Escape the Frictionless Circle

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speising
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Re: Escape the Frictionless Circle

Postby speising » Wed Feb 15, 2017 8:48 pm UTC

BigNose wrote:More:
Spoiler:
As stated, when the rope is spilled out by 40m, you negate any spin momentum by spinning on the spot, moving the rope end from hand to hand.
The momentum at the other end is the spin in the block and the directional momentum from where you pushed it.

Consider now that the energy on you is a zero value, (we can now ignore the spin momentum on me) the rope is taut, but the block contains both directional momentum and spin momentum, therefore, there is more energy at the gold end that at your end.

Thus, the block will continue to move in the same direction, albeit at a slower rate and start to unwind that spin momentum, which it has been doing all the time of its travel, ergo, the gold block will now start to pull you and you and the block are traveling in the SAME direction.

Who cares if the block continues to spin and coil the rope, YOU have directional momentum and will achieve the edge eventually.
Probably very dizzy from all that spinning.
PS
Spoiler:
gmalivuk wrote:I misunderstood what you meant by negating spin, but as it turns out spin isn't relevant to why it can't work anyway.

If the gold block does something to pull in the rope, or if you do something to pull in the rope, a tension is applied to the rope and it pulls on both you and the block equally. There's no way to get around that basic Newtonian fact of the matter.


tl;dr If A + B = C + D, but A is negated, then B <> C + D, ergo you are pulled.

How do you stop spinning without imparting angular momentum to something else?

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Re: Escape the Frictionless Circle

Postby gmalivuk » Wed Feb 15, 2017 9:17 pm UTC

BigNose wrote:Consider now that the energy on you is a zero value, (we can now ignore the spin momentum on me) the rope is taut, but the block contains both directional momentum and spin momentum, therefore, there is more energy at the gold end that at your end.

Thus, the block will continue to move in the same direction, albeit at a slower rate and start to unwind that spin momentum, which it has been doing all the time of its travel, ergo, the gold block will now start to pull you and you and the block are traveling in the SAME direction.
No, this won't happen. The block pulls on you as hard as you pull on the block. It pulls you towards itself, but in so doing it pulls itself toward you. You and the block start by moving away from each other with the same linear momentum (but you're moving faster because you're lighter). Then you both stop moving away from each other (because F=ma means your lighter mass will be decelerated more, and both you and the block will come to a stop after the same amount of force is applied for the same amount of time).

The fact that the block (and you) are spinning around your respective centers of mass has nothing to do with where those centers of mass move relative to each other. The fact that the block winds up the rope just means it transfers its own angular momentum into its own linear momentum. Your own linear momentum responds by changing in exactly an equal-and-opposite way, and you and the spinning block start moving back towards each other.

Exactly the same thing would happen if the gold block were another person who just started pulling in rope at the same rate the spinning gold block would wind it up.
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Re: Escape the Frictionless Circle

Postby SDK » Wed Feb 15, 2017 9:30 pm UTC

Everything in your post is correct, except this part.
gmalivuk wrote:but you're moving faster because you're lighter

Doesn't change the fact that spinning doesn't get you anywhere though.
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Re: Escape the Frictionless Circle

Postby gmalivuk » Wed Feb 15, 2017 10:01 pm UTC

How is that part not right?
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Re: Escape the Frictionless Circle

Postby elliptic » Thu Feb 16, 2017 11:42 am UTC

BigNose wrote:More:
Spoiler:
As stated, when the rope is spilled out by 40m, you negate any spin momentum by spinning on the spot, moving the rope end from hand to hand.
The momentum at the other end is the spin in the block and the directional momentum from where you pushed it.

Consider now that the energy on you is a zero value, (we can now ignore the spin momentum on me) the rope is taut, but the block contains both directional momentum and spin momentum, therefore, there is more energy at the gold end that at your end.

Thus, the block will continue to move in the same direction, albeit at a slower rate and start to unwind that spin momentum, which it has been doing all the time of its travel, ergo, the gold block will now start to pull you and you and the block are traveling in the SAME direction.

Who cares if the block continues to spin and coil the rope, YOU have directional momentum and will achieve the edge eventually.
Probably very dizzy from all that spinning.
PS
Spoiler:
gmalivuk wrote:I misunderstood what you meant by negating spin, but as it turns out spin isn't relevant to why it can't work anyway.

If the gold block does something to pull in the rope, or if you do something to pull in the rope, a tension is applied to the rope and it pulls on both you and the block equally. There's no way to get around that basic Newtonian fact of the matter.


tl;dr If A + B = C + D, but A is negated, then B <> C + D, ergo you are pulled.


Here's the set-up (ignoring your own spin which plays no part as you correctly note).

Image

What you apparently think you're doing is converting the gold block's spin into your own linear momentum.

What you're *actually* doing is exchanging the block's angular momentum for angular momentum of *both* bodies round the common c.o.m, while the linear momentum being imparted to you gets balanced by an opposite change in the block's linear momentum.

Here's a force diagram:

Image

The offset pull from the rope on the gold block resolves to a linear force through its c.o.m. and a couple acting against the block's spin. Also note that the tension in the rope doesn't act through the common c.o.m. so there's *another* couple acting on the combined system which opposes the first one. Thats how the angular momentum is accounted for - as the two bodies are pulled together the block's rate of spin slows down but instead we now have a rotation of the whole system around the common c.o.m. which wasn't there before.

Meanwhile the linear forces on the blocks are simply the tension on the rope, hence obviously equal and opposing. So there can be no change in the combined linear momentum and the c.o.m. *does not move*.

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Re: Escape the Frictionless Circle

Postby lordofthesnails » Thu Feb 16, 2017 5:09 pm UTC

LockeZ wrote:So here's a physics solution:
Spoiler:
Gold, like every other element, has a rate of radioactive decay. And the rope does not connect you to the radioactively expelled protons, neutrons and electrons that aren't gold any more.

The only thing that doesn't have a rate of radioactive decay is you. Because the rules explicitly state that you can't expel anything from your body. So you're not allowed to decay, even if the laws of physics want you to.

In theory you could get as far away from the gold as possible and then wait a very long time until the gold decays, which would shift your center of mass away from the center of the circle. The more the gold decays, the further your center of mass would get. But I think the furthest you can ever shift in this manner is 50 meters plus half the length of your body, since that's the furthest you can get from the gold. So, while this would be some sort of progress, it wouldn't be a solution, unless you're 50 meters tall.


If that's allowed...
Spoiler:
Even a single electron leaving a moving gold block is enough give your system momentum. Push the block away, stop at the end of the tether and then ride your miniscule momentum to the edge of the circle.

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Re: Escape the Frictionless Circle

Postby SDK » Thu Feb 16, 2017 10:15 pm UTC

gmalivuk wrote:How is that part not right?

Because I'm bad at math. This cube is, in fact, 160 kg, not 8 kg. That kickstarter just got a whole lot harder. :(
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Re: Escape the Frictionless Circle

Postby BigNose » Thu Feb 16, 2017 10:15 pm UTC

Last gasp
Spoiler:
Consider the point at which the rope is at 40m out.
During that period, the gold block with the rope wrapping around it, is storing energy.
As the rope is not wrapping around you, the total energy is in your spin and not in the spin and momentum.
The gold block will have slowed as it would have been pulling on against both you and the energy built into the rope as it wrapped around the gold block.
At that point, you have the same amount of energy as the gold block spinning and moving.

The rope is taut at 40m and you are spinning.
The rope has 10m wrapped around the block which is also spinning, but the golf block won't stop its momentum, as the energy released is a continuation of the momentum PLUS spin you applied, with now the reverse spin on the gold block, as it begins to unwind the stored energy in the wrapped rope.

The only way that the gold can unwind, is if it has momentum, otherwise it would simply spin on the spot, winding and unwinding the rope length of 10m.

If it does, then simply pull only the rope when it is fully coiled around the gold. As you have only spin energy and you are pulling against a static object with spin energy (the same as yours), then you will start to move back together. Simply split your legs as you pass the gold, out to the edge.

PS Don't forget to let go of the rope Bollox! You can't be separated from the rope!
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Re: Escape the Frictionless Circle

Postby gmalivuk » Thu Feb 16, 2017 10:50 pm UTC

BigNose wrote:During that period, the gold block with the rope wrapping around it, is storing energy.
No, it isn't.

The gold block will have slowed as it would have been pulling on against both you and the energy built into the rope as it wrapped around the gold block.
No, it won't. The block spinning with a rope that isn't taut isn't slowing down at all, because the rope doesn't impart any force until it's taut. (This is why there's no energy stored in the wound-up rope.)

The rope has 10m wrapped around the block which is also spinning, but the golf block won't stop its momentum, as the energy released is a continuation of the momentum PLUS spin you applied, with now the reverse spin on the gold block, as it begins to unwind the stored energy in the wrapped rope.

The only way that the gold can unwind, is if it has momentum, otherwise it would simply spin on the spot, winding and unwinding the rope length of 10m.
If the gold still has enough linear momentum to continue in its same linear direction and reverse its rotational direction to unwind, then it hasn't stopped you from moving away from the center, it's just slowed you down. It gives up linear momentum to reverse the direction of its rotational momentum, but it takes the same amount of linear momentum from you, because to reverse its rotational direction it applies a tension on the rope, which pulls both it and you toward the center with the same force.

The spin is a red herring. It doesn't magically let you avoid considering what actually happens to the linear momentum of the bodies involved.
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Re: Escape the Frictionless Circle

Postby Sableagle » Fri Feb 17, 2017 10:12 am UTC

I diguise myself as an unarmed black teenager. 30 x 7.45 g x 390 m/s = 87,165 gm/s, so a couple of magazines from a Ferguson PD hero ought to get me doing about 39 cm or 1¼ feet per second away from him. The magic keeps me in perfect health, and by the time he's finished deleting his body camera footage I'll have slid to the edge.
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Re: Escape the Frictionless Circle

Postby Freddino18 » Sun Feb 19, 2017 2:36 am UTC

Response to previous post: WHY?!

Going back to the "grow fingernails until they are long enough" approach, it would actually be easier (and faster) to grow out your hair.

Based on the CDC's estimate of about six inches' worth of hair growth per year for the average person, it would take about 116 years to grow your hair long enough to reach the edge (maybe less, depending on starting length and relative growth speed). This could be reduced to as short as four days to reach the edge with a single strand sourced from all the hairs on your body (based on another estimate that all of the hair on your body grows by a total of about 30 ft per day). Every four days, you could add another strand to your hair rope, and as they would still be attached to your body (by the base strand), this does not violate the rule against ejecting mass. In fact, if you have any sort of length of existing hair on your body, you can start out with a multi-strand "head start", drastically reducing your time to escape. This idea could be combined with the previous one, by creating a hair whip to aggravate the cop.

Another idea, stolen from another thread, would be to have a negative mass with equal but opposite mass as you+the gold block, which would serve to provide propulsion without violating the law of conservation of momentum (negative mass means negative momentum).
Nicias wrote:I actually did my doctoral dissertation on negative mass, but mostly on cosmological scales.

A matched pair of positive and negative mass particles would indeed accelerate forever. This doesn't violate conservation of energy, since the negative mass has negative kinetic energy.

Generalizing, a gas with negative mass and positive mass particles would have particles of both masses have their average particle speeds increase without bound. The temperature would not increase since a negative mass gas has negative temperature. Since we do not observe this, we can be confident that there are no negative mass particles in the universe. (I'm not super confident about the exact details of the gas situation. It's been years since I've looked at this stuff.)


Edit: After leaving the circle, I would get patents for/sell: the gold, the tether, the negative mass object, the circle, and the magic life/health stuff (as EverLife 9 3/4)
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Re: Escape the Frictionless Circle

Postby LaserGuy » Thu Mar 02, 2017 4:52 am UTC

I think the first response is probably basically right. You just need to fall over. Wrap the rope around yourself until you have a taut line connecting you and the block. You should have a large angle connecting the little block to your torso... your collective centre of mass, then, must be somewhat higher than the block, or, at least, higher than the centre of mass of the block alone. If you fall outward, your centre of mass will be reduced in height and will move away from the block. Since the centre of mass moves outward, it must have a small velocity in the radial direction, and since there's no friction, this should persist indefinitely and carry you out.

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Re: Escape the Frictionless Circle

Postby jaap » Thu Mar 02, 2017 9:35 am UTC

LaserGuy wrote:If you fall outward,

That's a big if. How do you fall outward if your feet slide inward?

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Re: Escape the Frictionless Circle

Postby Eebster the Great » Fri Mar 03, 2017 3:10 am UTC

The puzzle as phrased has only gimmicky solutions. Changing the center of mass of a system requires an unbalanced external force. In the idealized scenario, making reasonable assumptions (e.g. level ground, no magnetic field), the only external force on the system is gravity, which is exactly balanced by the normal force on the ground, and friction, which by hypothesis is zero within the circle. This leads to the conclusion that any solution requires a part of the system reaching the ground outside the circle while the system's center of mass remains at its center.

These are the "gimmicky" solutions, such as stretching the rope many times its original length, beating the gold into thread, or building a rope out of hair. By the dimensions given in the problem, these are highly implausible, so I think it is basically indisputable that the puzzle has no plausible solutions.

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Re: Escape the Frictionless Circle

Postby Sizik » Fri Mar 03, 2017 8:13 pm UTC

Stand to top of the block. Spin in place, causing the block to spin in the opposite direction, conserving angular momentum. Keep spinning faster and faster, until the gold block flattens and expands outwards far enough to reach the edge of the circle.

This assumes the rope won't be twisted up in the process.
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Re: Escape the Frictionless Circle

Postby Soupspoon » Fri Mar 03, 2017 11:35 pm UTC

From conservation of angular momentum, I predict that your rate of counterspinning will be by far the most notable and, edicts against permanent personal injuries aside, you would more obviously be centripetally flattened out (or worse!) beyond the frictionless surface limit.

So maybe that's when you succeed. Need to compare the elastic modulus and elastic limit of your magically-sustained physique to that of the gold, of course, to work out which event precedes...

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Re: Escape the Frictionless Circle

Postby Sizik » Sat Mar 04, 2017 12:05 am UTC

Soupspoon wrote:From conservation of angular momentum, I predict that your rate of counterspinning will be by far the most notable and, edicts against permanent personal injuries aside, you would more obviously be centripetally flattened out (or worse!) beyond the frictionless surface limit.

So maybe that's when you succeed. Need to compare the elastic modulus and elastic limit of your magically-sustained physique to that of the gold, of course, to work out which event precedes...


I was assuming that the "magically staying alive" bit would keep you're shape together, but it could also allow for personal flattening to happen as well.

Also, as the radius of the gold increases, you can jog around in a circle instead of spinning.
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King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.


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