Ball rolling down a slope without slipping

jaumzaum

First I want to introduce a exercise I tried to solve, and then the doubt I have at the final resolution.

"In a slope with θ angle to the horizontal plane, a homogen ball with radius R and mass m roll down without slipping. Calculate the friction force."

Force equilibrium ma = mg sinθ - Ff (Ff is friction force)
Second Newton La Ff.R = I.γ (γ is angular acceleration)

Ff = (2/5)γmR

Condition of non-slipping a=γR
Ff = (2/5)ma
Ff = mgsinθ-ma = (2/5)ma -> a =(5/7)g sin θ

And Ff = (2/7)mgsinθ

Ok, now is the doubt.

a) Doesn't friction force depends on static friction coefficient of the slope with the ball?

b) In case the static friction coefficient was μ, and μ< (2/7)tanθ ( in this case the maximum friction force would be < (2/7)mgsinθ, and we couldn't have the force above), what would happen?

arildno

a) The MAXIMUM static friction force depends on the static friction coefficient. The actually present static friction force can be anywhere between 0 and that value.

b) What do YOU think would happen if the force required to maintain rolling without slipping exceeds the maximum static force that can be produced?

jaumzaum

Would the ball start slipping?

If so, would it be possible to calculate variables such as angular velocity in function of time?

arildno

1.Yes, it would start slipping, or rather, would be slipping from the start in your problem.

2.Yes, you will be able to determine the angular and linear velocities of the ball in the slipping case.

The rate of change of the angular velocity is then simply found from the torque the KINETIC frictional force produces about the ball's centre, whereas the acceleration of the center of mass is found in the normal manner.
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A much, much nastier problem would be rolling on a CURVED surface, so that you can get to the point that it BEGINS to slip at some point, and if the problem gets even uglier, when the ball simply cannot follow the surface any longer, and spins off the surface..

jaumzaum

Thank you,

Now I think I'm starting to understand...

I have a last question

Can we say that a body "searches" for the non-slipping condition?
I mean, if we had μ=(3/7)tanθ (so we could have the friction force calculated before)

We could have the friction force


(3/7)mgsinθ (maximum friction force) and the ball would start slipping
(2/7)mgsinθ and the ball would not slip
(1/7)mgsinθ and the ball would slip too

All the 3 above values are possible values as they are <= (3/7) mgsinθ
So can we say that the system prefer the non-slipping one?

Thank you
John

arildno

I wouldn't say it "searches" for anything.

I'd rather say "rolling wirhout slipping" is a "middle thing" that occasionally is achieved by a system.

jaumzaum

But if we abandonate a ball in the situation above (μ=(3/7)tanθ ), would it start slipping or not?

arildno

Yes.
When you reach the maximum static force (say that the curve of the surface gets too steep), the kinematics sort of "snaps" (because of the extremely rapid reconfiguration of the binding electro-magnetic forces underlying the two different states of friction), and kinematic frictional force kicks in.
The motion will be jerky at that point, experiencing a "jump" in the acceleration of the system

I'm not sure if the simplistic, standard Coulomb force modelling of the friction is very accurate at such a juncture point.
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Basically, you now have begun to tread waters unswum by me.

jaumzaum

Hi Arildno, I think I did not understand so well your last post. What I mean is:

We need a (2/7)mgsinθ friction force for the ball roll without slipping

So if we had a μ=3/7tanθ, we could have this force (as 2/7 is < than 3/7)

So if in this situation we leave a ball at the slope, it will start rolling with or without slipping? I mean, how can we foresee what would happen? We know that in this case the friction could be from 0 to (3/7)mgsinθ, but can we calculate which value would it be? Can we know if the ball will or will not slip in the slope?