Does the Direction of Friction in Rolling Without Slipping Matter Physically?

[aaaa202]

I never really thought about it but it seems arbitrary in which way we calculate friction in terms of problems involving rolling without slipping. That is you will get the same results whether you say the friction helps your rotational acceleration or your translational acceleration.
For instance consider a cylinder rolling without friction due to some force being applied at R:
It is not hard to see you get the following equations for a (where f is the force due to friction):
F-f = ½ma
F+f = ma

Or if you chose the direction of the friction to be in the other direction:
F+f = ½ma
F-f = ma

It's not hard to see that these equations will give the same acceleration. Physically however I don't understand this. I would argue that because the "mass" is less in the equation with ½m it is beneficial to have the frictional force "help" you here rather than help the translational motion where the mass is bigger. What is wrong with my thinking??

 

[tiny-tim]

hi aaaa202!

It's not hard to see that these equations will give the same acceleration. Physically however I don't understand this.

they give the same acceleration because all you've done is to give the friction force a different name …

in one, you've called it f, and in the other, you've called the same force -f

obviously, that will give you the same value for acceleration (and a negative value for friction) ​

 

[aaaa202]

I have given a different name for it. Hmm okay but can't you say that I have reversed the direction of the friction force acting on the system? - so if it was before acting against the translation of the center of mass it is now helping it. If we say that f is the magnitude of the friction force acting on the system. And then my question on the physical intuition remains.

 

[tiny-tim]

… Hmm okay but can't you say that I have reversed the direction of the friction force acting on the system? - so if it was before acting against the translation of the center of mass it is now helping it.

no, you've only reversed the name

it's like calculating the currents in an electric circuit …

you arbitrarily call the currents I₁ I₂ I₃ etc, being careful to mark the direction of each current with an arrow …

you usually successfully guess the correct direction for each arrow, but if you're wrong it doesn't matter … that I simply comes out negative, and you know the current goes the opposite way to the arrow

 

[aaaa202]

okay but then what physical property assures that it has no effect to switch the minus sign?
I mean if you have something moving against friction then in general I would say the minus sign in Newtons law has an effect.
With the currents the physical property is clearly that ∫Edl=0 so Kirchoffs laws gives a kind of energy conservation which must always remain true - so a negative current would just yield a negative potential drop.
Here it would seem that the physical property is that the total work done on the cylinder remains the same. But again I am just back to the question: Do you think it PHYSICALLY makes sense that it doesn't matter if the frictional force is -f or f.
It seems you are going at a very mathematical thinking and indeed your reasoning is logical but I am trying to think in terms of the physical system and for that I am still curious why it wouldn't matter which of the parts of the motion the frictional force helps and works against.

 

[tiny-tim]

… It seems you are going at a very mathematical thinking and indeed your reasoning is logical but I am trying to think in terms of the physical system and for that I am still curious why it wouldn't matter which of the parts of the motion the frictional force helps and works against.

but what you're doing is mathematical …

you're re-defining something

you're not doing anything physical

(alternatively, if you insist on all forces being positive, if you define your friction the wrong way round, it comes out negative, which is impossible, thereby proving it was the wrong way round)