[hamdani yusuf (OP)]

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

The problem is like the title of this thread. What's the unit of torque? In Newton meter, how do we know that the distance is perpendicular instead of parallel to the force?
In Joule per radian, we know that it comes as the result of dot product between the force and distance of movement, which means they're parallel to each other. The radius of rotation can be calculated by dividing the arc length by angle of rotation.

[hamdani yusuf (OP)]

Which part of my post is confusing to you?

Now it is obvious that the tightening  torque at any point equals the elastic energy in the ring.

What is the ratio between them?

1:1, thanks to the  principle of conservation of energy.

Your post above clearly shows your confusion. Even if I end up fail to help you, I still hope I can prevent someone else from making a similar mistake.

[alancalverd]

In Newton meter, how do we know that the distance is perpendicular instead of parallel to the force?

Because we say "the applied torque was X Nm" or "the energy expended was Y joules" depending on what we mean.

[hamdani yusuf (OP)]

In Newton meter, how do we know that the distance is perpendicular instead of parallel to the force?

Because we say "the applied torque was X Nm" or "the energy expended was Y joules" depending on what we mean.

You need to add the word "torque" to specify that the meter is perpendicular to the Newton. Its information is less dense than Joule per radian, which already contains information about direction.

[alancalverd]

Fortunately most people to whom it matters, know what torque means. It's the reason than a set of spanners has their lengths proportional to the nut diameter, and torque wrenches are the length they are.

Joule per radian contains no information about direction, does not tell you how hard to pull on the wrench, and would include all the irrelevant frictional losses  up to the point where the tightening torque is what matters. And how big is a radian? How do you measure it on a torque wrench? I've seen vehicle maintenance specifications that say "hand tight plus 45 degrees" which is adequately precise and easy to achieve

.

Your post above clearly shows your confusion. Even if I end up fail to help you, I still hope I can prevent someone else from making a similar mistake.

Are you suggesting that conservation of energy is a mistake? Now that would really be a breakthrough!

[Bored chemist]

how do we know that the distance is perpendicular instead of parallel to the force?

Context.

[hamdani yusuf (OP)]

1:1, thanks to the  principle of conservation of energy.

You might have meant different things, but your answer there literally means that 1 Nm of energy equals 1 Nm of torque.

[hamdani yusuf (OP)]

Joule per radian contains no information about direction,

Joule doesn't have direction, but radian does.

[hamdani yusuf (OP)]

does not tell you how hard to pull on the wrench,

It does. The more energy you must exert to turn the wrench by an amount of rotational angle means the harder you have to pull it.

[hamdani yusuf (OP)]

would include all the irrelevant frictional losses  up to the point where the tightening torque is what matters.

Just exclude all the irrelevant frictional losses  up to the point where the tightening torque is what matters.

[hamdani yusuf (OP)]

How do you measure it on a torque wrench?

You can just read the display, if you use electronic torque wrench.

[alancalverd]

You have hand-turned your frictionless bolt into contact with the flange surface. No work has been done in this perfect universe. Now we tighten it. The torque wrench either clicks at or indicates the maximum applied moment that will not cause it to rotate. So θ is irrelevant.

If a massive body is free to rotate, then τ = I d²θ/dt² and if we apply an impulse then τθ = Iω, all by analogy with linear motion.

This is all perfectly straightforward. What's your problem?

[alancalverd]

You can just read the display, if you use electronic torque wrench.

I haven't seem a torque wrench that integrates τθ, nor one that measures θ.

[hamdani yusuf (OP)]

You can just read the display, if you use electronic torque wrench.

I haven't seem a torque wrench that integrates τθ, nor one that measures θ.

https://en.m.wikipedia.org/wiki/Torque_wrench
They essentially measure the angle between the head and the handle.

https://en.m.wikipedia.org/wiki/Battery_torque_wrench

[alancalverd]

But not the rotation of the object being torqued. And a click wrench doesn't indicate any angle at all. 

[hamdani yusuf (OP)]

But not the rotation of the object being torqued. And a click wrench doesn't indicate any angle at all. 

It does. The click indicates that the angle between the head and the handle has exceeded a threshold.

[alancalverd]

No. It indicates that the applied torque has exceeded the threshold, at which point the angle increases with no further increase in torque.

[hamdani yusuf (OP)]

No. It indicates that the applied torque has exceeded the threshold, at which point the angle increases with no further increase in torque.

Which angle are you talking about?

the angle between the head and the handle

Both are parts of the torque wrench.

[hamdani yusuf (OP)]

In the later picture, clicking means that the angle between head and handle has exceeded 30 degrees.

[alancalverd]

No, no, no

In the first picture, the angle indicates the applied torque, i.e. the flexion of the spring steel shaft, not the angle through which the bolt has been rotated.

In the second picture the head-handle angle remains at  < 30 deg until the applied torque exceeds the preset value, at which point the handle clicks through at least a further 30 degrees. This could be alarming and dangerous in a cramped working environment - my preset "clicker" doesn't have a hexagonal cam, just a single cam and a limit of about 5 degrees.

Don't confuse the torque indicator or limiter with the ratchet mechanism!