[hamdani yusuf (OP)]

In more general cases where the trajectory is not necessarily circular, the rotational radius isn't necessarily constant. In other words, there's a non-zero radial displacement. In these cases, current standard is no longer adequate to describe the rotational system.
I'm working on the problem with elliptical trajectory using my proposed standard. Qwen can solve it in a few prompts, even in older version. Newer AI models will be able too, IMO.

The first case of elliptical trajectory is when the tangential speed is constant. Consequentially, its kinetic energy is constant. The acceleration must be purely orthogonal. But the angular speed must vary, inversely proportional to the rotational radius at the moment.

This video could help you visualize what I described above. Imagine the track is elliptical instead of circular.

Superconductor at -196?C, Quantum Levitation | Magnetic Games

With the use of liquid nitrogen, the YBCO compound can be cooled until it becomes a superconductor, and a superconductor placed in a magnetic field has amazing behaviors.
Please activate the subtitles to get more info on the experiment.

The next case is when the angular speed is constant. This can be done by modifying the previous setup with a smooth pipe where the puck can slide inside without friction, and a motor/generator unit equipped with battery to control the rotation of the pipe at a constant angular speed. When accelerating, potential energy from battery flow to the motor, and converted to kinetic energy. When decelerating, kinetic energy from the puck is converted by the generator and stored back to the battery as potential energy.

Let's analyze this.
Angular speed ω is constant.
rotational axis is kept at the same position, ie. the center of motor/generator. This is different from the first case, where the position of rotational axis keeps changing. This emphasizes the difference between dynamic rotational radius and static geometric radius.

Tangential speed v = ω . r_rot, is thus proportional to the rotational radius.
kinetic energy is proportional to tangential speed squared.
Total energy must be conserved, thus changes in kinetic energy must be counter balanced by changes in potential energy.
In the diagram, semi minor axis is 1 meter, while semi major axis is 2 meter.
The kinetic energy at lowest speed is then only a quarter from its highest speed.
Let's say at highest speed, ie. highest kinetic energy at major axis, the potential energy is zero. Then at minor axis, 3/4 of the kinetic energy is converted to potential energy.

[alancalverd]

The formula E = τ. θ implies that standard unit for torque is Joule per radian.

which is why nobody uses it.

In the real world ΔE = τ.r.θ

[hamdani yusuf (OP)]

The formula E = τ. θ implies that standard unit for torque is Joule per radian.

which is why nobody uses it.

In the real world ΔE = τ.r.θ

Can you check with unit analysis?

The relationship between torque and energy is fundamentally about work done by torque. Work done by a torque is the product of the torque and the angular displacement it causes. The formula for work done by torque is: W = τθ, where τ is the torque and θ is the angular displacement in radians.
Google it.

[alancalverd]

Can you check with unit analysis?

I presume you mean dimensional analysis. No need.

rθ is the distance the weight has risen or fallen. ΔEnergy = weight x distance.

The relationship between torque and energy is fundamentally about work done by torque.

Exactly. But you can still have a static torque that does no work, just as you can have a static weight whose potential energy doesn't change.  So it's a bad idea to define torque as a product of energy, or indeed anything to do with movement.

[hamdani yusuf (OP)]

Can you check with unit analysis?

I presume you mean dimensional analysis. No need.

rθ is the distance the weight has risen or fallen. ΔEnergy = weight x distance.

The relationship between torque and energy is fundamentally about work done by torque.

Exactly. But you can still have a static torque that does no work, just as you can have a static weight whose potential energy doesn't change.  So it's a bad idea to define torque as a product of energy, or indeed anything to do with movement.

I do mean unit.
Meter and mile have the same dimension. So do degree and radian. But if you involve more than one unit, you need to include a conversion factor.

In your previous equation, Energy equals torque times distance times angle. Even if the angle is ignored, you still get torque times distance, which is not equal to energy.

The equation E = τ.θ is necessary for static torque. In this case, torque can be non-zero, but the angle is zero. Consequently, the work is zero.

[hamdani yusuf (OP)]

The third case of elliptical trajectory I want to address is planetary orbit.

In this case, the rotational axis is one focal point of the ellipse. The other focal point is empty and play no role in rotational dynamics. This is a strong argument to distinguish between rotational and geometric radius.
In this case, tangential speed as well as angular speed are not constant. But rotational axis is constant. So is the angular momentum of the planet, which is equal to I.ω.
I is angular inertia, which is minimum at perihelion and maximum at aphelion.
ω is angular speed, which is maximum at perihelion and minimum at aphelion.

[paul cotter]

Degree and radian are dimensionless as they refer to a certain fraction of a circle, ie they are a ratio of a part of a circle to the whole of the circle. Nowhere else do we ascribe dimensions to a ratio.

[alancalverd]

I do mean unit.

OK. The unit of energy is the joule and the unit of torque is the newton,meter. They are different entities so they have different units.

[paul cotter]

"I do mean unit.
Meter and mile have the same dimension. So do degree and radian. But if you involve more than one unit, you need to include a conversion"
This quote is what I was answering- Hamdani, you say you mean unit but you constantly hop from unit to dimension.

[hamdani yusuf (OP)]

Degree and radian are dimensionless as they refer to a certain fraction of a circle, ie they are a ratio of a part of a circle to the whole of the circle. Nowhere else do we ascribe dimensions to a ratio.

They are made dimensionless by convention. On the other hand,

The mole (symbol mol) is a unit of measurement, the base unit in the International System of Units (SI) for amount of substance, an SI base quantity proportional to the number of elementary entities of a substance. One mole is an aggregate of exactly 6.02214076?1023 elementary entities (approximately 602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, ion pairs, or other particles. The number of particles in a mole is the Avogadro number (symbol N0) and the numerical value of the Avogadro constant (symbol NA) expressed in mol−1.[1] The relationship between the mole, Avogadro number, and Avogadro constant can be expressed in the following equation:[1]
\
https://en.wikipedia.org/wiki/Mole_(unit)#

[hamdani yusuf (OP)]

I do mean unit.

OK. The unit of energy is the joule and the unit of torque is the newton,meter. They are different entities so they have different units.

Currently, they are distinguished only by convention. That's the cost of inconsistently allowing writing or erasing radian from the equations.

[hamdani yusuf (OP)]

"I do mean unit.
Meter and mile have the same dimension. So do degree and radian. But if you involve more than one unit, you need to include a conversion"
This quote is what I was answering- Hamdani, you say you mean unit but you constantly hop from unit to dimension.

The title of this thread is about unit. But it doesn't prevent anyone to talk about other things, including dimension, as long as they are relevant to the subject of discussion.

[alancalverd]

Currently, they are distinguished only by convention.

Rubbish. They are distinguished by name (energy, torque) and unit (joule, newton-meter). 1 joule of energy will heat a gram of water by about 0.25 K. 1 nm of torque will not.

"Convention" distinguishes between two things with the same name, like mole (carnivorous quadruped) and mole (6.022 x 10²³).

Ratios of the same entity are obviously dimensionless. The aspect ratio of a television screen is a length divided by a length, L/L = 16/9 usually. Hence the ratio of  circumference to radius of a circle is also dimensionless. We use the term grad, rad or deg to indicate whether the denominator is 400, 2π, or 360.
 

[hamdani yusuf (OP)]

Currently, they are distinguished only by convention.

Rubbish. They are distinguished by name (energy, torque) and unit (joule, newton-meter). 1 joule of energy will heat a gram of water by about 0.25 K. 1 nm of torque will not.

"Convention" distinguishes between two things with the same name, like mole (carnivorous quadruped) and mole (6.022 x 10²³).

Ratios of the same entity are obviously dimensionless. The aspect ratio of a television screen is a length divided by a length, L/L = 16/9 usually. Hence the ratio of  circumference to radius of a circle is also dimensionless. We use the term grad, rad or deg to indicate whether the denominator is 400, 2π, or 360.
 

That's the convention: Newton meter is used for torque, while Joule is used for Work and energy.
But the formula implies that 1 Joule equals 1 Newton meter. In current standard, they are distinguished by convention.
1 Newton meter is equal to 1 Joule when the quantity is Work or energy.
1 Newton meter is generally not equal to 1 Joule when the quantity is torque.

[alancalverd]

But the formula implies that 1 Joule equals 1 Newton meter. In current standard, they are distinguished by convention.

NO!

The difference is between a scalar product (energy = force x distance  moved in the line of action of the force) and a vector product (torque = force x distance perpendicular to the line of action of the force.) 

Hence energy is a scalar and torque is a vector.

Nothing to do with "convention" beyond the sensible fact that different quantities have different names and different units, so the rest of us don't get confused.

You will recall that "conservative forces do no work"; in other words, no work is done moving a mass at constant speed  along a frictionless horizontal plane, because the reaction force is perpendicular to the direction of movement. Orthogonality is very important in maths and physics.

[paul cotter]

Indeed, if one's math is correct and rigorous there will be a unit vector in the expression for torque, differentiating it comprehensively from similar but NOT equivalent expressions for energy.

[hamdani yusuf (OP)]

But the formula implies that 1 Joule equals 1 Newton meter. In current standard, they are distinguished by convention.

NO!

The difference is between a scalar product (energy = force x distance  moved in the line of action of the force) and a vector product (torque = force x distance perpendicular to the line of action of the force.) 

Hence energy is a scalar and torque is a vector.

Nothing to do with "convention" beyond the sensible fact that different quantities have different names and different units, so the rest of us don't get confused.

You will recall that "conservative forces do no work"; in other words, no work is done moving a mass at constant speed  along a frictionless horizontal plane, because the reaction force is perpendicular to the direction of movement. Orthogonality is very important in maths and physics.

The cross sign doesn't show up in the unit. That's why you need a convention.

[alancalverd]

The convention is to call energy "energy" and measure it in joules, and to call torque "torque", and measure it in newton-meters.

What more could you possibly want? 

The leg dimension of a cow is 4. The leg dimension of a cat is 4. One is a herbivore, one is a carnivore. We distinguish them with the conventional names "cow" and "cat", which is important because they both live on farms, but do different jobs. But to an octopus,  or, apparently,  a physics teacher, they are indistinguishable because they are both quadrupeds.

[hamdani yusuf (OP)]

The convention is to call energy "energy" and measure it in joules, and to call torque "torque", and measure it in newton-meters.

What more could you possibly want? 

The leg dimension of a cow is 4. The leg dimension of a cat is 4. One is a herbivore, one is a carnivore. We distinguish them with the conventional names "cow" and "cat", which is important because they both live on farms, but do different jobs. But to an octopus,  or, apparently,  a physics teacher, they are indistinguishable because they are both quadrupeds.

The equation states that torque equals work divided by angular distance of rotation.
It implies that unit for torque is Joule per radian.
Angular distance of rotation is not limited to 2π radian.
If you apply a constant torque to a rotating system, you will need 10 times of work to make it turn 20π radian.

[alancalverd]

The equation states that torque equals work divided by angular distance of rotation.

No. If an object rotates , the work done to accelerate it and oppose friction is  ∫τdθ.

In the absence of rotation τ can be > 0 but E = 0.