What Is Violated in Deriving Torque Equation for Rotations?

[aaaa202]

I am at the moment trying to do something different in the classical mathematical derivations of the equations for rotations and trying to find out why this would not yield a correct result.
This time I do this:

Consider an arbitrarily shaped rigid body rotating about an axis. Let us find out a relationship between its angular acceleration and the external forces acting on it.

For a small masselement of mass m_i we note that the tangential force on it is given by:

F_tan-i = m_ir_iα
To derive the torque equation you normally multiply by r_i and sum up all the torques to get:
Ʃτ = Iα
You can trust that Ʃτ is the sum of external torques only since a simple argument shows that all internal torques cancel.

But let us instead multiply by r_i² and sum up to get:

ƩF_tan-ir_i² = (Ʃm_ir_i³)α

Then you can still use the same argument for the internal quantities defined by the right hand side cancelling out. Yet, it does not yield the same result as the correct equation Ʃτ = Iα.
What is violated by this approach?

 

[arildno]

What is violated here is that in general, torquesare VECTORIAL quantities, and it is basically meaningless to multiply with the scalar r^2.

You are working with the special one-dimensional case, and hence this is not clear to you.

You are led astray by the SIMPLIFICATION of a general principle, and I advise you to simply accept, for now, that we define the quantities we do for GOOD REASONS but that those good reasons are best understood on a more advanced level.

 

[aaaa202]

More advanced levels as in what? Lagrangian mechanics and the fact that there must be exactly 2 conserved quantities in rotational systems?
I can see that the torques above would violate energy conservation, so I guess that is an argument. Also it would be meaningless to define a crossproduct with r² since you can't square a vector and get a vector if that is what you mean.
But in that case, do you know where I can find a more general derivation of Newtons second law for rotating systems? I study at the university and should be able to understand it, yet my book (Y&F - University physics) is very qualitative and pedagogical with little focus on making derivations as general as possible.

 

[arildno]

More advanced levels as in what? Lagrangian mechanics and the fact that there must be exactly 2 conserved quantities in rotational systems?
I can see that the torques above would violate energy conservation, so I guess that is an argument. Also it would be meaningless to define a crossproduct with r² since you can't square a vector and get a vector if that is what you mean.
But in that case, do you know where I can find a more general derivation of Newtons second law for rotating systems? I study at the university and should be able to understand it, yet my book (Y&F - University physics) is very qualitative and pedagogical with little focus on making derivations as general as possible.

Derivation??
What are you talking about??

We DEFINE a number a quantities in physics, because they are..USEFUL.

If you can show that your alternate ideas includes additional information or predictions than the ones used, then go ahead.

Or are you questioning the formal mathematical validity here?

 

[aaaa202]

Yes I am questioning the mathematical validity - or rather why the other mathematical approach does not work. As you can see it gives an expression for the angular velocity in terms of the sum of forces times the distance from the rotation centre at which they are applied, where you square that. A simple argument shows that the internal ones must cancel out. Yet something must be wrong since it is not mathematically equivalent to Ʃτ = Iα. What goes wrong?

 

[voko]

The reason for this is simple. It is because \frac {d} {dt} \vec{r} \times m\vec{v} = \vec{v} \times m\vec{v} + \vec{r} \times m\vec{a} = \vec{r} \times m\vec{a} = \vec{r} \times \vec{F} Which means we can relate the change of angular momentum with the torque. While with r^2 we would not have anything of the kind.

I suggest you get a good book on mechanics. Some are even available online. For example: http://archive.org/details/Mechanics

 

[aaaa202]

But don't you agree that for the derivation above, the logic could just as well be followed if you multiplied by r_i² as if you multiplied by r_i as you are to do. I will show you the specific derivation and you can tell me if it would violate any of the steps if you did that.
My problem is not that I don't understand the correct equations. I know everything about conservation of energy, torques, conservation of angular momentum etc. I'm just curious if that system of equations are the only ones that make up a system that can describe the world mathematically.

 

[gabbagabbahey]

The equation \sum \tau = I\alpha is a consequence of the way Torque and Moment of Inertia are defined. If you redefine Torque as something else, and don't redefine Moment of Inertia, why would you expect the equation to still hold true?

 

[aaaa202]

Well would my defining of the new quantity Ʃm_ir_i³ not make up for that problem?

But maybe all these speculations are lunatic. After all the definition of torque is the only thing that doesn't violate the fact that the work done by the rotating force should equal the change in kinetic energy:
Frdθ = ½I(ω_f²-ω_i²) = Iαdθ
=>
Fr = Iα
So I guess that somehow establishes a unique and crucial link between translation and rotation.

But it still bothers me that I always in derivations find quantities that would fit just as well into the logic and can't explain why they aren't instead used.

 

[gabbagabbahey]

But it still bothers me that I always in derivations find quantities that would fit just as well into the logic and can't explain why they aren't instead used.

Some of it is due to the usefulness of conserved quantities when it comes to analyzing physical systems and predicting their behaviour. For example, momentum is always conserved as it is conventionally defined, and conservation of momentum is frequently used to analyze physical systems (especially collisions!), so defining it in some other way would probably be less useful. Since conserved quantities are so useful, defining other quantities which can be easily related to conserved quantities can also be useful. Take, for example, the relationship between torque and angular momentum. Angular momentum as conventionally defined is conserved when there are no external forces present (and changing its definition to include an extra factor of |r| would change that!), so defining torque as the time rate of change of angular momentum makes it useful.

Some of it just boils down to convention, which is useful for allowing physicists to communicate effectively with each other.