A particle is connected to the end of a thin, weightless string, which has its other end connected to the cylinder, in such a way that motion of the particle causes the string to wrap around the cylinder. If we know the cylinder radius $R$, string length $L$ and particle speed &v&, I need to calculate the time it takes for the string to completely wrap around the cylinder. The velocity vector is always perpendicular to the string.
My attempt. Suppose we think of this problem as of circular motion, such that the particle always moves around a cylinder of different radius , $r= R+dR$. If we integrate this radius from $R$ to $R+L$ we wouldn't be able yo determine the time
This is supposed to be a simple problem, so i would like only a subtle hint just to get started.
EDIT:
I apologize for violating homework questions rule. So I will provide you with additional information about the concept that gives me trouble here. It is the combination of linear and curved motion of the particle and string. I understand that the particle will move along some curved part. I don't understand what kind of path that is. That's why I was trying to write position dependent equation, instead of time dependent as asked. Somehow I am trying to relate string length and particle path traveled. I want to understand the nature of this motion. Particle is moving with constant tangential velocity component. Are there other components? Since particle is traveling at constant speed, is it safe to assume that time it takes to complete this motion is $t=\frac{path traveled}{v}$.
While this is not exactly a homework question, more of something to keep me puzzled, I'd still like to stop being stuck on this problem. Also note that I did not ask for an explicit solution but a hint to help me understand the motion of this particle.
Best Answer
The path will be an Archimedean spiral.
I think the key to this is that the speed of the particle does not change. You have an equation for the angular velocity in terms of the radius. You also have an equation for the rate at which r is changing in terms of its angular velocity: r is changing by
$$-2 \pi R\omega $$
Which seems to be all you need.