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Probably a most horrible question title, please respond with suggestions for a better one.

In the example of my badge reel, I grab hold of my badge itself and pull on the reel until the string is extended as much as it can be. (If I understand correctly, there is now stored energy in the reel wanting to pull the badge back) If I then take the string starting closest to my badge and coil it around my finger until I reach the reel, and then let the reel go, I observe that the reel and the line begin to uncoil around my finger in larger and larger orbits. So, my question is, if there is an amount of stored energy E in the reel, how is it that the reel is allowed to fling freely around my finger in these orbits that (to my simple-minded self) must use more and more energy to continue orbiting without reel pulling into the center of the orbit (my finger).

I've attached a video of me doing this (because my lack of articulation is astounding)

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2 Answers 2

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This is an interesting phenomenon and one that can be explained as follow.

Inside the reel is a torsion spring that stores potential energy. When the string is fully furled up inside the reel (around some narrow cylinder), the potential energy is minimum (almost zero) because the torsion spring is almost tensionless. But when you pull the string out of the reel, the torsion srping starts exerting a reactive force and potential energy starts building up in the spring.

Now you wind all the unfurled string around your finger and then let go. Consider the following diagram:

Reel, string and finger.

This is the schematic situation when the rotating has been going for a little while.

The spring loaded with potential energy exerts the force $F$ on the reel and this provides the centripetal force to keep the reel in orbit around your finger. The centripetal force is given by $\frac{mv^2}{L}$ with $L$ roughly speaking the length of string that connects your finger and the reel (strictly speaking the formula calls for the distance between the centres of gravity of the reel and your finger). $m$ is the mass of the reel and $v$ the orbital velocity of the reel.

The rotations have successively increased radii because of the following. Each revolution increases $L$ by about $2\pi R$ where $R$ is the radius of your finger. But $L$ is also shortened somewhat with each internal revolution inside the reel, as the string gets refurled up inside the reel. As the cylinder inside the real is smaller than your finger, $L$ gets effectively longer with each rotation around your finger. So the reel effectively spirals around your finger.

When the system runs out of string around your finger, conservation of angular momentum causes the rotation to continue and the string refurls up around your finger.

So, my question is, if there is an amount of stored energy E in the reel, how is it that the reel is allowed to fling freely around my finger in these orbits that (to my simple-minded self) must use more and more energy to continue orbiting without reel pulling into the center of the orbit (my finger).

No energy is actually spent on keeping the reel in orbit because $F$ is always perpendicular to $v$. But the spring does eventually run out of potential energy because it spends it on internal rotational energy of the cylinder on which the string is gradually furled up.

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  • $\begingroup$ "But L is also shortened somewhat with each internal revolution inside the reel, as the string gets refurled up inside the reel. As the cylinder inside the real is smaller than your finger, L gets effectively longer with each rotation around your finger." Are you saying that if I were to grab the reel at an arbitrary time after the cycles begin, I would be able to observe that the reel has recoiled some amount of the cord? Because visual observation (as you've mentioned) suggests otherwise. $\endgroup$
    – MegaMark
    Oct 2, 2015 at 19:28
  • $\begingroup$ It's irrelevant what size the cylinder inside the real is, as there is only one exit, and the reel as a whole turns to point the exit towards the line. $\endgroup$ Oct 2, 2015 at 21:00
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The motion can be broken down into four phases. The first phase is simple: the reel starts pulling in on the line, forcing it to slide around your finger (as it would if you slowly let it reel in), with the reel pulling the line in as fast as it unwraps from your finger. There are two forces on the reel: the pull inward of the line, and the push outward of your finger. The two forces are not parallel, though, so there's a net force that's circumferential to your finger which adds to the momentum, and the reel quickly speeds up as it orbits.

The orbiting brings in a new force: the (pseudo-) centrifugal force. This is the apparent force pushing outward on the reel in its rotating frame of reference. As the reel speeds up this force increases until it is greater than the previous force from your finger, and the reel starts moving away from your finger, starting the second phase of motion.

The centrifugal force is proportional to $V^2\over{r}$, so as the orbital radius increases the force decreases, so that the net force is again purely circumferential, and the reel stops moving out. However, the reel is still increasing its velocity, again increasing the centrifugal force, so the balance point keeps getting farther and farther out.

At some point, the circle is so large and the velocity so high that air drag starts to take effect. When the drag becomes large enough to balance out the net force on the reel, preventing the reel from speeding up any further, the third phase begins, with the reel simply orbiting at a constant velocity, reeling in the line as fast as it's unwound from your finger.

Finally, when the line finishes unwinding from your finger and starts winding back up in the other direction, the force changes so as to reduce rather than increase the reel velocity. The reel slows down, the centripetal force goes down, the reel and your finger are both pulling on the line, until the reel quickly hits your finger and stops. The process then reverses, although due to the energy lost to drag it won't go as far or as quickly.

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