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The capstan equation will give me the relation between the tension force at both sides of a rope wrapped around a cylindrical object, but what happens in this situation where the 'rope' is something sticky like a tentacle that uses its entire length to wrap around the object? How would I define the tension at the "other end?"

The cylinder is rotating away from the tentacle. There is no additional wrapping of the rope so the body holding the tentacle could be considered to be pulled along with the rotation, gently slowing it until the rotation stops.

enter image description here

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    $\begingroup$ In order to be a well framed problem, you will need to define any masses and forces present. You should also clarify where you want to find the tension. Once the problem is well defined you may very well immediately see the answer, but it is hard to help with your problem of turning your vague mental problem into a concrete and answerable one, since we can’t read your mind. Try starting by deciding whether the tentacle has mass, and what forces are causing the changing rate of rotation. $\endgroup$
    – Duncan Harris
    Commented May 23, 2018 at 2:50
  • $\begingroup$ Is the end fixed on the pulley? If that is so, then infinite force can be held as the end of the rope is never going to slip on the pulley. $\endgroup$
    – John Alexiou
    Commented Jan 24, 2023 at 20:52
  • $\begingroup$ It is not fixed on the pulley, only "sticking to it" from static friction $\endgroup$
    – AAC
    Commented Sep 22, 2023 at 8:03

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I think that, in general, the idea is the same as in the equation with two tensions, but one of them will be replaced with the “stickiness force” on the end of the rope.

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If you solve the governing equation $$\frac{\rm d}{{\rm d}\theta} T = \mu T$$ and designate $T$ the tension on the free end, and $T_0$ the tension on the fixed end you will find that

$$ T_0 = T \exp\left( -\mu \theta \right) $$

fig1

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