# How is motor torque related to capstan equation?

I know that capstan equation relates $$T_{in}$$ and $$T_{out}$$ as $$T_{out} = T_{in}e^{u \cdot F_c}$$, where $$u$$ is friction coefficient and $$F_c$$ is wrap (slip) angle. In my situation, the capstan is driven by a motor. Input and output ends of the wrapped material are free (under tension but not supported). Both motor torque and the angular velocity of the wrapped material are positive (clockwise). The wrapped material is elongating and moving at the same time.

What I don't understand is the role of motor torque in the scenario. The equation does not take motor torque into consideration and I couldn't grasp how it has no effect at all.

• I don't understand much about capstan equation. But with a fast reading on wikipedia, looks like this equation is for imobile capstan and T(in) is holding torque (no moving). To correlate motor torque this equation isn't applicable, or at least not totally confiable. Jun 10 '20 at 13:22
• Thanks for answering @Leafk. I suppose you're right, it is for imobile situations. However I've seen situations where the equation is applied to mobile cases without great altercations. I just couldn't find a thorough explanation of application difference between mobile and imobile cases. The closest I could find was this work: yaor.me/slip-stick-web-roller-contact Jun 15 '20 at 10:19
• The friction of the rope on capstan means Tout will be significantly greater than Tin, which is static. Tout is a force and Torque = Force x distance. So there is a relationship, but not directly. Jun 18 '20 at 23:05