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Most related questions here specifically talk about torque, force, angular momentum/velocity etc. I just want to know how I can aptly describe the motion of the tip of some string as it wraps around a cylinder. So basically, the path of an object in circular motion with a decreasing radius. But I'd like to know how exactly I can define that in an equation (cartesian and polar) if I know the radius of the cylinder and the length of the string.

Previously I was just trying to describe that the end of the string will always stay tangent to the circular face of the cylinder, and I simply substracted the length of the string that was making contact with the string from the full length. I'm mostly just interested in 100 deg of motion, not the continual wrapping around the object.

I figure a spiral path would most aptly describe the motion, but I would like to know how I can specifically define that. I had also been toying around with the idea of just using the equation of a circle with a differential radius, but I don't think that would give me good results.

Any help/advice would be great.