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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.

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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.

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  • 3
    $\begingroup$ No, the path is not an Archimedean Spiral, which has the eqn $r=R\theta$. In this case $r^2=R^2(1+\theta^2)$. $\endgroup$ – sammy gerbil Aug 20 '16 at 0:14
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As the particle travels around the cylinder (having radius r) and the taught string wraps around the cylinder, the path of the particle will trace out a spiral with radius that decreases as a function of the cylinder radius. In other words, the radius of the spiral (let's call S) equals the string length at any given point. All you need to do is find the arclength of the spiral then divide that by the particles velocity to determine the time required for the string to fully wrap around the cylinder. Here's how:

For this problem it's helpful to work backwards (i.e. assume the string is initially fully wrapped and you want to unwind it at velocity, v). The amount of time required will be the same, whether your winding or unwinding.

Each time the particle completes a revolution about the cylinder, S (spiral radius) will change (increase in our unwinding case) by an amount equal to the circumference of the cylinder (2*pir). So, S is really a function of cylinder radius (constant) and the angular position of the particle (theta, in radians) about the cylinder. S = thetar. Using radians, at the start position theta is 0 (S = 0), at 1/4 revolution theta is pi/2 (S=r*pi/2), at 1 full revolution theta is 2pi (S=r2pi), 2 revolutions theta is 4pi, and so forth.

Now you need to know how many revolutions R the string of length (l) can possibly wrap around the cylinder of radius r. This is simply given by: R = l/(2*pir). Since one full revolution is 2pi, you can now find theta by: theta = 2piR.

Using the terms found above you can now calculate the arc length of the spiral. The general for for a spiral arclength L is an integral and is given below (tip: spend $1 or so and buy the wolfram alpha app to evaluate integrals, it's worth it).

To find L: From theta = a to b, integrate SQRT ((S^2 + (dS/dtheta)^2) with respect to theta.

where a is your start position (0 in your case) and b is your end position (2*piR). From the work we did above we know S = thetar. So simply plug this value into the spiral arclength equation and evaluate from 0 to 2*pi*R to obtain L.

Now just divide L/v (recall that v = particle velocity), to obtain the time required to fully wrap the string around the cylinder.

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