[hamdani yusuf (OP)]

I think you (and maybe others) have been misled by the notion of "rotational analogs of linear quantities".

I think you (and maybe others) have been misled by the deceptively simple formula for torque, without considering that it's correlated to other rotational quantities. As I mentioned in my video, torque is only one third of rotational quantities with problematic standard units. The other two are angular momentum and rotational inertia.

These three proposed units for new standard of rotational quantities are simply a logical consequences from changing the standard unit of rotational radius to meter per radian.

I marked this post as the best answer because it contains the key to solve the perennial problem of teaching mechanics. It requires acknowledging the difference between rotational radius and geometric radius, as well as their physical units.

[paul cotter]

No. Torque can have any arbitrary value in the absence of rotation. Mechanics has a long history of being taught in a most excellent manner.

[alancalverd]

Once again,  you have failed to answer the  simplest of questions by the application of your proposed new quantity.

This one?

What is the time derivative of angular momentum?
What is R when nothing is rotating?
BTW, 0! = 1, by definition.

No, the force required on the brake pad to prevent the car from rolling.

[alancalverd]

If  you want the moment of inertia to have the dimensions ML²θ^-2, what does this imply when θ = 0?

According to your system an object at rest cannot be rotated because it would have infinite rotational inertia. This means that we don't need parking brakes at all!

[alancalverd]

As for "acknowledging the difference between rotational and geometric radius" this is taught very effectively to people who play ball-and-stick games like cricket, baseball, or tennis, and also to the mechanics who balance your wheels in a tyre shop. What's the problem?

[alancalverd]

But here's the good news.

Elastic modulus, δ, is defined as stress/strain. This has obvious applications and units where the stress is linear, such as a stretching or shearing force, but it is also useful to assign a modulus to a torsion bar or clock spring, and that is indeed force x radius /angle, the torque required to produce unit rotation of the free end of the torsion element.

So Hamdani's unit does indeed have an application, from galvanometers and Cavendish's G balance (nanonewton meters per radian) to the suspension of the Leopard tank (mega....).

But it ain't the unit of torque!

[hamdani yusuf (OP)]

If  you want the moment of inertia to have the dimensions ML²θ^-2, what does this imply when θ = 0?

According to your system an object at rest cannot be rotated because it would have infinite rotational inertia. This means that we don't need parking brakes at all!

It implies that the object doesn't rotate.  Rotational distance d=0. Rotational radius = 0/0 which is indeterminate. It could be anything. This is consistent with the cases I mentioned earlier.
You can put a wedge under the tires and take an advantage of reactionary force.

What's the rotational inertia of a thin long stick? It depends on the position of the rotational axis.

[hamdani yusuf (OP)]

As for "acknowledging the difference between rotational and geometric radius" this is taught very effectively to people who play ball-and-stick games like cricket, baseball, or tennis, and also to the mechanics who balance your wheels in a tyre shop. What's the problem?

We need to assign different units for them, despite similar names.
Geometric radius has the same units as length.
Rotational radius has the same units as rotational arclength per rotational angle.
This distinction is necessary to get consistency in units of rotational quantities.

[alancalverd]

I can't get out of the car, run around it, and shove wedges under the tyres, without killing myself or crashing into the next car, if the slope exceeds  one or two degrees. Perhaps you have footmen waiting on every hill just in case you want to park there, but that's not how things work on my planet. You bring the car to a standstill with the footbrake (I won't ask you to do the calculation) then apply the parking brake.

The rotational radius of a centrally pivoted disc is, by symmetry, the same as its geometric radius. i.e. not zero.

So please, using your redefinition of torque, calculate the force between the brake pads and the disc.   

You can't. But the guys who designed my car could, using everyone else's definition of torque.

[alancalverd]

Wikipedia: The radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there. The radius of gyration has dimensions of distance [L] or [M⁰LT⁰] and the SI unit is the metre (m).


This is elementary school physics, and nobody but yourself finds it confusing or insufficient.

[hamdani yusuf (OP)]

Confusion, ad infinitum!

I'm sorry that I couldn't describe the problem simple enough for you to understand it. Let me try again.
Current standard units create inconsistencies when compared to the results from equations relating one rotational quantities to the others.



Compare them with the new proposed standard units, which are consistent with the relating equations.

For me, these tables show the simplest and most obvious way to describe the inconsistency in current standard units of rotational quantities. If you find any simpler or more obvious way, please let me know.
Power can also be expressed as torque times angular velocity. Their standard units must also be consistent.

[hamdani yusuf (OP)]

I can't get out of the car, run around it, and shove wedges under the tyres, without killing myself or crashing into the next car, if the slope exceeds  one or two degrees. Perhaps you have footmen waiting on every hill just in case you want to park there, but that's not how things work on my planet. You bring the car to a standstill with the footbrake (I won't ask you to do the calculation) then apply the parking brake.

The rotational radius of a centrally pivoted disc is, by symmetry, the same as its geometric radius. i.e. not zero.

So please, using your redefinition of torque, calculate the force between the brake pads and the disc.   

You can't. But the guys who designed my car could, using everyone else's definition of torque.

In case your brake is jammed, the excessive horizontal force can make the whole car to roll, instead of only the wheels. That's why the radius of rotation is undefined, until some rotation actually occurred, however small it is.

[hamdani yusuf (OP)]

∇

Wikipedia: The radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there. The radius of gyration has dimensions of distance [L] or [M⁰LT⁰] and the SI unit is the metre (m).


This is elementary school physics, and nobody but yourself finds it confusing or insufficient.

It seems like you haven't read the article from AAPT in my previous post, which described problems in current standard units of rotational quantities, especially the radian. Otherwise you won't say that I am the only one who can see the problem.

[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.

[alancalverd]

I did read the AAPT article and it is frankly rubbish. The radian has never been a problem in my learning, work, teaching or research, as long as it is properly taught. In fact the radian is a solution, not a problem, and neatly links physics and mathematics in a way that other measures of angle cannot.

There has been a misunderstanding caused by bad teaching of circular geometry at primary school level, if it is done in the context of elementary programming. A whole raft of kids believe that a circle consists of 360 straight lines, and whilst they might grow up to be fairly adequate navigators of small aircraft over short rhumb line distances, they will never understand science or engineering.

As I have pointed out throughout this debate, there is no problem in defining torque as force x distance, and you have discovered that problems arise if you want to invoke angular displacement in that definition as it prevents you from analysing stall or holding torque, two very important characteristics of real mechanical systems.

[hamdani yusuf (OP)]

I did read the AAPT article and it is frankly rubbish. The radian has never been a problem in my learning, work, teaching or research, as long as it is properly taught. In fact the radian is a solution, not a problem, and neatly links physics and mathematics in a way that other measures of angle cannot.

It means you've never been assigned with tasks that required consistency in units of rotational quantities.

[hamdani yusuf (OP)]

As I have pointed out throughout this debate, there is no problem in defining torque as force x distance,

You need to specify which distance. If it's the displacement affected by the force, you get work. The rotational radius is the ratio between the arclength displacement and rotational angle.
Multiplying the force with rotational radius equals to Multiplying the force with arclength displacement divided by rotational angle.

[hamdani yusuf (OP)]

When a wheel is reversing its direction of rotation, there will be an infinitesimally short moment where it's stationary. But the rate of change of angular momentum is not zero.
Thanks to L'Hopital's Rule which tells us that if we have an indeterminate form 0/0 or ∞/∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.
torque = Force . rotational distance / rotational angle
rotational radius = rotational distance / rotational angle
torque = Force . rotational radius

Now please answer the question, what is the radius of rotation, when nothing is rotating?

[hamdani yusuf (OP)]

Rolling car.
You can't take rotational radius for granted.

[alancalverd]

It means you've never been assigned with tasks that required consistency in units of rotational quantities.

On the contrary, I have used torsion balances and suspensions of various kinds, used torque wrenches to assemble clocks and trucks, measured stall torque of electric and pneumatic motors, been involved with dynamic balancing of cricket bats, wheels and propellors, applied in-flight countertorques to the various gyroscopic, vortex, thrust and differential drag torques associated with powered aircraft and gliders, and repaired and used the brakes on all sorts of road vehicles and aircraft.

At no time has it been appropriate to describe torque as anything but force x distance. A thorough understanding of torque is essential when taking off in a tail-heavy aircraft, recovering from a spin, or teaching the principles of attitude, turn, magnetic compass, direction indicator and tachometer instruments, d'Arsonval meters, piano tuners, or magnetic resonance imaging. So far, neither I nor any of my students, pianos or patients have suffered from the correct definition of torque.