[alancalverd]

Rotational inertia is the same as when it is stationary: I = 0.4 mr² for a homogeneous solid sphere.  Since it is not a function of ω or θ, I prefer the term "moment of inertia" as less likely to confuse a student.

Rotational kinetic energy is 0.5 I ω² = 1J

You know ω = 2π rad/sec so you can deduce I and hence its angular momentum I ω.

[hamdani yusuf (OP)]

The same as when it is stationary: I = 0.4 mr² for a homogeneous solid sphere. 

Your problem is in guessing that rotational inertia is an analog of momentum, which is why the phrase "moment of inertia" is less confusing.

The homogeneity of the ball wasn't specified in the question. Nor its radius. What's known is the angular velocity and kinetic energy.

Let me remind you that rotational inertia is analogous to linear inertia, or inertial mass. It's clearly shown in the comparison table.

[alancalverd]

See my modified and expanded reply above.

Since I = ∑mr², you can calculate it for any distribution of mass about any point you choose. I just gave a solid homogeneous sphere as an example.

[paul cotter]

"The force is proportional to the rotational angle. But the energy is proportional to the rotational angle squared"                Agreed, but we are not discussing energy, we are talking about torsional stiffness. This is just a pointless digression. The fact remains that your interpretation of torsional stiffness, ie Nm/rad squared, cannot be correct and hence torque being described as Nm/rad is equally incorrect. No amount of digression into parameters such as work , energy or moment of inertia can rescue your spurious analysis.

[hamdani yusuf (OP)]

Here's the updated table for proposed new standard units, now including kinetic and potential energy, as well as torsional stiffness.


If you can find inconsistency in this table, please let me know.

[alancalverd]

Your formula for rotational inertia is wrong and misguided, as are all the following formuale that mention rad or rad^2.

The question you must ask in each case is "what is the value when nothing moves?". If it tends to infinity, you are talking nonsense. 

[paul cotter]

Remove all "rads" and the table becomes consistent.

[hamdani yusuf (OP)]

Remove all "rads" and the table becomes consistent.

What's your unit for rotational angle, angular velocity, and angular acceleration?

[hamdani yusuf (OP)]

"The force is proportional to the rotational angle. But the energy is proportional to the rotational angle squared"                Agreed, but we are not discussing energy, we are talking about torsional stiffness. This is just a pointless digression. The fact remains that your interpretation of torsional stiffness, ie Nm/rad squared, cannot be correct and hence torque being described as Nm/rad is equally incorrect. No amount of digression into parameters such as work , energy or moment of inertia can rescue your spurious analysis.

In a linear system, its kinetic energy is half of mass times velocity squared, while the potential energy in a linear spring is half of spring constant times displacement squared.

In a rotational system, its kinetic energy is half of rotational inertia times angular velocity squared. While potential energy of a circular spring is half of its torsional stiffness times displacement angle squared.
Thus the unit for torsional stiffness must be the unit for energy divided by angle squared.

[paul cotter]

Your errors have been explained in a manner a child could understand and yet you double down on these errors and refuse to learn. This is, to me, a really sad situation. 

[hamdani yusuf (OP)]

Your errors have been explained in a manner a child could understand and yet you double down on these errors and refuse to learn. This is, to me, a really sad situation. 

Can you identify what is my biggest error?
What is the correct information?
Do you still think that the unit radian should be removed?

Remove all "rads" and the table becomes consistent.

What's your unit for rotational angle, angular velocity, and angular acceleration?

[hamdani yusuf (OP)]

I proposed a new standard unit for torque because I got inconsistent units when it was derived from different equations.

torque = force x rotational radius
current standard unit = Newton . meter = kg . meter^2 / second^2

torque = rotational inertia . angular acceleration
current standard unit = (kg . meter^2) . (radian / second^2)

torque = rotational work / angular displacement
current standard unit = Joule / radian = kg . meter^2 / (second^2 . radian)

If these inconsistencies don't bother you, then perhaps you don't need to make any change. You can just continue using current standard units as usual.

For those who are curious about this "perennial problem of teaching mechanics", and want to find a solution, you can read my proposed changes to standard units of rotational quantities as shown in the table above. The unit of torque is just one of several proposed changes. All of those changes are direct implications from simply changing the standard unit of rotational radius from meter to meter per radian, on the basis that the length is measuring the arc length of rotational displacement, while the radian is measuring the rotational angle produced by said rotational displacement.

This simple change brings back all the consistencies we all expected from a set of standard units. I realize that some of you might be reluctant to accept a change to something that you have already learned in most of your lifetime to the level of mastery. You might have even taught these current standard units with accompanying equations to many pupils. But life goes on, and if the new standard brings more benefits than its costs, then sooner or later someone will use and adopt it. And if it shows some competitive advantage, then it will be inevitable that everyone will follow suit.

So, my next target is to spread this information to wider audience, and let them make side by side comparison between the current standard and the proposed new standard. At least, the proportion of population who are ignorant to even the existence of this problem could be reduced. At least more people will put some efforts to solve it. If it turns out that there is even a better solution to this problem, I'll be glad to see it.

[hamdani yusuf (OP)]

Incidentally the quantity with SI units Nm/rad is known as the "torsional stiffness" of an object. It is the key characteristic of spiral springs, torsional suspensions, taut-band meters, and suchlike.

In the new proposed units, it would be Nm/rad^2.

Which would obviously be nonsense. The torsional equivalent of Hooke's Law makes force linearly proportional to deflection. 

What's obvious to you may not be obvious to someone else. What you think is obviously true might be considered obviously false by someone else. What's important is the justification for your conviction. In these cases at least one of you must have a false assumption.

When the heliocentric model was first proposed, it was thought to be obviously false by those who are familiar with geocentric model, because celestial objects were obviously revolving around the earth. Only further investigations can show that observations of objects in our solar system are more consistent with heliocentric model. Even after a few centuries, we can still find people who firmly believe in flat earth model, which is even more primitive than geocentric  spherical earth model.

So I guess expecting everyone to accept my proposed new standard right away is unrealistic.

So, my next target is to spread this information to wider audience, and let them make side by side comparison between the current standard and the proposed new standard. At least, the proportion of population who are ignorant to even the existence of this problem could be reduced. At least more people will put some efforts to solve it. If it turns out that there is even a better solution to this problem, I'll be glad to see it.

[alancalverd]

I proposed a new standard unit for torque because I got inconsistent units when it was derived from different equations.

You have misled yourself by including the word "rotational". You have then repeatedly failed to address the question of what is the value of applied torque if nothing moves.

There are no inconsistencies in newtonian mechanics, but, it seems, quite a bit of bad teaching.

[hamdani yusuf (OP)]

I proposed a new standard unit for torque because I got inconsistent units when it was derived from different equations.

You have misled yourself by including the word "rotational". You have then repeatedly failed to address the question of what is the value of applied torque if nothing moves.

There are no inconsistencies in newtonian mechanics, but, it seems, quite a bit of bad teaching.

You misled yourself by excluding the word rotational, while it is exactly the main topic of this thread. Torque, which is rotational force, is only one of many rotational quantities. It corresponds to force in linear quantities as shown by comparison tables by many scientific sources.

You repeatedly failed to identify what stopped the force from accelerating the object's rotation in the first place. Let me help you out. It can be the stiffness of the object, friction, or opposing field forces like gravity, electricity, and magnetic force. Or some combination of them.

You seemed to unconsciously imagined the object with infinite stiffness, where applying force doesn't change the shape of the object in any way. This unrealistic imagination has prevented you from understanding the more general and fundamental concepts of rotational quantities.

I proposed a new standard unit for torque because I got inconsistent units when it was derived from different equations.

torque = force x rotational radius
current standard unit = Newton . meter = kg . meter^2 / second^2

torque = rotational inertia . angular acceleration
current standard unit = (kg . meter^2) . (radian / second^2)

torque = rotational work / angular displacement
current standard unit = Joule / radian = kg . meter^2 / (second^2 . radian)

Why do you think these equations give different units for torque? Why should we choose one over the others?

[paul cotter]

Rubbish. Torsional stiffness can be easily shown to be linear in the variable of rotation and this introduces an inconsistency that you cannot overcome.

[hamdani yusuf (OP)]

Rubbish. Torsional stiffness can be easily shown to be linear in the variable of rotation and this introduces an inconsistency that you cannot overcome.

Linear in what respect?
All of the units of rotational quantities that I proposed are consistent with all other units of rotational quantities, based on all related equations. If they're somehow inconsistent with your understanding of one particular quantity, it's more likely that your understanding is incorrect.

[alancalverd]

Why do you think these equations give different units for torque? Why should we choose one over the others?

Because they display a deliberate misunderstanding of newtonian physics..

The first equation would be the correct defintion of torque if you removed the word "rotational"

The second equation describes the initial acceleration of a body that is free to rotate, but its layout implies that it is a definition of torque, which it isn't. α = dω/dt =τ/I maps effect to cause.

Work done against friction, or work done by a windlass, is again expressed as  effect ← cause , W = τθ.

The strength of these conventional equations is the fact that they remain true when θ = ω = α = 0

All of the units of rotational quantities that I proposed are consistent with all other units of rotational quantities,

and give absurd answers when you use them to determine e.g the force on a brake pad.

[hamdani yusuf (OP)]

The difference between rotational radius and geometric radius.

Rotational radius is the ratio between arclength rotational displacement of a point in a rotating system and the corresponding rotational displacement angle.

The displacement arc is the trajectory of a single point in the rotating system. It represents the position of the same point at different time.

Rotational radius is a vector.

On the other hand,
Geometric radius is the distance  between the perimeter and the center of a circle.

The perimeter is made of different points at the same time.

Geometric radius is a scalar.

The pictures below shows some different values of geometric radius in rounded squares.

[hamdani yusuf (OP)]

Why do you think these equations give different units for torque? Why should we choose one over the others?

Because they display a deliberate misunderstanding of newtonian physics..

The first equation would be the correct defintion of torque if you removed the word "rotational"

The second equation describes the initial acceleration of a body that is free to rotate, but its layout implies that it is a definition of torque, which it isn't. α = dω/dt =τ/I maps effect to cause.

Work done against friction, or work done by a windlass, is again expressed as  effect ← cause , W = τθ.

The strength of these conventional equations is the fact that they remain true when θ = ω = α = 0

All of the units of rotational quantities that I proposed are consistent with all other units of rotational quantities,

and give absurd answers when you use them to determine e.g the force on a brake pad.

If your definition of torque leads you to inconsistent units of rotational quantities, you must have defined it incorrectly.

If nothing is rotating, there's no rotational radius, ie. it's undefined.
For the frictional force to be generated, there must be a force acting on the object in the first place. Which mean some displacement must be in effect, due to finite stiffness of real objects, including the brake disc and pads.
If this displacement has zero curl, then rotational angle is zero, and rotational radius is infinite.

Newton's third law says that "To every action, there is always opposed an equal reaction". If your actions somehow different from the reaction, you must have missed something.