[hamdani yusuf (OP)]

https://www.quora.com/Why-do-we-use-only-Nm-as-a-unit-for-torque-but-not-J-although-Nm-J-in-case-of-work-energy

https://physics.stackexchange.com/questions/37881/why-is-torque-not-measured-in-joules

It's not too surprising that current AI models got confused by the question, since they were trained using human generated data, which also showed a widespread confusion.

[hamdani yusuf (OP)]

I asked Gemini about Example of Radians Squared.

An Example of Radians Squared: Moment of Inertia
Moment of inertia is a physical quantity that measures an object's resistance to rotational motion. For a point mass, the moment of inertia (I) is given by:
 * I = mr^2
Where:
 * m is the mass of the object
 * r is the perpendicular distance from the axis of rotation
As you can see, the unit of moment of inertia involves kilogram-meter squared (kg m^2). While this isn't explicitly radians squared, it's closely related as the meter part represents a distance, which is the basis for defining a radian.
It's important to note that this is a simplified example for a point mass. For more complex objects, the calculation of moment of inertia involves integration, and the results can involve more intricate expressions.

I asked how it's derived.

The moment of inertia of a rigid object is derived from its mass distribution and axis of rotation. It is a measure of an object's resistance to changes in its rotational motion. The moment of inertia of a point mass is given by I = mr^2, where m is the mass of the object and r is the perpendicular distance from the axis of rotation.
The moment of inertia of a rigid object is the sum of the moments of inertia of all of its constituent point masses. The moment of inertia of a rigid object is a scalar quantity, which means that it has magnitude but no direction.
The moment of inertia of a rigid object is used in the calculation of its angular momentum, which is a measure of its rotational motion. The angular momentum of a rigid object is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a rigid object is also used in the calculation of its kinetic energy, which is the energy that it has due to its motion. The kinetic energy of a rigid object is given by K = 0.5 Iω^2, where K is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
 * https://www.numerade.com/ask/question/a-uniform-circular-disk-of-mass-10-kg-and-radius-10-m-is-free-to-rotate-about-a-fixed-axle-perpendicular-to-it-passing-through-the-center-it-is-acted-upon-by-two-forces-in-the-clockwise-dire-13564/

The answer could be a basis for assigning a dimension to the unit.
In this case, it's useful to think about the radius of rotation as arc length divided by angle of rotation.
dθ = dl/R
R = dl/dθ

[hamdani yusuf (OP)]

Here's an example to distinguish between torque and force.

A hammer-like metal object consists of a cube sandwiched between a pair of half cylinders. It's equipped with a handle with negligible mass. A force is applied to the end of the handle.

When the floor is frictionless, the force will only make the hammer slide horizontally, which means the net torque is zero.
But when the floor has high friction, the hammer can tumble, which means that the hammer has been rotated.

It's worth noting that radius of rotation can be different from radius of the geometrical object, as shown in the example. Instead of just meter, radius of rotation is better expressed in meter per radian. The meter is for the arc length traveled by the rotation, while the radian is for angle of rotation.

[alancalverd]

How does it explain this equation?

Up to the elastic limit of a bolt, the tightening torque τ is a linear-ish function of θ, so the integral from 0 to θ equals the final torque.

[alancalverd]

If you put down Gemini and pick up an elementary physics textbook, it will tell you that I = Σmr², as every schookid knows.
Why do you waste your time consulting a long-winded mechanical idiot?

[hamdani yusuf (OP)]

If you put down Gemini and pick up an elementary physics textbook, it will tell you that I = Σmr², as every schookid knows.
Why do you waste your time consulting a long-winded mechanical idiot?

That's basically what it said.

The moment of inertia of a rigid object is derived from its mass distribution and axis of rotation. It is a measure of an object's resistance to changes in its rotational motion. The moment of inertia of a point mass is given by I = mr^2, where m is the mass of the object and r is the perpendicular distance from the axis of rotation.
The moment of inertia of a rigid object is the sum of the moments of inertia of all of its constituent point masses.

[hamdani yusuf (OP)]

It's worth noting that radius of rotation can be different from radius of the geometrical object, as shown in the example. Instead of just meter, radius of rotation is better expressed in meter per radian. The meter is for the arc length traveled by the rotation, while the radian is for angle of rotation.

Here are some other examples.

[hamdani yusuf (OP)]

How does it explain this equation?

Up to the elastic limit of a bolt, the tightening torque τ is a linear-ish function of θ, so the integral from 0 to θ equals the final torque.

If the torque starts with 0 and ends with τ in linear increment, the integral is θτ/2.

[hamdani yusuf (OP)]

Here's an example to distinguish between torque and force.

A hammer-like metal object consists of a cube sandwiched between a pair of half cylinders. It's equipped with a handle with negligible mass. A force is applied to the end of the handle.

When the floor is frictionless, the force will only make the hammer slide horizontally, which means the net torque is zero.
But when the floor has high friction, the hammer can tumble, which means that the hammer has been rotated.

It's worth noting that radius of rotation can be different from radius of the geometrical object, as shown in the example. Instead of just meter, radius of rotation is better expressed in meter per radian. The meter is for the arc length traveled by the rotation, while the radian is for angle of rotation.

This example is a case where radius of rotation isn't a constant through out the rotation.

[alancalverd]

If the torque starts with 0 and ends with τ in linear increment, the integral is θτ/2.

This would of course be the work done against friction, not the tightening torque of a clean bolt.

[hamdani yusuf (OP)]

If the torque starts with 0 and ends with τ in linear increment, the integral is θτ/2.

This would of course be the work done against friction, not the tightening torque of a clean bolt.

To eliminate the factor of 1/2, the torque must be constant throughout the tightening, which is not realistic in how a torque wrench works.

[Bored chemist]

If the torque starts with 0 and ends with τ in linear increment, the integral is θτ/2.

This would of course be the work done against friction, not the tightening torque of a clean bolt.

Ignoring friction, a nut and bolt is just a complicated way of stretching a spring.
But if you ignore friction, it's not much use.

[alancalverd]

Good engineers like to use a clean bolt or lubricate an old one to minimise sliding friction and maximise the value of our sweat in tensioning the bolt. If you are really fussy you can use a locking washer to stop it unwinding!

[Bored chemist]

Without friction, a bolt simply wouldn't hold.
So the analysis of what work is done in tightening a bolt is rather complicated.
As the bolt tightens, the force holding the sliding faces together rises and increases the frictional force;  and the force that is needed to increase the tension in the bolt by stretching it also increases.
One of those is a conservative force and the other isn't.

[alancalverd]

Without friction, a bolt simply wouldn't hold.

Unless you use a cotter pin or a tab washer. Where the structure is subject to a lot of vibration, you can't rely on inter-face friction to stop it unwinding. We also use them where there is negligible tightening force, i.e. where the bolt shank is actually the pivot of a hinge.

[hamdani yusuf (OP)]

Good engineers like to use a clean bolt or lubricate an old one to minimise sliding friction and maximise the value of our sweat in tensioning the bolt. If you are really fussy you can use a locking washer to stop it unwinding!

Thread-locking fluid or threadlocker is a single-component adhesive, applied to the threads of fasteners such as screws and bolts to prevent loosening, leakage, and corrosion.
https://en.wikipedia.org/wiki/Thread-locking_fluid

[hamdani yusuf (OP)]

Concept of torque is not limited to bolt tightening, which must stop at some maximum value. Other rotating equipment like pumps, mixers, and escalators also use the same concept. We need to understand the concept in a more general usage.

[Bored chemist]

We need to understand the concept in a more general usage.

As I pointed out, people already do.

These may cause our understanding on the concept of torque incomplete.

Unless you ask someone who talks about car engines.
https://spicerparts.com/calculators/horsepower-torque-calculator
The units there are particularly scrambled.
But it shows that they don't just use torque for bolts.

[hamdani yusuf (OP)]

As I pointed out, people already do.

Some of us haven't. I just tried to help them out by examining the same concept from different angles. Who knows one of them can be easier to understand because it's more familiar to them.

[alancalverd]

You are just adding to your own confusion and helping nobody. Force x moment arm = torque. What's the problem?