# Find the force P such that the disk rolls without slip

In summary: The answer is contained in the more general answer to question a).In summary, the conversation discusses the conditions for a disk to roll without slip, including the relationship between its velocity and friction force. The equations of second Newton's law and torque are also referenced to determine the maximum value of the force P and the angular acceleration of the disk. It is determined that the maximum angular acceleration can be achieved by pushing horizontally at the center of the disk, and the maximum force P can be found by using the formula P=3Fr.

## 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

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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.

## 1. What is the concept of "rolling without slip" in physics?

The concept of "rolling without slip" in physics refers to the motion of a rolling object, such as a disk, where the point of contact between the object and the surface it is rolling on does not slip or slide. This means that the object's rotational motion and its linear motion are perfectly coordinated, and the object moves smoothly without any slipping or skidding.

## 2. Why is it important to find the force P for a disk to roll without slip?

It is important to find the force P for a disk to roll without slip because it allows us to understand and control the motion of the object. Without the proper force, the disk may slip or slide, leading to inaccurate results and potentially dangerous situations. By finding the correct force, we can ensure that the disk moves in a predictable and controlled manner, making it easier to conduct experiments and make accurate calculations.

## 3. How is the force P calculated for a disk to roll without slip?

The force P can be calculated using the equation P = μmg, where μ is the coefficient of friction between the disk and the surface it is rolling on, m is the mass of the disk, and g is the acceleration due to gravity. This equation takes into account the frictional force between the disk and the surface, which is necessary for the disk to roll without slip.

## 4. What factors can affect the force P needed for a disk to roll without slip?

The force P needed for a disk to roll without slip can be affected by various factors, such as the coefficient of friction between the disk and the surface, the mass and size of the disk, and the speed at which the disk is rolling. Other factors, such as the surface material and any external forces acting on the disk, can also play a role in determining the necessary force.

## 5. Can the force P be negative in the context of a disk rolling without slip?

No, the force P cannot be negative in the context of a disk rolling without slip. This is because a negative force would cause the disk to move in the opposite direction, leading to slipping or sliding. The force P must always be positive and in the same direction as the motion of the disk for it to roll without slip.

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