[hamdani yusuf (OP)]

post 10 (a wiki quote) identifies the Newton-meter (with force being a vector) as the unit of torque.
End of story.  The rest seems to be yet another blog seeking to get off track in as many ways as possible, which is not the intent of the main sections of the forum.

The rest demonstrates how the confusion is already widespread, and needs to be addressed to prevent it from spreading even further.

[hamdani yusuf (OP)]

Once you have started a circularly symmetric body rolling on a horizontal frictionless surface,

If the surface is frictionless, the wheel will probably just slide and not roll at all. Which way the body rotates (if at all) depends on where relative to the CoM the impulse is applies.

The initial state is determined to be already rolling, which means a combination of linear and angular motion. Thus the body will keep rolling since no net force nor torque is being applied to it.

[Bored chemist]

The units of both are mass length squared per time squared.

If you want to calculate kinetic energy from velocity my multiplying the square of the speed by the mass you have the right dimensions but there's a factor- a half- that you need to multiply by.

If you want to calculate it from a torque and an angle  you have the right dimensions but there's a factor - the angle-  that you need to multiply by.

The 1/2 and the angle are both dimensionless.

Even with a consistent set of units like the SI, you still end up with constants in your calculations.

The 1/2 there in calculating kinetic energy from mass and velocity is a constant,
while the angle in calculating work from torque and the angle of rotation is a variable.

And pi is a constant.
So what?

So you were comparing apple to orange.

No.

[hamdani yusuf (OP)]

I asked Gemini.
Why Radians are Dimensionless?

A radian is a measure of an angle, defined as the ratio of the arc length of a circle to its radius.
Since both the arc length and the radius are measured in units of length (like meters or centimeters), these units cancel out when calculating the radian.
Therefore, a radian is essentially a ratio of two lengths, resulting in a dimensionless quantity.
This is unlike other units like meters, seconds, or kilograms, which have specific dimensions (length, time, and mass, respectively).
In essence, a radian is a pure number that represents an angle, without any associated physical dimension.

I continued.
Why Amount of Substance Has a Dimension?

The amount of substance, measured in moles, does indeed have a dimension. It's one of the seven base units in the International System of Units (SI).
Why?
 * Distinct from Counting: While it might seem like simply counting particles, the mole is a macroscopic unit used to quantify microscopic entities (atoms, molecules, ions, etc.). It's analogous to using "dozen" to count eggs rather than individually counting each egg.
 * Relating to Other Quantities: The mole is essential for connecting microscopic properties (like atomic mass) to macroscopic properties (like mass). For example, the ideal gas law (PV = nRT) uses the amount of substance (n) to relate pressure, volume, temperature, and the gas constant.
 * Consistency in Calculations: Assigning a dimension to the amount of substance ensures dimensional consistency in equations. It helps prevent errors in calculations and ensures that the units on both sides of an equation match.
Key Points:
 * The mole is a base unit, independent of other base units like mass, length, or time.
 * It's a measure of the quantity of a substance, not its mass or volume.
 * Avogadro's number (approximately 6.022 x 10^23) defines the number of particles in one mole.
By recognizing the amount of substance as a distinct dimension, we can accurately quantify and manipulate matter at both the microscopic and macroscopic levels.

Essentially, the angle of rotation can be thought as the amount of waves. If an object is rotated by 6π radians, it undergoes 3 waves.

On the other hand, the amount of substance can be thought as ratio between the mass of the whole substance and the mass of single individual substance.

[alancalverd]

[Quote ]from: Bored chemist on 11/08/2024 23:44:06
The units of both are mass length squared per time squared.

this is so wrong.  It seems to presume that energy and torque are the same thing, and then applying a relativistic notion of mass-energy equivalence to torque.

No, it's purely classical.

Work = energy = force x distance, MLT^-2 x L
Torque = force x distance MLT^-2 x L

Which is why you can measure torque with a spring.

[alancalverd]

Essentially, the angle can be thought as the amount of waves. If an object is rotated by 6π radians, it undergoes 3 waves.

You are confusing yourself - as if Gemini wasn't doing it already!

[alancalverd]

If the surface is frictionless, the wheel will probably just slide and not roll at all. Which way the body rotates (if at all) depends on where relative to the CoM the impulse is applies.

Not according to Newton. I said "rolling" , not "spinning". Both linear and angular momentum are conserved.

[paul cotter]

Okay, I got that wrong too.

[hamdani yusuf (OP)]

Work = energy = force x distance, MLT-2 x L
Torque = force x distance MLT-2 x L

The distance used to calculate work is different from the distance used to calculate torque. The ratio between those distances is the angle of rotation.

[hamdani yusuf (OP)]

Which is why you can measure torque with a spring.

A spring can be used to measure force. But that alone isn't enough to measure torque. You also need a fulcrum. And the direction of the force does not intersect with position of the fulcrum.

[hamdani yusuf (OP)]

Essentially, the angle can be thought as the amount of waves. If an object is rotated by 6π radians, it undergoes 3 waves.

You are confusing yourself - as if Gemini wasn't doing it already!

OK, I've revised the angle into the angle of rotation in my post. I hope it can reduce your confusion.
I thought it was already clear that the context of the discussion is about rotation, and not the angle in Euclidean geometry.
When there's no propagation involved, the wave can be replaced by oscillation.

[hamdani yusuf (OP)]

Okay, I got that wrong too.

But you're right about the epic confusion.

[Origin]

But you're right about the epic confusion.

My observation of your posts is that you are always epically confused.

[alancalverd]

Essentially, the angle of rotation can be thought as the amount of waves. If an object is rotated by 6π radians, it undergoes 3 waves.

No! You can generate a single sine curve as the projection of a point on a rolling cylinder, but deBroglie waves can have any integer number in a circle.

[hamdani yusuf (OP)]

According to ChatGPT

The concept of torque, often associated with the rotational force or the moment of force, was first studied in the context of ancient mechanics. The earliest formal studies of torque can be traced back to the work of Archimedes of Syracuse (circa 287-212 BCE).

Archimedes made significant contributions to the understanding of levers and the principles of balance, which are closely related to the concept of torque. He formulated the principle of the lever, stating that "magnitudes are in equilibrium at distances reciprocally proportional to their weights." This principle essentially captures the relationship between force and distance that defines torque.

However, the modern understanding and terminology of torque, as used in physics and engineering today, were developed later, particularly during the Scientific Revolution of the 16th and 17th centuries. The term "torque" itself comes from the Latin word "torquere," meaning "to twist," and was adopted in the study of mechanics in the 19th century.

So, while the concept of torque has ancient roots, the systematic study and formalization of torque as we understand it today evolved over many centuries.

The ancient roots in the concept of torque contributed to current confusion. The usage of weight and distance in the principle of lever have limited the direction of force and the magnitude of rotational angle. In this narrowly special conditions, the concept of energy can be mostly ignored without losing significant loss of functionalities. It's a different story when the rotation involves multiple turns like in the case of steam engines.

[alancalverd]

Thus demonstrating that ChatGPT is about as much use as Gemini or a chocolate teapot.

[hamdani yusuf (OP)]

Essentially, the angle of rotation can be thought as the amount of waves. If an object is rotated by 6π radians, it undergoes 3 waves.

No! You can generate a single sine curve as the projection of a point on a rolling cylinder, but deBroglie waves can have any integer number in a circle.

Number of waves in wave mechanics can be expressed in cycle as well as radian, with 2π as ratio between them.

[alancalverd]

The number of waves in unit length, i.e. the reciprocal of wavelength, is called "wavenumber" and has nothing to do with π.
The number of waves in unit time is called frequency, again nothing to do with π.

[hamdani yusuf (OP)]

But you're right about the epic confusion.

My observation of your posts is that you are always epically confused.

Which part of my post is confusing to you?

[hamdani yusuf (OP)]

The number of waves in unit time is called frequency, again nothing to do with π.

Angular frequency is 2π times frequency.