Solving Spool Acceleration Problem Using Newton's Laws

[lua]

Homework Statement

On the spool of the inner radius r and the outer R, a light unstretchable thread is wound, the end of which is drawn by the force F, as shown in the figure. The moment of inertia of the coil with respect to the horizontal axis passing through the center of mass is I. Determine the acceleration of the center of mass of the spool if it is known that the spool is rolling without sliding to the right and comment on the solution.

Relevant Equations

Second Newton law for translation and rotation

I solved this problem using second Newton law for translational motion and the same law for rotational motion, and got $$a= \frac {F} {m+ \frac {I} {R^{2}}} (cosϕ−rR)$$ where m is spool mass.
Now, we have three cases:
(a) $cos\phi>\frac{r}{R}$, when spool is accelerating to the right,
(b) $cos\phi<\frac{r}{R}$, when spool is accelerating to the left and
(c) $cos\phi=\frac{r}{R}$, when $a=0$ and $\alpha=0$ ($\alpha$ is rotational acceleration).

In the (c) case, if fraction force is $F_f \le \mu N$ ($\mu$ is a friction coefficient), from $mg=Fsin\phi+N$ we get that for
$$F\le \frac{\mu}{cos\phi+\mu sin\phi}mg$$
spool is resting.
My question is: what is going on for $F\gt \frac{\mu}{cos\phi+\mu sin\phi}mg$?

 

[lua]

I forgot the figure...

 

[haruspex]

My question is: what is going on for $F\gt \frac{\mu}{cos\phi+\mu sin\phi}mg$?

As you wrote, you considered "if friction force is $F_f \le \mu N$".
What if it is not?

Btw, you get an interesting simplification if you substitute tan(α) for μ.

 

[lua]

First, for the case (c), I would say that the zero values of translational and rotational acceleration mean one of the following three things:
1. If the spool was accelerating (to the left or right) before meeting requirement (c), it will keep moving with constant linear and rotational speed equal to these speeds just before meeting (c);
2. If the spool was resting before (c), it will keep standing still after meeting (c).
Is this OK?

On the other hand, maximum force of friction is $\mu N$. It can't be greater, as I understand it.

I tried to do this simplification, but I just complicated it. I got $F \le \frac{tan \alpha}{cos \phi} \frac{1}{1+tan \alpha tan \phi} mg$.

P.S. I can't edit my first post anymore. Can anybody correct expression for acceleration? There should be $\frac{r}{R}$ instead of $rR$.

 

[haruspex]

maximum force of friction is μN.

Right, so what happens if you pull harder on the string?

For the "simplification ", multiply the cos in the denominator through the other factor there, and multiply numerator and denominator by cos(α). You should get a familiar expression in the denominator.

 

[lua]

If you pull harder, I assume that either string will brake or the spool will start to move with constant speed because of zero acceleratios. But I can't prove that.

When I look once more through equations, $F_{max}=T_{max}= \frac{\mu}{cos \phi + \mu sin \phi} mg$ is maximum straining force of the string. In equations I assumed that it has to be $F=T$, otherwise the string would break. So, if I apply force to the string greater then $F_{max}$, the string will break.

 

[lua]

For the "simplification ", multiply the cos in the denominator through the other factor there, and multiply numerator and denominator by cos(α). You should get a familiar expression in the denominator.

I got: $$F_{max}= \frac{1}{2} \frac{sin 2 \alpha}{cos(\alpha - \phi)}mg$$

 

[haruspex]

I assume that either string will break

Assume it doesn't. What happens if you attach a string to a box on the floor and pull horizontally, harder and harder? The string breaks or...

I got: $$F_{max}= \frac{1}{2} \frac{sin 2 \alpha}{cos(\alpha - \phi)}mg$$

I get $$ \frac{sin \alpha}{cos(\alpha - \phi)}mg$$
You seem to have an extra factor cos(α) in the numerator.

 

[lua]

I think I understand the source of my confusion. I assumed that the condition $F_{f} \le \mu N$ was closely related with the (c) case. But it is not the case.
I first considered the case of system dynamics in the middle of the movie. I found parameters of system dynamics (translational and rotational acceleration) and discussed what happens with various values of $\phi$.
Then I went to the movie beginning when system was at rest. Friction force opposes increasing external force to its maximum value of $F_{max}$. When the external force $F$ exceeds $F_{max}$, if $cos \phi \ne \frac{r}{R}$, the system will accelerate to the right or to the left with the acceleration calculated in the first post.
But when the external force $F$ exceeds $F_{max}$, and $cos \phi = \frac{r}{R}$, it seems that system has to start moving, but regardless of increasing external force, because of such value of $cos \phi$, there can't be any dynamics and system starts to move with constant translational and rotational speed. But it is very weird conclusion.

 

[ehild]

But when the external force $F$ exceeds $F_{max}$, and $cos \phi = \frac{r}{R}$, it seems that system has to start moving, but regardless of increasing external force, because of such value of $cos \phi$, there can't be any dynamics and system starts to move with constant translational and rotational speed. But it is very weird conclusion.

Your formula is valid for the "rolling without sliding"case. What happens if the horizontal component of the external force exceeds the maximum static friction? Is the assumption "rolling without sliding" valid then? Think of @haruspex 's hint about pulling a box on a rough floor.

 

[haruspex]

When the external force F exceeds $F_{max}$

When the applied force exceeds max static friction, what happens? It ...

 

[lua]

Your formula is valid for the "rolling without sliding"case. What happens if the horizontal component of the external force exceeds the maximum static friction? Is the assumption "rolling without sliding" valid then? Think of @haruspex 's hint about pulling a box on a rough floor.

When the applied force exceeds max static friction, what happens? It ...

Are we talking only about the case (c) or any case listed above?

I think I understand what you are trying to say, but I don't understand why system has to start sliding.
I understand that when system is rolling, if the resulting velocity of the point of contact with the ground isn't zero, system will begin to slide. It seems to me that if applied force exceeds maximal static friction, then the total force applied to the system is $F-F_{max}$ and system is accelerating from rest.

Also, I would like to ''see the full movie'': system is at rest - the external force is applied (any $\phi$) - something happens - in the end, system accelerates (one way or the other) and while accelerating, if it becomes $cos \phi = \frac{r}{R}$, system keeps moving in the same way with constant velocity.
I'm trying to understand that ''something happens'' part.
If sliding is what happens when $F$ exceeds $F_{max}$, how does the system come into acceleration without sliding? What would be the flow of the experiment to drive the system from rest to acceleration without sliding?

P.S. I even tried to do experiment with the object in the attached file. It mostly did the sliding but it seemed to me that it was because of its lightness and smooth surface it was on.

 

[ehild]

Are we talking only about the case (c) or any case listed above?

In any case, you have two equations, one for the translation and one for the rotation. Both equations contain the force of friction. You got one equation by eliminating the friction, and using the rolling condition: $a=\alpha R$. So you assumed that the motion was pure rolling which means that the contact point did not move with respect to the ground, the friction was static. But it is not true for big enough force.
If you apply such force, so as $F\cos(\phi)>\mu N$the spool will slide, and can also rotate, but the rotation and translation are not coupled. You have one equation for the linear acceleration and an other one for the angular acceleration, both containing the force of sliding friction $\mu N$.
Remember that the coefficient of static friction is usually bigger than that for kinetic friction. In your very smart experiment, you have found an angle when the spool did not roll, and did not move at all. Increasing the force, you just overcame static friction, but that force was bigger than kinetic friction: so the spool started to accelerate, gaining some speed. But you - without noticing- decreased the force, so it just balanced friction so the spool maintained constant velocity.