[hamdani yusuf (OP)]

The fourth row shows standard unit for rotational inertia, which is currently set in kg.meter^2. But the equation related to torque and angular acceleration suggests that its unit is kg.meter^2/rad, which creates inconsistency.
You can continue for the next rows in the first table, where every row contains inconsistency.

In the second table, standard units for radius of rotation is set to meter per radian. Applying the equations, every row produce consistent results.

In the fourth row of second table, the proposed unit for rotational inertia is kg.meter^2/rad^2. It's consistent with the result from equation related to torque and angular acceleration. It's also consistent with the result from equation related to angular momentum and angular velocity.

Rotational kinetic energy is 1/2.I.ω^2. Thus
I = Ek/2/ω^2.
And the unit is Newton.sec^2/rad^2
Which is equal to kg.meter^2/rad^2.
This is consistent with other equations involving rotational inertia.

[alancalverd]

Which laws?

The rotational equivalent of Hooke's law, as I stated earlier. Torsional force is linear with deflection. That's how all the devices I listed, work.

[hamdani yusuf (OP)]

The new standard can also have the same benefit of brevity as option 0. Like the unit for power, which we usually state in Watt instead of Newton meter per second, we can introduce a new unit equals to Newton meter per radian.
What would it be? Here are some options.
Wenn
Woo
Wyy
Werr
You might see a pattern here.

Alternatively, we can use someone's name with significant contributions to the understanding of torque. According to various AI models, they are:
Archimedes
da Vinci
Newton
Leibniz
Euler
d'Alembert
Lagrange
Hamilton
Poisson
Thomson
Thompson

Various AI models consistently mentioned Archimedes as the earliest contributor to the concept of torque.

Archimedes explained the principle of leverage, which is closely related to torque. He understood that:

- The force applied to a lever, multiplied by the distance from the fulcrum (pivot point), determines the rotational force (torque).
- The ratio of the distances from the fulcrum to the points where the forces are applied determines the mechanical advantage of the lever.

In other words, Archimedes discovered that:

Torque = Force ? Distance

This fundamental principle is still widely used today in mechanics, engineering, and physics to calculate torque and understand rotational motion.

Archimedes' work on levers and torque was presented in his book "On the Equilibrium of Planes," which is considered one of the greatest works of ancient Greek mathematics and engineering.

There is no evidence that Archimedes mentioned a specific unit for torque.

In fact, the concept of torque as we understand it today, with its associated units, developed much later.

Archimedes did discuss the principles of leverage and rotational motion, but he didn't use the same mathematical framework or units that we use today to describe torque.

Meta AI

Unfortunately his explanation is limited to the concept of leverage or mechanical advantage, and didn't cover the whole range of modern understanding of torque and its relationship with other rotational quantities.

His practical approach was useful for some applications, thanks to its simplicity. But when it's used more generally to describe other rotational quantities such as rotational inertia, angular velocity, angular momentum, angular acceleration, rotational kinetic energy and power, we get inconsistency in their units. A rotational quantity can get different standard units if they were derived using different equations.

To be fair, it's not Archimedes' fault that caused this confusion. He only showed the equation, without mentioning the unit for torque. The unit for torque was set much later by people who developed the concept further. It's unfortunate that they naively used Archimedes' equation for torque to determine its unit without considering the types of its operands, as well as its implications to the units of other rotational quantities.

[hamdani yusuf (OP)]

Which laws?

The rotational equivalent of Hooke's law, as I stated earlier. Torsional force is linear with deflection. That's how all the devices I listed, work.

The proposed new standard units essentially come from distinction between rotational radius and geometric radius. By expressing rotational radius as length per rotational angle, we get the following results :
Rotational inertia = mass times radius of rotation squared = mass times (distance per rotational angle) squared
Angular momentum = momentum times radius of rotation = momentum times (distance per rotational angle)
Torque = force times radius of rotation = force times (distance per rotational angle)

Linear stiffness = force / linear displacement

Rotational stiffness = rotational force / rotational displacement
= Torsional stiffness = torque / displacement angle

Proposed unit for Torsional stiffness = (Nm/rad) / rad
= Nm/rad^2

[hamdani yusuf (OP)]

There's another difference between geometric radius and rotational radius. Geometric radius is a scalar, while rotational radius is a vector. So it's understandable that they have different units. As I proposed earlier, standard unit for geometric radius is meter, while standard unit for rotational radius is meter per radian.
This distinction has been shown to solve all inconsistencies in the units of rotational quantities derived from every valid equation.
People tend to forget that the operation used in the equation for torque is a cross product, which means that the operands are both vectors, including the radius of rotation.

The cross product is a mathematical operation that combines two vectors to produce another vector.

Given two vectors:

Vector A = (a1, a2, a3)
Vector B = (b1, b2, b3)

The cross product of A and B is:

A ? B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

The resulting vector:

- Is perpendicular to both A and B
- Has a magnitude equal to the area of the parallelogram formed by A and B
- Points in a direction determined by the right-hand rule

The cross product is used in various fields, including:

- Physics: to calculate torque, angular momentum, and magnetic fields
- Engineering: to describe rotational motion, forces, and stresses
- Computer graphics: to perform 3D transformations and calculations

Meta AI

[paul cotter]

Torsional stiffness is a linear function of rotation and you have it as quadratic. That is a major fail that you cannot square.

[alancalverd]

From the magic and utterly reliable AI source quoted in Reply #582

In other words, Archimedes discovered that:

Torque = Force x Distance

To nobody's surprise, that definition has been in current use for the last 2200 years and has allowed the  successful design and use of everything from a ship's capstan to space probes, Mars landers, hip replacements.......

Yes, force x distance / angular displacement is a useful concept, called torsional stiffness, also used in civil and mechanical engineering.

I strongly advise against pretending (never mind teaching) that the two are identical, because they aren't. 

[hamdani yusuf (OP)]

Torsional stiffness is a linear function of rotation and you have it as quadratic. That is a major fail that you cannot square.

You need to be careful in using unit for measurements. The first radian comes from radius of rotation. While the second one comes from increasing reactionary force as the rotational displacement increases.

[paul cotter]

You need to be careful in using unit for measurements. The first radian comes from radius of rotation. While the second one comes from increasing reactionary force as the rotational displacement increases.
??

[hamdani yusuf (OP)]

From the magic and utterly reliable AI source quoted in Reply #582

In other words, Archimedes discovered that:

Torque = Force x Distance

To nobody's surprise, that definition has been in current use for the last 2200 years and has allowed the  successful design and use of everything from a ship's capstan to space probes, Mars landers, hip replacements.......

Yes, force x distance / angular displacement is a useful concept, called torsional stiffness, also used in civil and mechanical engineering.

I strongly advise against pretending (never mind teaching) that the two are identical, because they aren't. 

The inconsistency found in current standard units of rotational quantities is essentially caused by treating a vector of rotational radius the same way as a scalar of geometric radius. The failure to make the distinction has caused the inconsistency, known as perennial problem in the teaching of mechanics.
In current practice, the common solution is to ignore the unit analysis altogether, and only assign the unit for the end result from a lookup table. Some of scientists have expressed their distaste of this.

[hamdani yusuf (OP)]

You need to be careful in using unit for measurements. The first radian comes from radius of rotation. While the second one comes from increasing reactionary force as the rotational displacement increases.
??

I've written the equations clearly, plain and simple that any average high school student will be able to understand. Tell me which part of them is still unclear to you?

Which laws?

The rotational equivalent of Hooke's law, as I stated earlier. Torsional force is linear with deflection. That's how all the devices I listed, work.

The proposed new standard units essentially come from distinction between rotational radius and geometric radius. By expressing rotational radius as length per rotational angle, we get the following results :
Rotational inertia = mass times radius of rotation squared = mass times (distance per rotational angle) squared
Angular momentum = momentum times radius of rotation = momentum times (distance per rotational angle)
Torque = force times radius of rotation = force times (distance per rotational angle)

Linear stiffness = force / linear displacement

Rotational stiffness = rotational force / rotational displacement
= Torsional stiffness = torque / displacement angle

Proposed unit for Torsional stiffness = (Nm/rad) / rad
= Nm/rad^2

[hamdani yusuf (OP)]

Torsional stiffness is a linear function of rotation and you have it as quadratic. That is a major fail that you cannot square.

You can also find squared unit of distance in the unit for work, even though it is linear with the displacement. It renders your objection invalid.

[hamdani yusuf (OP)]

...

Yes, force x distance / angular displacement is a useful concept, called torsional stiffness, also used in civil and mechanical engineering.

I strongly advise against pretending (never mind teaching) that the two are identical, because they aren't. 

I also distinguish between them. You seemed to be missing my previous post.

Linear stiffness = force / linear displacement

Rotational stiffness = rotational force / rotational displacement
= Torsional stiffness = torque / displacement angle

Proposed unit for Torsional stiffness = (Nm/rad) / rad
= Nm/rad^2

In case you've already forgotten, my proposed unit for torque is Nm/rad, which is different from my Proposed unit for Torsional stiffness, Nm/rad^2

Some of you might feel that I am being too repetitive, by restating things that you already understand. But bear with me, since different people have different learning rate.

[hamdani yusuf (OP)]

According to their awareness of this problem, people can be classified into some categories.
1. Those who are completely ignorant of the problem. Most kids and illiterate people fall into this category. Until high school, I was also included here.
2. Those who are aware of the problem, but haven't found the solution. I was here until a few years ago.
3.Those who are aware of the problem, as well as the solution. Currently, it's the fewest.

By spreading the information about this problem and solution through social media, I hope to change the composition in the classification above to reduce the proportion of people in the first category, and at least increase the proportion of people in the second category.

[hamdani yusuf (OP)]

What is the time derivative of angular momentum?

irrelevant in the case of the parking brake since the angular momentum is zero.

For those who doesn't know it yet,
the time derivative of angular momentum is called torque.

[hamdani yusuf (OP)]

It seems like we also need to distinguish between individual torque and action-reaction pair of torque. They are analogous to tension and compression in linear motion.

Tension is the pulling or stretching force transmitted axially along an object such as a string, rope, chain, rod, truss member, or other object, so as to stretch or pull apart the object. In terms of force, it is the opposite of compression. Tension might also be described as the action-reaction pair of forces acting at each end of an object.

https://en.m.wikipedia.org/wiki/Tension_(physics)

Our errors can be classified as false positive and false negative. We can make erroneous distinction between two identical concepts. We can also fail to distinguish between two different concepts.
Let me first describe the linear force because it is simpler and easier to understand. Force is needed in some cases, which can be classified as follow.

1. From Newton first law, to change velocity of an object. Rate of change in velocity is called acceleration. Assuming that mass is constant, Force equals acceleration times mass of the accelerating object.
In the more general case where the mass doesn't have to be constant, Force is equal to rate of change in momentum.
F = dp/dt

[hamdani yusuf (OP)]

2. To maintain velocity of an object under the effects of friction, or other non-conservative effects. Cruising terrestrial vehicles are some examples.

[hamdani yusuf (OP)]

3. To increase the potential energy of an object. Like lifting things against gravitational field. Compressing a spring or other elastic objects.

In contrast, the first category described change in kinetic energy of an object.

[alancalverd]

For those who doesn't know it yet,
the time derivative of angular momentum is called torque.

So when you rotate an object at constant angular velocity against friction, the applied torque is zero.

Great! Free heat!

By simply inventing an absurd definition of a term that everyone else understands, you have solved the world's energy problems and destroyed the economies of several evil dictatorships!

[paul cotter]

Specifying torsional stiffness as Nm/rad squared will produce multiple errors in any subsequent application. It is a linear function of rotation, not a quadratic. I have to come to the unfortunate conclusion that Hamdani is either unwilling or incapable of learning.