What Is the Maximum Angle Theta for Directional Control of a Rolling Spool?

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The discussion centers on determining the maximum angle theta (θ) for the directional control of a rolling spool when a string is pulled from its inner radius. At θ = 0, the spool rolls forward without slipping, but as θ increases, a critical angle θ-max is reached beyond which the spool rolls in the opposite direction. Participants suggest starting with a diagram to illustrate the forces involved, including the pulling force and friction. One user shares a practical experiment with a sewing machine bobbin, noting that they observed the spool rolling in the intended direction at an angle of about 80 degrees. The problem highlights the relationship between the angle of force application and the resulting motion of the spool.
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Consider a spool of inner radius r and outer radius R.
A string is wrapped around the inner cylinder. The spool is
motionless on a table. The string is pulled so that it feeds
off the bottom side of the spool. If the angle feta=0 so that
the string is pulled horizontally, it is possible to make the spool
roll without slippingin the direction it is being gently pulled.
However, as the angle feta is increased it is found if feta is
greater than a maxiumum value feta-max this becomes impossible
and instead the spool rolls in the opposite direction.
Find the value of feta max.

I'm lost on how to start on this problem. Your helps is appreciated.
 
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Start with a diagram! Mark the angle θ (BTW pronounced "theta") on it, along with the pulling force F and the friction force. Write up the Torque = I*ω for the spool.

Fascinating problem; it is really difficult for me to believe it works but we just tried it with a sewing machine bobbin and it does! The angle was about 80 degrees.
 
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