Path of the end of a string wrapping around cylinder

[PKU]

Hey,
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.

 

[phinds]

I think it's a simple logarithmic spiral but that may only work if the string were rotating around a point and getting shorter. Wrapping around a cylinder is more complicated.

 

[A.T.]

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.

Note, that this equivalent to motion of a point on the ground, in the reference frame of a rolling wheel. So basically rotation and translation coupled via the radius.

 

[PKU]

Note, that this equivalent to motion of a point on the ground, in the reference frame of a rolling wheel. So basically rotation and translation coupled via the radius.

How does that include the shortening radius?

 

[CWatters]

Haven't done it myself but...

Perhaps start with an equation for the point (circle) where the string meets the cylinder (perhaps in terms of the angle θ to the x axis) and add a displacement to get to the free end. The displacement will be orthogonal to the "radius" and the length of the displacement can be calculated as the initial length of the rope minus a fractional part of the circumference.

 

[rcgldr]

The path is "involute of circle". Take a look at post #4 in this thread:

https://www.physicsforums.com/threads/understanding-angular-momentum.328121

and post #32 of this thread:

https://www.physicsforums.com/threa...rgy-considerations.185178/page-2#post-1435942