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

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SUMMARY

The maximum angle theta (θ) for directional control of a rolling spool is critical for understanding its motion dynamics. When a string is pulled horizontally (θ = 0), the spool rolls without slipping. However, as θ increases beyond a specific threshold, known as θ-max, the spool begins to roll in the opposite direction. Experimental observations with a sewing machine bobbin indicate that θ-max is approximately 80 degrees, confirming the theoretical predictions regarding the spool's behavior under varying angles of force application.

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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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