which can produce a reliable signal with only 1 micro-Joule input. If the torque stop as soon as the sensor is active, then the energy transferred is only 1 micro-Joule. But if it continuously applied until it turns by 1 radian, then the energy transferred is 1 Joule.

[hamdani yusuf (OP)]

Here's a real world experience, although the numbers are a bit rounded to simplify the arithmetic. A grinding machine is used to break down granule raw material into powder. The agitator is driven by a 100 kW induction motor with 1500 rpm (= 25 rotation/second) angular speed equipped with a gear box. It has a Variable Speed Drive that can inform about several parameters, including power and torque. The power is expressed in kW, while the torque is in % of stated motor capacity.
At the beginning of the process, the material is still coarse, and the motor gets higher load. The torque is kept at 80% by controlling the frequency of VSD, which can go as low as 300 rpm (=5 rps). As the process goes on, the material gets finer and the speed can progressively increase to the maximum, and then the the measured torque will steadily decrease. At the end of the process, the material is fine and smooth, and the motor gets lower load. The torque meter shows around 10% at maximum frequency of VSD.
Here's the graph for illustration.

The red curve is for torque, while the blue one is for angular speed. The vertical axis is percent of motor capacity for both torque and angular speed, while horizontal axis is processing time.

[hamdani yusuf (OP)]

Let's say that during the constant 80% torque phase, the motor speed steadily increase from 300 rpm to 1500 rpm (20% to 100%) in 1 hour. The average power consumed by the motor is 100kW * 80% * (20+100)/2 % = 48 kW. The total energy consumed is 48 kWh.

Let's say that during the constant 100% speed phase, the torque steadily decrease from 80% to 10% in 1 hour. The average power consumed by the motor is 100kW * 100% * (80+10)/2 % = 45 kW. The total energy consumed is 45 kWh.

By multiplying torque with angular speed, we get power. While multiplying torque and rotational angle gives the consumed energy. It implies that SI unit of torque is Joule/radian.

[Bored chemist]

Q What's the unit of Torque?
A The  unit of torque is mass* length squared / time squared

(Not sure this needed a thread)

[hamdani yusuf (OP)]

Q What's the unit of Torque?
A The  unit of torque is mass* length squared / time squared

(Not sure this needed a thread)

Your answer makes me sure that this thread is needed.
Let's have a simpler case for illustration.
An elevator motor drives a 1 meter radius pulley to lift 1 Newton of weight. After a brief momentary initial kick, a constant torque is applied which lifts the weight at 1 mm/s constant speed. A thousand seconds later, the motor stops and the weight has been elevated by 1 m from initial position. The work done to the weight is 1 Joule. The rotation angle is 1 radian.
The second case, the motor doesn't stop until 2000 seconds from initial kick. With the same torque and lifting speed, the weight has been elevated by 2 m from initial position. The work done to the weight is 2 Joule. The rotation angle is 2 radian.

In the first case, 1 Joule of energy is used to rotate the pulley system by 1 radian, while In the second case, 2 Joule of energy is used to rotate the pulley system by 2 radian. In both case, the torque is the same, i.e. 1 Joule/radian.

[Bored chemist]

Q What's the unit of Torque?
A The  unit of torque is mass* length squared / time squared

(Not sure this needed a thread)

Your answer makes me sure that we need this thread.
Let's have a simpler case for illustration.
An elevator motor drives a 1 meter radius pulley to lift 1 Newton of weight. After a brief momentary initial kick, a constant torque is applied which lifts the weight at 1 mm/s constant speed. A thousand seconds later, the motor stops and the weight has been elevated by 1 m from initial position. The work done to the weight is 1 Joule. The rotation angle is 1 radian.
The second case, the motor doesn't stop until 2000 seconds from initial kick. With the same torque and lifting speed, the weight has been elevated by 2 m from initial position. The work done to the weight is 2 Joule. The rotation angle is 2 radian.

In the first case, 1 Joule of energy is used to rotate the pulley system by 1 radian, while In the second case, 2 Joule of energy is used to rotate the pulley system by 2 radian. In both case, the torque is the same, i.e. 1 Joule/radian.

So what?

Get back to me if this stops being true.

The  unit of torque is mass* length squared / time squared

[hamdani yusuf (OP)]

So what?

Get back to me if this stops being true.

The  unit of torque is mass* length squared / time squared

The angle of rotation is missing.
Torque is rotational force, analogous with force is linear force.
Both measures the energy needed to move an object to 1 unit of distance: rotational and linear, respectively.

[hamdani yusuf (OP)]

Here's another example. We need to dismantle a machinery. It has 4 bolts to secure it to the floor, and each of them needs to be rotated 10 turns to release.
The first bolt is still in good condition. After initial kick, 1 Joule of energy can be used to rotate it for 10 turns.
The second bolt is rusty. After initial kick, 1 Joule of energy can only be used to rotate it for 2 turns. To release the bolt, 5 Joule of energy is needed.
I hope this example can help to understand the importance of rotational angle in measuring torque.

[hamdani yusuf (OP)]

Let's examine the Wikipedia article about torque.
https://en.wikipedia.org/wiki/Torque

In physics and mechanics, torque is the rotational analogue of linear force.[1] It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically τ, the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M. Just as a linear force is a push or a pull applied to a body, a torque can be thought of as a twist applied to an object with respect to a chosen point; for example, driving a screw uses torque, which is applied by the screwdriver rotating around its axis. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum.


Relationship between force F, torque τ, linear momentum p, and angular momentum L in a system which has rotation constrained to only one plane (forces and moments due to gravity and friction not considered).

https://en.wikipedia.org/wiki/Torque#Relationship_with_power_and_energy

Relationship with power and energy
The law of conservation of energy can also be used to understand torque. If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass, the work W can be expressed as

How can the angle of rotation in second derivation be missing in the first one?

[hamdani yusuf (OP)]

https://en.wikipedia.org/wiki/Torque#Units

Torque has the dimension of force times distance, symbolically T−2L2M and those fundamental dimensions are the same as that for energy or work. Official SI literature indicates newton-metre, is properly denoted N⋅m, as the unit for torque; although this is dimensionally equivalent to the joule, which is the unit of energy, the latter can never used for torque.[14][15] In the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar. This means that the dimensional equivalence of the newton-metre and the joule may be applied in the former but not in the latter case. This problem is addressed in orientational analysis, which treats the radian as a base unit rather than as a dimensionless unit.

The last sentence has offered a solution.

[alancalverd]

In the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar.

That's the nub of the question.

torque is the rotational analogue of linear force

Analogous does not mean identical.

[alancalverd]

The first bolt is still in good condition. After initial kick, 1 Joule of energy can be used to rotate it for 10 turns.
The second bolt is rusty. After initial kick, 1 Joule of energy can only be used to rotate it for 2 turns. To release the bolt, 5 Joule of energy is needed.

The "initial kick" refers to releasing the potential energy stored in the tension of the bolt, i.e. the tightening torque. Some mechanical assembly instructions actually specify the initial and final length of the bolt where both ends are accessible, e.g. on a pipe flange. This exemplifies the torque-energy equivalence and can be more precise than simply using a torque wrench as it ignores turning friction and effectively determines the mating force in the flange.
Subsequent movement against friction is about kinetic energy, force x distance.

[hamdani yusuf (OP)]

How can the angle of rotation in second derivation be missing in the first one?

It's in the history.
https://en.wikipedia.org/wiki/Torque#History

The concept originated with the studies by Archimedes of the usage of levers, which is reflected in his famous quote: "Give me a lever and a place to stand and I will move the Earth". The term torque (from Latin torquēre, 'to twist') is said to have been suggested by James Thomson and appeared in print in April, 1884.[2][3][4] Usage is attested the same year by Silvanus P. Thompson in the first edition of Dynamo-Electric Machinery.[4] Thompson motivates the term as follows:[3]

Just as the Newtonian definition of force is that which produces or tends to produce motion (along a line), so torque may be defined as that which produces or tends to produce torsion (around an axis). It is better to use a term which treats this action as a single definite entity than to use terms like "couple" and "moment", which suggest more complex ideas. The single notion of a twist applied to turn a shaft is better than the more complex notion of applying a linear force (or a pair of forces) with a certain leverage.

An implicit assumption in the derivation of torque through force and distance from the axis of rotation, is the existence of immovable fulcrum. It requires effectively infinite inertia.

The missing of rotational angle from the previous derivation through force and radius can be identified by comparing with the derivation below.

https://en.wikipedia.org/wiki/Torque#Derivation

For a rotating object, the linear distance covered at the circumference of rotation is the product of the radius with the angle covered. That is: linear distance = radius x angular distance. And by definition, linear distance = linear speed x time = radius x angular speed x time.

By the definition of torque: torque = radius x force. We can rearrange this to determine force = torque / radius.

In general, the distance used should be the linear distance covered at the circumference of rotation. In case the angle is 1 radian, this length of arc is equal to radius.
A more explicit version of the red sentence is:
torque = energy / angle of rotation
torque = arc length of motion . force / angle of rotation
torque = radius x angle of rotation . force / angle of rotation
torque = radius x angle of rotation . force / angle of rotation
torque = radius x force

[hamdani yusuf (OP)]

Whether the unit of torque is Nm or Nm/rad depends on which distance we are using. We need to recognize that are two distinct kind of distance used here: the radius and the arc length of rotation.
In the former, the multiplication between force and distance is a cross product, while in the later it's a dot product.
Thus, we can avoid confusion by distinguishing the type of products being used. That would give us the unit of torque as N x m, or N.m/rad.

To calculate the work done in Joule, both cases must be multiplied by the angle of rotation. In both cases, the result is N.m.

[Bored chemist]

Thus, we can avoid confusion by...

Thus far, the list of people who are confused by this only seems to have one entry, and that's you.

[paul cotter]

I sense another epic study of confusion is beginning.

[hamdani yusuf (OP)]

Thus, we can avoid confusion by...

Thus far, the list of people who are confused by this only seems to have one entry, and that's you.

I've shown that even Gemini gets confused. I won't be surprised if many people are confused too.
Perhaps they don't tell about their confusion because they haven't thought about it much so they don't even realize that they have been confused by inconsistent explanations. Some people might already spotted the inconsistency, but they don't tell us because they don't think that we can help them.

[Bored chemist]

Thus, we can avoid confusion by...

Thus far, the list of people who are confused by this only seems to have one entry, and that's you.

I've shown that even Gemini gets confused.

Do you  know that Gemini isn't a person?
Its a prototype undergoing testing.

[hamdani yusuf (OP)]

I also asked ChatGPT the same question. And it concluded that 1 N.m = 1 J/rad
It seems like the confusion is widespread among chatbots.
 Claude Sonnet also gives similar answer.

You're right to ask for clarification on this. Let me provide a clear and accurate answer:

No, energy is not equal to torque times angle of rotation.

What you're referring to is actually the formula for work done in a rotational system:

Work = Torque ? Angle of rotation

This equation calculates the work done by a constant torque over a given angular displacement. While work is a form of energy transfer, it's not the same as energy in all its forms.

Key points:
1. This equation gives the work done, not energy in general.
2. Work is measured in joules (J), which is the same unit as energy.
3. Energy exists in many forms (kinetic, potential, thermal, etc.) that aren't always related to this specific calculation.

The similarity in units and the fact that work is a form of energy transfer can lead to confusion, but it's important to distinguish between work done in a specific rotational context and the broader concept of energy.

Thank you for asking this important question. It helps clarify a common point of confusion in physics.

Unfortunately I reached the limit when I asked for further question.