Car's maximum acceleration on a road is proportional to what?

[Steve4Physics]

I did not make that diagram nor originally post it but I reposted it to argue I thought it was wrong. Here is another free-body diagram I found of a wheel slowing down by rolling resistance. It looks like it would have a torque that would speed the wheel up but we know it does not so there is something about the way friction works at the point of contact that I'm just not getting. It is easy to see how $F_{rr}$ will act to slow down the wheel but I don't see how the torque $RF_{rr}$ acts in the correct direction. I wonder if it has something to do with the point of contact actually being the center of rotation? And where is the static friction in this diagram? The brain fog continues...
https://www.physicsforums.com/attachments/296631
https://archive.thepocketlab.com/educators/lesson/rolling-resistance-physics-lab

The Post #85 diagram shows rolling resistance as a simple horizontal force acting at the base of the wheel, vertically below the wheel's centre. This is an inaccurate simplfication and (as has already been noted) gives a torque which would tend to accelerate the wheel!

A better representation is this: https://www.lockhaven.edu/~dsimanek/scenario/rollres3.gif

For this diagram and accompanying explanation, see the section entitled “Rolling Resistance” about halfway down this link: https://www.lockhaven.edu/~dsimanek/scenario/rolling.htm

 

[jack action]

Ok, I resolved my brain fog issue over the free body diagram of the wheel alone. What bugged me in the diagram is that it looked like the blue Axle force to the right was exactly equal to the static friction force to the left which could not be true under acceleration. The total forces left and right are slightly different and total to a net force of the exact amount necessary to accelerate the wheel of mass $m$ by acceleration $a$. Likewise the driving torque at the axle is greater than the counter torque due to all other torques by exactly the amount consistent with an angular acceleration $\alpha= \large \frac{a}{R}$.View attachment 296601

The diagram is not a true free body diagram (FBD).

It's a simplified schematic version to explain the relationship between force and torque directions.

What is important to understand is that a rolling wheel can reverse the friction force direction by reversing the wheel torque, even if the rolling direction stays the same.

In an accurate FBD, yes, there would be wheel accelerations to consider (both linear and angular). BUT THIS IS IRRELEVANT TO THE PROBLEM AT HAND. It would only confuse the OP to make an FBD for each wheel and the car's body. Just imagine the wheel is massless and the car's body is heavier by the equivalent of the wheels' masses.

Rolling resistance should also be ignored for this discussion. IT IS IRRELEVANT TO THE PROBLEM AT HAND and can only confuse the OP.

I can't believe we have 91 posts in this discussion.

 

[kuruman]

Here are 3 FBDs related to the one-wheel skateboard mentioned in post #21. To make things simple, only horizontal forces are shown.

$f_{\text{S}} = ~$ force of static friction exerted by the road on the wheel.
$f_{\text{WB}}=~$ contact force exerted by the wheel on the board at the axle.
$f_{\text{BW}}=~$ contact force exerted by the board on the wheel at the axle.

To @rudransh verma:
The sum of the board and wheel FBDs is the FBD of the one-wheel skateboard. The board, the wheel and the combined board + wheel have a common acceleration $a$. What is the net force in each case that provides this acceleration?

A picture is worth 10³ words.

 

[sysprog]

It would be nice if there was a simple drawing tool here at PF.

If you have access to PF, you presumably also have access to some great physics-friendly drawing tools available on the net, e.g. GeoGebra $-$ https://geogebra.org/m/NvzumAMV, LaTeX/TIKZ $-$ https://texample.net/tikz/examples/all/, Inkscape $-$ https://inkscape.org/gallery/?tags=physics.

 

[hutchphd]

A picture is worth 10³ words.

All preamble aside, what if one wishes to use the term "rolling friction"? Personally I would be inclined not to use it ever. Is there a reason to put it in the elementary curriculum (as yet another frictional force??) Just askin'