Find the force P such that the disk rolls without slip

[Andres Padilla]

Homework Statement

When a disk rolls in a surface whitout slip, the velocity of the disk's edge (where it contacts the surface) is zero with respect the surface and the friction force is less than the maximum allowable of Us.N, where Us is the coefficient of static friction and N is the normal force exerted by the surface against the disk. Determine:

A)the maximum value of the force P such that the disk rolls without slip
B) angular acceleration of the disk for this maximum value of P

DATA:
m=1.8 kg
R=20cm
Us=0.25
The force P is placed in the center of disk and it goes to the right.

Homework Equations

Second netown's law
Torque= I.α
Vcm= w.R
Acm=αR

The Attempt at a Solution

I know that when a disk rolls without slip, the condition Vcm= w.r must be satisfied, and also I read that in that condition, the friction force must be zero (But I am not sure about that last thing).

So setting my second Newton's law we have the next (Ignoring the last thing I wrote):

P-fr= m.Acm (Where Acm is the acceleration in the mass center)

Now, I use the next equation:
Torque= I.α

Where Torque= Fr. R

So I would have: Fr. R = I.α

the moment of inertia for a disk is I=1/2 MR^2
So replacing I have:

Fr. R = 1/2MR^2.α

Canceling one R in each side I have:

Fr = 1/2MR.α

I also know α= Acm/R (Due the condition of rolling without slip)

So I have

Fr =1/2 MR.Acm /R (Where again I can cancel out both R's)

Fr =1/2 M.Acm
Solving for Acm: Acm= 2Fr/M
Now replacing this thing in the second Newton's law I have this:

P- Fr = M. (2Fr/M)

Canceling both M and solving por P i have this:
P=3FR
Obiously I can find Fr with Fr= N.Us, But I am not sure if all that process is okay.

And also, I don't why angular acceleration depends of P, because I think I could find it with

Fr = 1/2MR.αI would be happy if someone could help me with that

 

[haruspex]

in that condition, the friction force must be zero

No. You may have misunderstood the context.

P=3FR
Obiously I can find Fr with Fr= N.Us, But I am not sure if all that process is okay.

Looks good, but you mean 3Fr.

And also, I don't why angular acceleration depends of P, because I think I could find it with

Fr = 1/2MR.α

Yes, but so what? They are just asking for the maximum angular acceleration that can be achieved, without slipping, by pushing horizontally at the centre. As I read question b), it does not imply that it is necessary to find P first.