Static equilibrium | Uniform sphere on incline | Determining friction

[erobz]

The model is obviously not accurate for a vertical wall.

But then an infinitesimal displacement away from vertical it is?

 

[Orodruin]

But then an infinitesimal displacement away from vertical it is?

Within the model yes. But as I have already stated - repeatedly - it is obvious that the model breaks down well before that. Just as the rope model does before the rope becomes horizontal.

 

[erobz]

Within the model yes. But as I have already stated - repeatedly - it is obvious that the model breaks down well before that. Just as the rope model does before the rope becomes horizontal.

Ok, I wasn't picking up on that. My point was this model has a window of validity, but it isn't obvious where exactly that is with what is exposed to us in manipulatable variables. I feel a "better" model is the alternative to this, which is what I think you (ultimately) believe too.

 

[Orodruin]

My point was this model has a window of validity

But that is exactly what you have been arguing against!

Most if not all models have limited applicability. Divergences often appear if those limits are taken without keeping this in mind.

 

[jbriggs444]

That’s not a frictionless surface though.

Worse, he is inadvertently providing some upward force with his hand in addition to the intended horizontal force.

 

[erobz]

But that is exactly what you have been arguing against!

Most if not all models have limited applicability. Divergences often appear if those limits are taken without keeping this in mind.

To me it feels worse than what I've encountered in the past... infinite forces from infinitesimal deviations seems unsavory. I'm not an academic. I'm still trying to probe my parents to see if Santa Clause is real or not.

Worse, he is inadvertently providing some upward force with his hand in addition to the intended horizontal force.

I'm aware of that too, but I don't think that helps the argument against reducing the amount of idealization, does it?

 

[jbriggs444]

To me it feels worse than what I've encountered in the past... infinite forces from infinitesimal deviations seems unsavory. I'm not an academic. I'm still trying to probe my parents to see if Santa Clause is real or not.

I'm aware of that too, but I don't think that helps the argument against reducing the amount of idealization, does it?

It helps clarify matters. We do not need to model exotic things like deformation and off center contact forces if the reality primarily involves equal vertical support forces on either side.

I am fine with the idealization and the resulting singularity as the wall becomes vertical.

 

[erobz]

It helps clarify matters. We do not need to model exotic things like deformation and off center contact forces if the reality primarily involves equal vertical support forces on either side.

I am fine with the idealization and the resulting singularity as the wall becomes vertical.

Yeah, friction at both contact point works that of the wall and the ball and the friction of the thing applying the force. My point was its like the model is like fine,fine,fine,...adinfinitum. Suddenly friction is necessary at 90.000000000000000... I still want to spit it out, instead of chewing it. Its too philosophical in flavor to be good food IMO.

I get the model breaks well before that, but I suspect if we add a touch more realism the divergence would vanish?

 

[jbriggs444]

I get the model breaks well before that, but I suspect if we add a touch more realism the divergence would vanish?

In reality there is no such thing as a vertical wall. Or an exact angle of the wall. Or even an unambiguous definition of the "wall". There is no point in pushing reality that far.

We work with models that are good enough.

 

[erobz]

There is no point in pushing reality that far.

But that's the whole ball game!

We work with models that are good enough.

Yeah, so long as it is admitted that good enough is mostly fuzzy in this sport, and new stuff can be found in the fuzz that is groundbreaking. I'm not suggesting that here, but to say you guys don't adjust/fine tune is a bit defeatist. Maybe we teach what is "good enough", but I don't think we truly ever except what is good enough.

 

[Orodruin]

But that's the whole ball game!

It most definitely isn’t. The whole ball game is making useful predictions. That includes both accuracy (good enough for the purpose), reliability, and ease of performing the prediction.

 

[Orodruin]

I'm not suggesting that here, but to say you guys don't adjust/fine tune is a bit defeatist. Maybe we teach what is "good enough", but I don't think we truly ever except what is good enough.

I think you are missing the entire point here. This is a post in the introductory homework forum. The point is helping the OP solve the problem within the constraints of the given model. A discussion of what refinements the model could use in particular limits (or even how the model behaves in those limits) is not part of the problem. It might have been in a more advanced setting, but here that discussion is not productive to the OP because the entire purpose is understanding the simplified model, not to consider where it breaks down and why or what possible refinements could be made to resolve the issues.

 

[Lnewqban]

I should have clarified that it depends on the idealisations being used.
....So if we knew the details of the deformations and the moduli of the materials we could determine all the forces. So the indeterminacy arises from denying ourselves some of the information.

Understood.
In agreement with your valid explanation.
Thank you much, @haruspex and @jbriggs444

Therefore, is there a correct response for the last question in the OP?
"What is the frictional force of the incline on the sphere?"

The way I see it, the OP situation could be considered to be equivalent to a perfectly rigid sphere resting on two perfectly rigid rollers (frictionless axes).

The right-side roller would be rigidly anchored to Earth.

The left-side roller (which center is horizontally aligned with the center of the sphere) would be free to slide on a horizontal guide and being pushed to the right by the exact amount of force needed to keep the theta angle (being equivalent to keep the sphere from rolling or sliding up or down the incline in the OP diagram).

The sphere can be in static equilibrium only for one unique value of force F, just like an airplane remains in horizontal flight only for one value of lift force.

 

[haruspex]

is there a correct response for the last question in the OP?

Yes, the one already established: there is no frictional force.