Rolling Without Slipping

[kuruman]

You are right, of course. A ring has the maximum moment of inertia ($mR^2$) for any shape bounded by radius $R$.

However, there is nothing preventing us from contemplating a wheel rolling on an elevated track with a larger coaxial wheel attached to the same axle and extending below the tracks. We can get arbitrarily large $q$ by such measures. This sort of thing is almost the defining property of a yoyo.

Sure, that is why I specified rolling about the largest (outer) radius.

 

[TSny]

at $\kappa = 0$ as a special case, somewhere between $0 \leq q < \infty$, there is a flipping point of friction.

I suppose I'm at an impasse that these results are flawed unless someone can explain how there can be a $q$, effectively changing the mass of the yo yo that can make the wheel roll without any static fiction, on any selected mating surfaces (frictionless surfaces included), at any acceleration. Choose a mass such that it executes pure rolling without friction. Thats a paradox to me.

For $\boldsymbol{\kappa} \mathbf {= 0}$ (no force at the top), you should find that the friction force required for rolling without slipping is $$ f = \frac{5/7 - q}{q+1}F.$$ Positive $f$ corresponds to the friction force acting to the left in the same direction as $F$.
If $q$ happens to have the value 5/7, $f = 0$. So if $\kappa = 0$ and $q = 5/7$, no friction force is required for rolling without slipping.

We can see this from basic equations. For $\kappa = 0$ and no friction, $F$ is the only horizontal force. The acceleration of the center of mass is $$a_{cm} = \frac F M$$ and the angular acceleration of the wheel is $$\alpha = \frac {\tau_{cm}}{I_{cm}} = \frac{F R_1}{qMR_2^2}$$ Using $\dfrac F M = a_{cm}$, $R_1 = 5/7R_2$, and $q = 5/7$, this reduces to $\alpha = \dfrac{a_{cm}}{R_2}$. Or, $$a_{cm} = R_2 \alpha.$$ Integrating both sides with respect to time : $$v_{cm} = R_2 \omega.$$ This is the condition for rolling without slipping, and it is achieved without friction.

 

[erobz]

Would you mind explaining that "flipping point of friction" a little more detailed, please?

If we allow $\kappa$ to go to zero, the problem reduces to this situation. If you solve for friction force you will find that the is a very special $q$ that can be selected ( perhaps outside of the intent/scope of this problem ) such that the static friction can apparently be pointing to the left, disappear, and point to the right. Its a singularity.

 

[erobz]

This is the condition for rolling without slipping, and it is achieved without friction.

It's completely flawed. I can't trust the following flop either.

It implies that the micro surfaces suddenly begin to mesh perfectly at a special $q$ such that nary an horizontal impulse exist between the micro surfaces as they mate.

Rolling without slipping without static friction is certainly too much for me to swallow.

 

[jbriggs444]

It implies that the micro surfaces suddenly begin to mesh perfectly at a special $q$ such that nary an horizontal impulse exist between the micro surfaces as they mate.

What else are they supposed to do? In this case, we've arranged for no shearing force to exist. We've arranged for a normal force to exist. Naturally the surfaces are going to tend to mesh.

If you want to object that real contact like this will involve some dissipative interaction, that's true. But it is not at all what this exercise is about.

 

[erobz]

What else are they supposed to do? In this case, we've arranged for no shearing force to exist. We've arranged for a normal force to exist. Naturally the surfaces are going to tend to mesh.

I stress, that for this to be true they must not be receiving any net horizontal impulses from the micro surfaces as they mate. Why should the mass of the object parameterize when this singularity occurs for any mating set of micro surfaces? To me the result calls into question the existence of the directional flop described by the model. I think that is why talking about it is important.

 

[erobz]

Lets say that they don't perfectly mesh - in other words we tell the truth. What options could we have to talk about the "horizontal" micro impacts that we refer to as static friction in this case? Maybe we say every micro collision is elastic for this singularity, but then again, why should that depend on the mass of the object?

Taken to its limits it quantum mechanical in nature. I was just hoping there is something to sink ones teeth into in the classical "large scale" micro world - statistically speaking.

 

[erobz]

But it is not at all what this exercise is about.

My exercise here was to try to sus out my unrelenting objection. It consumes me sometimes. Also, I don't like it when people tell me (who I feel\think are my teachers) "It's obvious that ...yada yada". I have a bit of a problem with authority; a chip on my shoulder from past experiences with that phrase.

 

[A.T.]

I stress, that for this to be true they must not be receiving any net horizontal impulses from the micro surfaces as they mate. Why should the mass of the object parameterize when this singularity occurs for any mating set of micro surfaces?

It's not the mass, but the mass-distribution, that the parameter q describes here. It determines the ratio of angular to linear inertia.

And it's not a 'singularity'. Static friction, as function of q (with all other parameters constant) is a smooth, well behaved function at and around the point where static friction is zero.

Why q affects static friction in this particular scenario has been explained to you multiple times in this and the previous thread:
https://www.physicsforums.com/threads/rolling-without-slipping-problem.1078971/post-7247129
https://www.physicsforums.com/threads/rolling-without-slipping-problem.1078971/post-7247254

Here is another way to put it:

Static friction is a constraint force, which takes whatever direction and magnitude necessary (within certain limits) to constrain the kinematics in a certain way (prevent relative tangential motion at the contact). If other forces and moments, in combination with linear and angular inertia (here q comes in), already generate kinematics that satisfies the above constraint, then there is no need for any additional constraint forces, and static friction is zero.

To me the result calls into question the existence of the directional flop described by the model.

If your ideas about the micoscopic situation don't match this macroscopic result, then your micoscopic model is wrong, or you don't translate it to the macro scale correctly.

 

[erobz]

It's not the mass, but the mass-distribution, that the parameter q describes here. It determines the ratio of angular to linear inertia.

And it's not a 'singularity'. Static friction, as function of q (with all other parameters constant) is a smooth, well behaved function at and around the point where static friction is zero.

Why q affects static friction in this particular scenario has been explained to you multiple times in this and the previous thread:
https://www.physicsforums.com/threads/rolling-without-slipping-problem.1078971/post-7247129
https://www.physicsforums.com/threads/rolling-without-slipping-problem.1078971/post-7247254

Here is another way to put it:

Static friction is a constraint force, which takes whatever direction and magnitude necessary (within certain limits) to constrain the kinematics in a certain way (prevent relative tangential motion at the contact). If other forces and moments, in combination with linear and angular inertia (here q comes in), already generate kinematics that satisfies the above constraint, then there is no need for any additional constraint forces, and static friction is zero.


If your ideas about the micoscopic situation don't match this macroscopic result, then your micoscopic model is wrong, or you don't translate it to the macro scale correctly.

I consider pure rolling without slipping on a frictionless surface to be complete contradiction. I can select any material surfaces that mate. You are really going to tell me that I can construct a wheel that will execute pure rolling on a frictionless surface and/or I can construct a wheel that will make a frictionless surface. The wheel will roll and translate together, instead of rolling and translating independently in the limit as friction goes to zero. Yikes.

Let me see how this goes "well you can't select a frictionless surface and get a wheel to execute roll without slipping, but you can select a wheel and it will not use the friction that was required for the theory to begin with. Static friction clearly goes to zero when we put this special value in it- you can't argue with the model. No one has ever argued with a model in physics - it's just not acceptable"

 

[A.T.]

You are really going to tell me that I can construct a wheel that will execute pure rolling on a frictionless surface.

Just match the other external forces and moments to mass and moment of inertia, such that the resulting linear and angular accelerations satisfy the kinematic condition of pure rolling.

What about this don't you understand?

How many 'Are you really telling me ...' questions will you post about the same trivial thing?

 

[jbriggs444]

I consider pure rolling without slipping on a frictionless surface to be complete contradiction.

You can consider what you like. No one can prevent you.

I can contrive a wheel that is negligibly different from circular that rolls at a rate which involves negligible slipping on a surface which offers negligible friction. This exercise involves just such a contrivance.

Yes, any real situation will involve a not-quite-circular wheel on a not-quite-frictionless surface with a not-quite-zero-but-perhaps-immeasurably-close slip rate. That does not render the idealization worthless.

 

[erobz]

You can consider what you like. No one can prevent you.

I can contrive a wheel that is negligibly different from circular that rolls at a rate which involves negligible slipping on a surface which offers negligible friction. This exercise involves just such a contrivance.

That same wheel once contrived cannot match every surface once it's conceived like the model dictates. There is a big difference there. What is needed is the creation of a truly universal lock and key here. You are describing the creation of a single lock and key.

 

[A.T.]

You are describing the creation of a single lock and key.

You are denying that a "lock and key" can exist. A single example is enough to disprove this.

 

[jbriggs444]

That same wheel once contrived cannot match every surface once it's conceived like the model dictates. There is a big difference there.

There is a small difference there. The same wheel, once contrived, will roll [approximately] without slipping regardless of the surface.

Look at the problem description. It does not mention the surface properties.

 

[erobz]

There is a small difference there. The same wheel, once contrived, will roll [approximately] without slipping regardless of the surface.

Look at the problem description. It does not mention the surface properties.

It doesn't mention them "because apparently a surface is a surface is a surface"? Where does the idealization break down.

A single example is enough to disprove this.

I'd imagine there might be some physics the quantum guys will throw out, but No.

You are denying that a "lock and key" can exist.

False. Look carefully at what the model is saying. I must design the perfect wheel that rolls across all surfaces in this perfect meshing to disprove this. I single example is a drop in the ocean.

I am denying anything, but the creator could design such a universal key for all locks like the model describes.

 

[erobz]

Look at the problem description. It does not mention the surface properties.

That is my point.

 

[jbriggs444]

That is my point.

Your point is that models are not 100% accurate. But as @A.T. points out, that is trivially true and was never in dispute.

 

[erobz]

Your point is that models are not 100% accurate. But as @A.T. points out, that is trivially true and was never in dispute.

No my dispute is "where should we stop believing it" Has anyone found a single "Golden Wheel", let alone the practically infinite set of them - there is "apparently" a "Golden Wheel" for every set of parameters we selected. Plot it out on Desmos if you don't believe it. I did.

I say when the model tells us there is a "Golden Wheel" we say meh....

 

[A.T.]

No my dispute is "where should we stop believing it"

You don't have to believe anything. You can do your own experiments, to check if static friction changes direction, as the solution suggests.

 

[erobz]

You don't have to believe anything. You can do your own experiments, to check if static friction changes direction, as the solution suggests.

I don't have a PhD...no one will believe me when I tell them I've found the "Golden Wheel". I just have an Bs MET (Mechanical Engineering Technology) from a low rent university. Maybe you can?

 

[jbriggs444]

No my dispute is "where should we stop believing it"

That does not clarify things for me.

Has anyone found a single "Golden Wheel", let alone the practically infinite set of them - there is "apparently" a "Golden Wheel" for every set of parameters we selected.

You are agreeing that for any set of parameters ($M$, $F$, $R$, $\kappa$ there is a $q$ for which $F_s$ vanishes according to the model?

Plot it out on Desmos if you don't believe it. I did.

You are saying that Desmos agrees as well?

I say when the model tells us there is a "Golden Wheel"

You wish to consider a wheel with the requisite $M$, $R$ and $q$ so that $F_s$ goes to zero for a given $\kappa$ to be "Golden" for the given $\kappa$. That is fine. It is just a word. You are free to use it. Though it would be polite to define a word before using it.

we say meh....

Yes indeed.

Meh.

 

[erobz]

You wish to consider a wheel with the requisite $M$, $R$ and $q$ so that $F_s$ goes to zero for a given $\kappa$ to be "Golden" for the given $\kappa$. That is fine. It is just a word. You are free to use it. Though it would be polite to define a word before using it.

"Golden Wheel": A wheel of parameters $R_1,R_2,q,\kappa$ that accelerates - executing rolling without slipping on every surface without applying a frictional force to any of said surfaces. Its what we've been talking about this whole time.

 

[jbriggs444]

"Golden Wheel": A wheel of parameters $R_1,R_2,q,\kappa$ that accelerates - executing rolling without slipping on every surface without applying a frictional force to any of said surfaces.

And you will not settle for negligible frictional force? You want it to be exactly zero frictional force and exactly zero slippage regardless of rolling resistance?

Nobody is claiming that a physical Golden Wheel exists. That is a claim that you have invented yourself.

The reasonable claim is simpler. We can design a wheel to have negligible static friction and negligible slippage regardless of the reasonable surface on which it is placed. "Reasonable" means that you are not allowed to use things like mud, sand or fly paper.

You tell us how negligible you want the difference between the physical behavior and the model and we can tell you how reasonable we need the surface and wheel to be.

 

[erobz]

Nobody is claiming that a physical Golden Wheel exists. That is a claim that you have invented yourself.

I have not invented anything of the sort...the mathematics invented it. I'm just giving it the flippant name it deserves when I call it "The Golden Wheel" given everything what has been discussed on a profound creation of a universal key for all locks.

 

[A.T.]

"Golden Wheel": A wheel of parameters $R_1,R_2,q,\kappa$ that accelerates - executing rolling without slipping on every surface without applying a frictional force to any of said surfaces.

In reality any measurement has limited accuracy, and anything you measure is subject to noise. Expecting to reliably measure any exact value of friction (not just zero) is silly.

But someone with an B.S. in engineering should be easily able to do an experiment, that checks if the friction has different direction, for the predicted combinations of the above parameters.

 

[erobz]

But someone with an B.S. in engineering should be easily able to do an experiment, that checks if the friction has different direction, for the predicted combinations of the above parameters.

I think "easily" is a relative term.

I never said I was a good engineer. I'm a stay at home dad, I mostly just clean stuff and help my kids with homework and take them to sports/arts/girl scouts/ etc...

You guys probably have all the stuff laying around from your demos on the subject.

 

[A.T.]

... on every surface ...

That part is also wrong. If the problem doesn't say anything about rolling resistance, then it's implied to be negligible, which excludes a lot of surfaces.

 

[PeterDonis]

I don't like it when people tell me (who I feel\think are my teachers) "It's obvious that ...yada yada". I have a bit of a problem with authority; a chip on my shoulder from past experiences with that phrase.

Please note that these comments are irrelevant to the topic.

 

[PeterDonis]

This thread has run its course and is now closed. Thanks to all who participated.