[hamdani yusuf (OP)]

Let's compare another pair of rotational quantity and its linear quantity counterpart.
Angular velocity is measured in radian per second.
Linear velocity is measured in meter per second.
The conversion factor is radian per meter.

[hamdani yusuf (OP)]

Another important quantity to describe rotational motion is angular momentum.
Angular momentum is also related to linear momentum.
For a point particle moving with linear momentum p at a distance r from a reference point, the angular momentum L relative to that point is given by the cross product:
L = r x p.
The formula above is similar to relationship between torque (rotational force) and linear force.

[alancalverd]

Tighten your bolt to 10 Nm/rad. Now apply a torque of 8 Nm/rad. Does the bolt turn? No. So what is the meaning of rad in this context?

[hamdani yusuf (OP)]

Tighten your bolt to 10 Nm/rad. Now apply a torque of 8 Nm/rad. Does the bolt turn? No. So what is the meaning of rad in this context?

What's the NET torque? If it's zero, it won't change its angular velocity.

To solve your confusion, let's start from the most basic relationship between angular and linear motion, which is distance. Linear distance is measured in meter, while angular distance is measured in radian. Thus, the convertion factor is radian per meter.
Torque is angular force. Linear force is measured in Newton. To convert it to angular force, divide it by the convertion factor. Thus the unit of torque is Newton times meter per radian.

[hamdani yusuf (OP)]

The table below shows the comparison between angular and linear quantities.

Here are the equations conversion, where d = arclength of the circumference corresponding to rotational angle.
θ = d * (θ/d)
ω = v * (θ/d)
α = a * (θ/d)
I = m * (d/θ)^2
τ = F * (d/θ)
L = p * (d/θ)

[paul cotter]

NONSENSE. Alan is not confused, you are.

[hamdani yusuf (OP)]

NONSENSE. Alan is not confused, you are.

Can you help him answer my questions?

Tighten your bolt to 10 Nm/rad. Now apply a torque of 8 Nm/rad. Does the bolt turn? No. So what is the meaning of rad in this context?

What's the NET torque? If it's zero, it won't change its angular velocity.

Do you agree with my post here?

The table below shows the comparison between angular and linear quantities.

Here are the equations conversion, where d = arclength of the circumference corresponding to rotational angle.
θ = d * (θ/d)
ω = v * (θ/d)
α = a * (θ/d)
I = m * (d/θ)^2
τ = F * (d/θ)
L = p * (d/θ)

[Bored chemist]

And also, I don't trust a system which can't count the Rs in "strawberry".

I don't either. Perhaps you used an outdated version.

How do you know that the version you use is sufficiently up to date not to produce nonsense?

[hamdani yusuf (OP)]

When asked to compare between Newton meters (Nm) and joules per radian (J/rad), Gemini answers.

Both Newton meters (Nm) and joules per radian (J/rad) are units of torque, but they emphasize different aspects of the concept.
Newton meter (Nm)
 * Focus: Force and distance
 * Definition: The torque produced by a force of 1 Newton applied at a distance of 1 meter from the axis of rotation.
 * Emphasis: Practical measurement of torque using a force and a lever arm.
Joule per radian (J/rad)
 * Focus: Energy and rotation
 * Definition: The torque required to rotate an object by 1 radian when the energy transferred is 1 joule.
 * Emphasis: Relationship between torque and energy in rotational motion.
Comparison
 * Dimensional equivalence: Both Nm and J/rad are dimensionally equivalent, meaning they can be converted to each other using a constant factor (1 Nm = 1 J/rad).
 * Conceptual difference: Nm highlights the force and distance involved in creating torque, while J/rad emphasizes the energy required for a certain amount of rotation.
 * Usage: Nm is more commonly used in practical applications, such as specifying the torque of a motor or tightening bolts. J/rad is more often used in theoretical analysis of rotational dynamics.
Analogy
Think of it like this:
 * Nm: How hard you push on a wrench and how long the wrench is.
 * J/rad: How much energy it takes to turn a screw a certain amount.
Both describe the same thing (torque), but from different perspectives.

[hamdani yusuf (OP)]

And also, I don't trust a system which can't count the Rs in "strawberry".

I don't either. Perhaps you used an outdated version.

How do you know that the version you use is sufficiently up to date not to produce nonsense?

The model I use can correctly answer the number of r.

[Bored chemist]

And also, I don't trust a system which can't count the Rs in "strawberry".

I don't either. Perhaps you used an outdated version.

How do you know that the version you use is sufficiently up to date not to produce nonsense?

The model I use can correctly answer the number of r.

And do you think that's proof that all it's errors have been fixed?

[hamdani yusuf (OP)]

And do you think that's proof that all it's errors have been fixed?

It's an evidence that when errors are found, they will be quickly fixed.

[Bored chemist]

It gets better; I just checked...
Guess what I found.

gemini error.png (17.19 kB . 1184x314 - viewed 376 times)

[hamdani yusuf (OP)]

Conceptual difference: Nm highlights the force and distance involved in creating torque, while J/rad emphasizes the energy required for a certain amount of rotation.

My focus here is not in involving energy specifically. But to explicitly write the unit of angular rotation, especially when it's not measured in radian.
Just look at all relationships between linear and angular quantities, also among angular quantities themselves. Removing the unit of angle doesn't make sense for angular distance, velocity, nor acceleration. Even though angle is a dimensionless quantity.

[hamdani yusuf (OP)]

It gets better; I just checked...
Guess what I found.

gemini error.png (17.19 kB . 1184x314 - viewed 376 times)

Have you tried Gemini advanced?
Or try other models, such as Meta AI, or Deep Seek.

[hamdani yusuf (OP)]

In linear motion, F=m.a
Analogous for angular motion,
τ = I.α
Unit of angle must be explicitly stated in α. Why shouldn't it in the other quantities?

[paul cotter]

A linear force may or may not produce motion- even if it does not produce motion the force is still present. Similarly a torque may or may not produce rotation and in the absence of rotation the torque is still present. In the absence of rotation what possible meaning or application is there for the rad? If torque produces rotation one can calculate the energy expended by the torque over the rotation angle but this is the only time the rotated angle has significance and it does arise in specifying the torque.

[hamdani yusuf (OP)]

In linear motion, p = m.v
momentum = mass . velocity
Analogous for angular motion,
L = I.ω
Unit of angle must be explicitly stated in ω. Why shouldn't it in the other quantities?
Can angular velocity be stated without explicitly mention the unit of rotational angle?

[hamdani yusuf (OP)]

In the absence of rotation what possible meaning or application is there for the rad?

Rad comes from the ratio between arc length of a circle and it's radius.
That's why it was given no dimension. When the arc length of distance is zero, the rotational angle is zero. Except if the radius is also zero.

Especially in levers, measuring radius is much easier than measuring the arc length or rotational angle.

[paul cotter]

We know what a rad is, I used the term because you were using it. There is no angle in torque.