# A yo-yo pulled by a rope: limit angle for the yo-yo to go to the left or to the right

1. Jan 18, 2012

### dany-vai

1. The problem statement, all variables and given/known data
We have a yo-yo, formed by a little cylinder tied to a little rope, and two big disks as wheels, one for each side of the central cylinder. We do a force T on the rope; its slope is an angle θ over the horizontal x-axis.
We are given the mass m of the cylinder, the mass M of each wheel, the radius of the cylinder R1 and the radius of each wheel R2. We want to know the limit angle θ , for the acceleration to be positive or negative.
I try to explain it better: under a certain angle, the yo-yo have a pure roll in the positive direction, over this angle it goes "back" and rolls towards the negative direction.

3. The attempt at a solution
I put in all the forces, the weight $mg$, the reaction $N$, and the friction force (static, since it's roling) towards the left side.
My equations are:
T*sinθ + N = m*g
T*cosθ - f$_{s}$ = m*a$_{cm}$
f$_{s}$*R2 - T*R1 = I$_{G}$*α = I$_{G}$ * a$_{cm}$ / R2

But I'm always obtaining a non-sense answer solving the system, I guess I'm doing wrong something about the signes. Can someone help me?
Thanks a lot.

2. Jan 18, 2012

### lightgrav

Re: A yo-yo pulled by a rope: limit angle for the yo-yo to go to the left or to the r

when it rolls, its rotation is around the point where it contacts the ground.
If the torque around this point is zero, it does not roll (alpha = 0).

3. Jan 18, 2012

### dany-vai

Re: A yo-yo pulled by a rope: limit angle for the yo-yo to go to the left or to the r

Thank you very much| Now I got it :D