Solve Unwinding Yo-Yo Homework: No Slipping Allowed

[ChrisBrandsborg]

Homework Statement

We consider a yo-yo rolling down an incline, sloping with angle θ relative to horizontal. The yo-yo starts a distance d up the incline. The string is attached to a hook further up the incline, in such a way that it unwinds as the yo-yo rolls down. The yo-yo can be thought of as three uniform discs: a small one of mass m and radius r, sandwiched between two larger ones, each of mass M and radius R. The string is initially wrapped around the smaller middle disc. The string unwinds without slipping. To start with, we will assume that there is no friction between the yo-yo and the incline.

a) Is it possible to roll down without slipping on the incline? Why/why not? (Carefully think of the rolling-without slipping conditions).

b) What is the speed of the yo-yo when it reaches the bottom of the incline?

c) What is the acceleration of the yo-yo? The angular acceleration? How large is the string tension? Now imagine that we turn on kinetic friction between yo-yo and incline, with a coefficient µk.

d) What is then the speed at the bottom of the incline? (Tricky! Carefully consider the motion of the point of application). For each question, provide an algebraic expression.

Homework Equations

α = R∝
ν = Rω
τ (torque) = I∝
Moment of Inerta (of uniform disc) = MR²
h = d⋅sinθ
Fg_x = Mg⋅sinθ
Fg_y = Mg⋅cosθ

The Attempt at a Solution

a) For the yo-yo (disc) to not slide down, we need friction on the surface, right? But in the problem info it says that we assume there are no friction between the yo-yo and the incline. Does that mean that in this case, it is not possible to roll down without sliding?

 

[TomHart]

The string is attached to a hook further up the incline

For the yo-yo (disc) to not slide down, we need friction on the surface, right?

Are you sure?

 

[ChrisBrandsborg]

Are you sure?

Maybe not, because it is attached to a string, but if it was an unattached disc, wouldn't it need friction between the surface and disc to roll without sliding?)

 

[TomHart]

You may be right. I originally didn't think that it needed friction, but now . . .

 

[TomHart]

Okay, I think the problem is saying that the string does not slip around the disk. It's not saying that the disk does not slip on the surface. Then the problem says "to start with we will assume there is no friction." Then later on they add in friction. My mistake initially. Sorry.

 

[ChrisBrandsborg]

Okay, I think the problem is saying that the string does not slip around the disk. It's not saying that the disk does not slip on the surface. Then the problem says "to start with we will assume there is no friction." Then later on they add in friction. My mistake initially. Sorry.

Yeah, so first, when there are no friction, the yo-yo will slide down, and when they later add friction, it will roll down without sliding?

 

[TomHart]

You are right that at first, the yo-yo will slide. But when they later add friction, you have to look at how much string unwinds relative to how far the yo-yo has to travel down the incline. Is it even possible for the yo-yo to unwind without sliding based on the distance the yo-yo has to roll relative to how much string is unwound. In other words, how far does the yo-yo travel in one revolution of the yo-yo versus how much string has unwound in that same one revolution?

This one is confusing me. I think I should make it a policy that I have my morning coffee before I get on this site. :/

 

[rcgldr]

Yeah, so first, when there are no friction, the yo-yo will slide down, and when they later add friction, it will roll down without sliding?

From the problem statement: "Now imagine that we turn on kinetic friction between yo-yo and incline, with a coefficient µk." Kinetic friction is sliding / slipping friction, so the yo-yo is still sliding / slipping.

 

[ChrisBrandsborg]

From the problem statement: "Now imagine that we turn on kinetic friction between yo-yo and incline, with a coefficient µk." Kinetic friction is sliding / slipping friction, so the yo-yo is still sliding / slipping.

Can you help me understand a)

Condition for the object to roll without slipping is that the contact-point does not move.
ω = ν/r
What does this mean in this problem? The object is moving, but is it sliding or is it rolling? Do we need friction between the surface and the yo-yo to make the yo-yo roll down the incline?

 

[TomHart]

For a yo-yo of radius R, what distance does it travel if it rolls down the incline one revolution? How does that compare with the amount of string that has unrolled in that same one revolution if the disk that the string is wrapped around is radius r? Edit: Does the amount of string released allow it to travel that distance?

This has been hard for me to visualize this. I almost went out and bought a yo-yo yesterday to play around with it.

 

[ChrisBrandsborg]

For a yo-yo of radius R, what distance does it travel if it rolls down the incline one revolution? How does that compare with the amount of string that has unrolled in that same one revolution if the disk that the string is wrapped around is radius r?

This has been hard for me to visualize this. I almost went out and bought a yo-yo yesterday to play around with it.

Distance = 2π⋅R?
Amount of string = How do you calculate the amount of string?

And also, how is related to the rolling without slipping conditions? I want to start with understanding the conditions, because if it was a ball, then it would need friction for the ball to rotate, else it wouldn't have a torque (which makes the object rotate). In this case we have a string, which have a force? Does it rotate without slipping because of the string?

 

[TomHart]

You calculated the distance that the yo-yo moves in one rotation correctly. You calculate the amount of string unrolled the same way, except that you use its radius, r. So is that enough string to allow the yo-yo to move a distance of 2πR? Edit: Corrected 2πr to 2πR.

 

[TomHart]

Oops, I meant to say, "Is that enough string to allow the yo-yo to move a distance of 2πR?

 

[ChrisBrandsborg]

Oops, I meant to say, "Is that enough string to allow the yo-yo to move a distance of 2πR?

2πR > 2πr, so is that enough?

 

[TomHart]

Maybe real numbers would help. Let's say the yo-yo has a radius of 1.6 inches. So if it rolls down the incline one revolution, it will travel 2π(1.6) = 10 inches. Now let's say that the disk that the string is wrapped around has a radius of 0.25 inches. So if the yo-yo completes one revolution, the amount of string that has unrolled is 2π(0.25) = 1.57 inches.

What is the problem with that?

 

[ChrisBrandsborg]

Maybe real numbers would help. Let's say the yo-yo has a radius of 1.6 inches. So if it rolls down the incline one revolution, it will travel 2π(1.6) = 10 inches. Now let's say that the disk that the string is wrapped around has a radius of 0.25 inches. So if the yo-yo completes one revolution, the amount of string that has unrolled is 2π(0.25) = 1.57 inches.

What is the problem with that?

Hmm.. but if the yo-yo moves 10 inches, wouldn't we need 10 inches of string as well?

 

[TomHart]

Hmm.. but if the yo-yo moves 10 inches, wouldn't we need 10 inches of string as well?

Exactly!

 

[ChrisBrandsborg]

But how is this related to problem a, about the conditions?

"Is it possible to roll down without slipping on the incline? Why/why not? (Carefully think of the rolling-without slipping conditions)."

 

[ChrisBrandsborg]

Conditions:

* Point of contact -> not moving
* w = v/r (if it is rotating, then w > 0)
* There is a torque, which makes the yo-yo rotate. First I thought friction made it rotate, but it is the string tension, right?

 

[TomHart]

In order for the yo-yo to move 10 inches down the incline, you have to unroll 10 inches of string. But it takes (10 inches)/(1.57 inch/rev) = 6.4 revolutions of the yo-yo to release 10 inches of string. And for the yo-yo to move 10 inches without slipping, it will only have completed one revolution.

Does that make sense?

 

[ChrisBrandsborg]

In order for the yo-yo to move 10 inches down the incline, you have to unroll 10 inches of string. But it takes (10 inches)/(1.57 inch/rev) = 6.4 revolutions of the yo-yo to release 10 inches of string. And for the yo-yo to move 10 inches without slipping, it will only have completed one revolution.

Does that make sense?

Yes, since r < R,

1 revolution won´t take as long time for the smaller disc compared to the bigger?

 

[TomHart]

It takes the same amount of time for both of them to complete one revolution because they are mechanically tied together; they don't move independently.

 

[ChrisBrandsborg]

It takes the same amount of time for both of them to complete one revolution because they are mechanically tied together; they don't move independently.

hmm, true.. But then how do we get 10 inches of string in 1 revolution?

 

[TomHart]

hmm, true.. But then how do we get 10 inches of string in 1 revolution?

That is the whole point. It is impossible. So with that in mind, is it possible for the yo-yo to roll down the incline without slipping when the string is tied to a hook on the other end?

 

[ChrisBrandsborg]

That is the whole point. It is impossible. So with that in mind, is it possible for the yo-yo to roll down the incline without slipping when the string is tied to a hook on the other end?

no it´s not :) So that´s the answer to a?
But if we had friction in opposite direction of the motion, then it would rotate?

 

[TomHart]

If you cut off the string from the yo-yo, and placed it on an incline that has friction, it would rotate as it moved down the incline because the friction produces a torque. Depending on the coefficient of friction and the angle of the incline, it may rotate without slipping or it may slip. If it was a high coefficient of friction, and a slight angle, the yo-yo would roll down the incline without slipping. If it was a very slippery surface and steep angle, the yo-yo would slip as it moved down the incline. It would still rotate because the friction does produce torque, but not without slipping.

For the case of the string attached to the yo-yo, it simply is not possible for the yo-yo to move down the incline without slipping. If there is friction on the incline, and the incline angle starts out at 0 and increases, the yo-yo will not move until the angle gets steep enough to where the yo-yo slips on the surface of the incline.

Edit: At least that is my theory. I still feel like I need to go buy a yo-yo to verify - because like I said, this one is hard for me to visualize. :)

 

[ChrisBrandsborg]

If you cut off the string from the yo-yo, and placed it on an incline that has friction, it would rotate as it moved down the incline because the friction produces a torque. Depending on the coefficient of friction and the angle of the incline, it may rotate without slipping or it may slip. If it was a high coefficient of friction, and a slight angle, the yo-yo would roll down the incline without slipping. If it was a very slippery surface and steep angle, the yo-yo would slip as it moved down the incline. It would still rotate because the friction does produce torque, but not without slipping.

For the case of the string attached to the yo-yo, it simply is not possible for the yo-yo to move down the incline without slipping. If there is friction on the incline, and the incline angle starts out at 0 and increases, the yo-yo will not move until the angle gets steep enough to where the yo-yo slips on the surface of the incline.

Okay, but to show that, do I only show what we did with the string length etc..
Don´t I need to show that Στ = 0 → α = 0 → ω = 0

 

[ChrisBrandsborg]

in c) "What is the acceleration of the yo-yo? The angular acceleration?",
Is that just to trick us into believing that there is an angular acc, but in reality it is = 0?

 

[TomHart]

Don´t I need to show that Στ = 0 → α = 0 → ω = 0

I would probably have to think hard about that to try to come up with something in equation form, and I just don't have the time right now to do that. However, based on the way the question is worded, I don't believe it is asking for that level of answer.

in c) "What is the acceleration of the yo-yo? The angular acceleration?",
Is that just to trick us into believing that there is an angular acc, but in reality it is = 0?

Up until the 2nd part of c), you are still working with no friction. So the yo-yo will be moving down the incline, and will be slipping, and will be rotating due to the torque produced by the string. You should be able to do those calculations. In the second part of c), friction is added in.

Chris, I have to go for now, so I won't be online for a while. Hopefully, if you have any other questions, someone else can chime in and help.

 

[ChrisBrandsborg]

I would probably have to think hard about that to try to come up with something in equation form, and I just don't have the time right now to do that. However, based on the way the question is worded, I don't believe it is asking for that level of answer.Up until the 2nd part of c), you are still working with no friction. So the yo-yo will be moving down the incline, and will be slipping, and will be rotating due to the torque produced by the string. You should be able to do those calculations. In the second part of c), friction is added in.

Chris, I have to go for now, so I won't be online for a while. Hopefully, if you have any other questions, someone else can chime in and help.

The friction isn't added before d, so is there angular acceleration even if it is slipping? Yeah, I´ll probably figure it out eventually :) thanks for your help!

 

[rcgldr]

To clarify the issue here, the yo-yo outer surface has to be spinning faster than it's moving down the incline. The kinetic friction would have to be relatively low and/or the angle relatively steep in order for the yo-yo to move at all. So if there is kinetic friction, what is the direction of friction force exerted by the incline onto the yo-yo?

 

[ChrisBrandsborg]

To clarify the issue here, the yo-yo outer surface has to be spinning faster than it's moving down the incline. The kinetic friction would have to be relatively low and/or the angle relatively steep in order for the yo-yo to move at all. So if there is kinetic friction, what is the direction of friction force exerted by the incline onto the yo-yo?

F_fr = μk⋅Mgsinθ ? (Perpendicular to the normal force and in the direction opposite to the motion)

 

[TomHart]

Okay, I went out today and bought a yo-yo - $3 Duncan. I was kind of afraid to try it because, if I was wrong . . .

What I found was that, on a surface with friction, the yo-yo would not roll down an incline when I held the string as shown in the original figure for this problem. (And yes, I made sure that the yo-yo was positioned such that the string was on the lower side of the small (center) disk, and not on the upper side, as that would be a much different situation.) As a matter of fact, and it was somewhat surprising (although it shouldn't have been), I could roll (without slipping) the yo-yo uphill with the string as shown in the figure - the string winding itself up on the yo-yo as it went. Of course, if the angle was too steep, or if I pulled too hard, the yo-yo would slip.

 

[haruspex]

For a), it might be easier to think in terms of the instantaneous centre of rotation. When a wheel rolls, at any given instant, the wheel is rotating about the point of contact with the ground. So for the purposes of part a) you can replace the yo-yo with a stick standing normally to the slope. At the bottom it is in contact with the slope, at the top tied to the string. Can it move without slipping?

For b and c, as you note, there is no friction. So yes, there will certainly be angular acceleration. What will be the instantaneous centre of rotation now, i.e. what part of the yo-yo is, for the instant, unable to move?

 

[ChrisBrandsborg]

The Torque comes from the string force, right? But how do we calculate the angular velocity when it rolls and slides? Because then energy isn't conserved?