# Unwinding Yo-Yo

Tags:
1. Nov 5, 2016

### ChrisBrandsborg

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
α = R∝
ν = Rω
τ (torque) = I∝
Moment of Inerta (of uniform disc) = MR2
h = d⋅sinθ
Fgx = Mg⋅sinθ
Fgy = Mg⋅cosθ

3. 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?

Last edited: Nov 5, 2016
2. Nov 5, 2016

### TomHart

Are you sure?

3. Nov 5, 2016

### ChrisBrandsborg

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?)

4. Nov 5, 2016

### TomHart

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

5. Nov 5, 2016

### 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.

6. Nov 5, 2016

### ChrisBrandsborg

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?

7. Nov 5, 2016

### 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. :/

8. Nov 5, 2016

### rcgldr

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.

Last edited: Nov 5, 2016
9. Nov 7, 2016

### ChrisBrandsborg

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?

10. Nov 7, 2016

### 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.

Last edited: Nov 7, 2016
11. Nov 7, 2016

### ChrisBrandsborg

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?

12. Nov 7, 2016

### 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.

13. Nov 7, 2016

### TomHart

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

14. Nov 7, 2016

### ChrisBrandsborg

2πR > 2πr, so is that enough?

15. Nov 7, 2016

### 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?

16. Nov 7, 2016

### ChrisBrandsborg

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

17. Nov 7, 2016

### TomHart

Exactly!

18. Nov 7, 2016

### 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)."

19. Nov 7, 2016

### 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?

20. Nov 7, 2016

### 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?