[alancalverd]

Then where does the radian go in the unit of torque?

The radian does not appear in the unit of torque, any more than the kilogram appears in the unit of currency.

If you want to twist a torsion bar, you need to apply a torque of x newton meters per radian that you want to twist it. If you want to buy apples they cost y dollars per kilogram that you want to buy.

[alancalverd]

Here's a list of units for angular distance,

"Angular distance" is meaningless. Angle is a measure of rotation, distance is a measure of linear separation.

[hamdani yusuf (OP)]

Here's a list of units for angular distance,

"Angular distance" is meaningless. Angle is a measure of rotation, distance is a measure of linear separation.

It's never too late to learn a new thing.

https://en.m.wikipedia.org/wiki/Angular_distance
Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors in three-dimensional space, or the central angle subtended by the radii through two points on a sphere. When the rays are lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation.

Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque.

[hamdani yusuf (OP)]

Then where does the radian go in the unit of torque?

The radian does not appear in the unit of torque, any more than the kilogram appears in the unit of currency.

If you want to twist a torsion bar, you need to apply a torque of x newton meters per radian that you want to twist it. If you want to buy apples they cost y dollars per kilogram that you want to buy.

Then why you use radian for unit of torque?

[hamdani yusuf (OP)]

The intended reference point was the vice screw at, say, r₁, the eventual  reference point was the foot of the bench at r₂. Ignoring the mass of the vice, if the bench length was L and it weighed B newton then the applied torque to lift it was Fr₂ > BL/2, roughly.

Human intentions should not determine the measure of a physical quantity. What if my intention were to bend the wrench half way?

[alancalverd]

Then why you use radian for unit of torque?

I didn't. A torsion bar requires a specified torque in order to twist a specified amount. The unit of torque is the newton meter, the unit of twist is a radian, so you specify the torsion bar (or clock spring, or torsion suspension of a galvanometer....)  as newtonmeters per radian (in a car suspension)  or, more likely when talking about the sensitivity of  taut-band meter (yes, I like analog displays!) , radians per newtonmeter.

[alancalverd]

Human intentions should not determine the measure of a physical quantity.

Of course they do. If my intention is to unscrew a bolt, the torque I apply will be quite different (like negative) from the torque I applied to tighten it.

What if my intention were to bend the wrench half way?

Then you'd need to increase B to prevent the bench lifting, so that BL > the limiting torque of the wrench.

[alancalverd]

Angular distance or angular separation is the measure of the angle between the orientation of two straight lines

In other words, angle. Why use two long words when the meaning is conveyed by one short word? 

Worse still, "angular distance" implies some measure of length such as 2r sin (θ/2) or rθ.

[hamdani yusuf (OP)]

I didn't.

If you want to twist a torsion bar, you need to apply a torque of x newton meters per radian

You can gaslight us in verbal communication. But it's harder when it's written.

[hamdani yusuf (OP)]

A torsion bar requires a specified torque in order to twist a specified amount. The unit of torque is the newton meter, the unit of twist is a radian, so you specify the torsion bar (or clock spring, or torsion suspension of a galvanometer....)  as newtonmeters per radian (in a car suspension)  or, more likely when talking about the sensitivity of  taut-band meter (yes, I like analog displays!) , radians per newtonmeter.

In Newton meter unit, the meter specifies the length of the trajectory in the direction of the force. If that's not the case, you need to determine the conversion factor. In the case of rotational motion, the trajectory is the arclength, which is equal to rotational angle times the radius of rotation. In other words, radius of rotation equals length of trajectory divided by angle of rotation. The numerical value of this conversion factor is exactly 1 only if the angle of rotation is measured in radian, and they use the same unit for length.

[alancalverd]

Torque has nothing to do with trajectory. It is simply the product of force x distance. The torque applied to a bolt can be the same whether the bolt rotates or not, depending on how rusty it is.

[Bored chemist]

Here's a list of units for angular distance, according to chatGPT.

Radians are commonly used in mathematics and physics, while degrees, arcminutes, and arcseconds are often used in astronomy and navigation.
Angular distance can be measured in rotations (revolutions, turns). Rotations are often used in engineering, mechanics, and rotational motion physics, especially when dealing with cycles and periodic motion (e.g., gears, wheels, and circular motion in physics).

Wiki does a lot better- it doesn't forget grads etc.
https://en.wikipedia.org/wiki/Angle#Units

Perhaps more interestingly, that page also reminds us that "In the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. "

[hamdani yusuf (OP)]

Torque has nothing to do with trajectory. It is simply the product of force x distance. The torque applied to a bolt can be the same whether the bolt rotates or not, depending on how rusty it is.

What happens to the equation
τ= I. α, when α=0?

Torque is defined as rotational counterpart of force.
F= m. a
τ= I. α

[hamdani yusuf (OP)]

Perhaps more interestingly, that page also reminds us that "In the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. "

Dimensionless doesn't mean unitless. 1 radian is different from 1 microradian. It's different from 1 rotation.

[alancalverd]

What happens to the equation
τ= I. α, when α=0?

Perhaps you can tell me, as it's your definition. I know for certain that α = 0 when the bolt is fully tightened, but I'm still applying τ > 0.

[hamdani yusuf (OP)]

Cursed Units 2: Curseder Units

Corrections: the force F in Ampere's law should be force per unit of length, and the denominator factor should be 2 instead of 4.

It describes rotational quantities at around 2:00 time stamp.

[alancalverd]

Not very good. kWh/1000 hours or kWh/year are useful units because we pay for electricity per kWh but we use appliances sporadically, and automatic appliances vary their consumption even when we aren't looking, so if you want to know how much it will cost you to run a freezer, heat your house, or keep the lights on at night, you need kWh/year.

Also much confusion between units and dimensions.

You would do better to buy a physics textbook rather than consult videos and chatbots that claim at the outset to be confused.  SI units make perfect sense to those who use them every day. Indeed so did CGS and Imperial units. All absolutely selfconsistent and logical. Imperial (plus US gallons) was good enough to fly to the moon - first time. SI eventually got machines to Mars and Venus.

[hamdani yusuf (OP)]

What happens to the equation
τ= I. α, when α=0?

Perhaps you can tell me, as it's your definition. I know for certain that α = 0 when the bolt is fully tightened, but I'm still applying τ > 0.

The equation has been around long before I was born.

[hamdani yusuf (OP)]

SI units make perfect sense to those who use them every day. Indeed so did CGS and Imperial units. All absolutely selfconsistent and logical. Imperial (plus US gallons) was good enough to fly to the moon - first time. SI eventually got machines to Mars and Venus.

You can cherish the good things in what we currently have. But don't forget to look for things that can be made better.
Standards are primarily intended for consistency.

[alancalverd]

The equation has been around long before I was born.

So if I apply a torque to a bolt that doesn't move, α= 0 so τ= 0? My torque wrench thinks not.

Standards are primarily intended for consistency.

And there seems to be some inconsistency here. Or maybe not?