Escape the Frictionless Circle

[PTGFlyer]

Is there gravity in this situation?

[Jeff_UK]

I wonder...

Spoiler:

I wonder whether you could use the fact that the speed of light is finite to your advantage.. say you set yourself up in 'orbit' of the block such that you're both orbiting a fixed point (probably somewhere just outside the block). I'm sure this could be arranged with kicking yourself away from the block at eccentric angles...

If you could then somehow provide constant acceleration, by slowly pulling yourself toward the block for instance, and you could reach the speed of light, the block (on a smaller radius orbit) would be going slower than you, and still be able to accelerate, whilst you stayed at the same speed. This would then have the effect of unbalancing the system and throwing you both out of the circle.

I don't know what would happen when you left the no-friction/vacuum zone travelling at or around e.

[SDK]

Just because you're going very close to the speed of light, doesn't mean you can't still accelerate. You can't reach the speed of light, ever, period. There will always be more room to accelerate, from 0.99c to 0.999c to 0.9999c and so on.

I'm also not sure it's possible to start your "orbit" anyway. Since everything is frictionless except your interaction with the block, it gets hit by exactly the same force you do, moves directly away from you, then bounces back to meet you in the middle again. I don't think there's any way around that.

[jmose]

Here is my guess:

Spoiler:

Somehow, climb on top of the gold cube by pulling the rope.
Stand upright
Spin your arms in reverse vertical circles
This should induce a spin (precession?)
Stop spinning 1 arm to produce a torque
This should move you perpendicularly to your spinning arm, like a gyroscope.

Jim

[mfb]

@Jeff_UK: You cannot start such an orbit, unless you spin in the opposite direction very fast (or let the block spin). But it would not help, conservation of momentum (both linear and angular) is still exact in special relativity.

[PeteP]

Talking about light speed, can you do anything with length contraction?

[jasondrake]

Sort of a trick answer:

Spoiler:

Pick apart the rope. You should be able to acquire three 50-metre strands without breaking your connection to the gold brick. Form these into one long, thin rope, and throw it to the outside of the circle where it can make contact with the friction area (a loop at the end will speed things along, but technically any friction outside the circle will put you on the path to escape).

And an answer that quintopia already touched upon:

Spoiler:

Radiation.

Cradle the brick between your legs, reach around to the far side (which we'll call "East") and start pounding a dent into the middle of it. The dent is not important itself, but it gives you a sense of accomplishment. What is important is that the force of your limbs is imparting momentum to the gold molecules, which must necessarily take the form of oscillations a.k.a. "heat".

The brick's net momentum is zero, but at any given time there are molecules vibrating eastward and others vibrating westward.

When you feel you need a break, huddle up to the west side of the gold brick. Most of the heat energy which flows out the west side (robbing the brick of some of its westward momentum) should transfer to your body. But, thanks to your biological circulatory system, the energy lost to the whole Brick+You system via infrared radiation should flow outward in all directions equally, ergo the momentum change from your body's radiation is zero.

The east side of the brick, where you have been pounding on it, is hotter. And it is not insulated by a biological (keep your hands off the dent). The dented part will emit infrared radiation towards the East, and electromagnetic radiation has momentum. This process means that individual molecules lose energy and momentum while they are on the eastward vector of their vibration cycle, leaving the whole brick with a net westward momentum.

That momentum was provided by your limbs (pounding on the brick). You are not ejecting anything from your body (or if you are, it is infrared with a net momentum vector of zero), because the winning radiation is coming from the gold brick.

Wait patiently, or repeat as desired, until you reach the uttermost West.

(You could also achieve this, I think, by pulling the rope across one side of the brick to create friction. That risks breaking the rope-- which might not be a bad thing. You lose the challenge if you break your tether, but at least you can escape, and, if you hurry, contrive a means to recover that big chunk of gold.)

[mfb]

@jasondrake: That approach is so much easier:

Spoiler:

Go to one side of the block. You are probably warmer than the block. You emit radiation roughly isotropically, but the block will absorb some part and emit it isotropically again. In total, you get a bit of net thrust, and you move (block first). Acceleration is (order of magnitude) 10^-10 m/s², so after about two weeks you reach the border.
What you suggest would reduce the effect by heating both sides.

[lorb]

You can create angular momentum in various ways, like grabbing opposing corners of the cube and exerting force of them. Such processes can be repeated to make you and the block spin very very fast, and they whole no-friction etc. doesn't ever make you slow down. As you get closer to c relativistic effects will provide a solution, assuming you magically survive this.

[mfb]

Momentum is still perfectly conserved in special relativity.
Also, the block flies apart long before you reach relativistic velocities. Also, you fly apart long before the block does.

[lorb]

Is flying apart not a kind of escaping the frictionless circle?
Apart from that. If you actually did spin at relativistic speed by length contraction the circle we are trapped on would have to shrink and maybe enable us to escape?

[Soupspoon]

I had wondered if there could be some sort of trick regarding the COM of a spinning block/you system that could be manipulated by drawing the block closer to you (and you to the block, far more) at one part of the spin and then letting it out in another part of the spin, but:
a) The COM wouldn't change, that's the whole point,
b) How are we getting the spin, anyway, to uselessly exploit thus?

My final answer, though, to ruthlessly pinch from the solution to another (less impossible) problem of physics:

Spoiler:

Find some way to communicate with someone outside the circle (through sign-language, hence sticking to limb forces only...) and suggest that if they help out, they shall get a share of the (I think) $6million lump of gold currently in your possession. (Bonus points if it's Steve Austin!)

No?

[LockeZ]

The important thing to me is that the rope connects you to the gold. So you can't throw a piece of gold, because any piece you break off will still be connected to you.

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.

What is a solution is to wait for the gold to start to radioactively decay, and then watch the protons and neutrons very closely as they're expelled. (Don't worry about the electrons, they're not a big deal and they're a lot harder to see.) Every time you see a proton or neutron flying off of the east side of the gold bar, punch the proton or neutron as hard as you can. The subatomic particle will be sent forward and you will be sent backward, to the west.

After a few octovigintillion years, you should make it to the edge with that plan.

But if you have a hot date you don't want to miss, here's a much faster solution, using logic instead of physics:

Spoiler:

Break your finger open so that the broken bone is exposed, and use the sharp broken bone to cut partway across the rope in a zigzag pattern so that it becomes 200 meters long. Then just push yourself away from the gold block.

[LjSpike]

Well, I've only read through some solutions but...

Spoiler:

We can't create energy alright, so the net movement must be nothing. Now, the problem was very specific in a block of gold, and a rope of 50 meters. So, using our finger nails (were in perfect health! A googleplexian of years should give us time to grow nails, and then do the next step, although the growing of nails would give us momentum if we held our hands and legs in one direction? But, anyway, lets keep that out of the equation).

As I said, using our finger nails, we then cut both the gold block, and rope in half (it doesn't really matter the masslessness of the rope, but, hey ho?). We now have 100M of rope, attached to either hand (we can't break the tether, but can move it round our body!), then we have 2 chunks of gold, and really horrible nails which should be in the guiness book of records.

Currently this is getting us nowhere, we have 100M of rope, the radius of the circle, kind of pointless?
We do however have the distance created by the block itself. Thats not to crucial, but if you wish to factor it in, you can now be infinitely small and solve the problem. I'll factor in the width of me with my arms spread out. Lets say 0.8 Meters for arguements sake, so as not to over-estimate. If we fling one cube in a direction, so that it heads to the edge of the circle, and we fling the other cube the other way to the center, we now should have a cube hit the center, so what, it holds us in place, and the other cube hit the edge, which ISN'T FRICTIONLESS! Woo, now we just slowly clamber towards that cube, every time it slips back in, repeat the process.

In many years time, you'll escape, to kill whichever cruel person trapped you in there!

[gmalivuk]

But you're 100m from the edge, and I don't see how your scenario gets anything past that or how you move your center of gravity away from the center of the disc in the first place.

[Neil_Boekend]

Well, I've only read through some solutions but...

Spoiler:

We can't create energy alright, so the net movement must be nothing. Now, the problem was very specific in a block of gold, and a rope of 50 meters. So, using our finger nails (were in perfect health! A googleplexian of years should give us time to grow nails, and then do the next step, although the growing of nails would give us momentum if we held our hands and legs in one direction? But, anyway, lets keep that out of the equation).

As I said, using our finger nails, we then cut both the gold block, and rope in half (it doesn't really matter the masslessness of the rope, but, hey ho?). We now have 100M of rope, attached to either hand (we can't break the tether, but can move it round our body!), then we have 2 chunks of gold, and really horrible nails which should be in the guiness book of records.

Currently this is getting us nowhere, we have 100M of rope, the radius of the circle, kind of pointless?
We do however have the distance created by the block itself. Thats not to crucial, but if you wish to factor it in, you can now be infinitely small and solve the problem. I'll factor in the width of me with my arms spread out. Lets say 0.8 Meters for arguements sake, so as not to over-estimate. If we fling one cube in a direction, so that it heads to the edge of the circle, and we fling the other cube the other way to the center, we now should have a cube hit the center, so what, it holds us in place, and the other cube hit the edge, which ISN'T FRICTIONLESS! Woo, now we just slowly clamber towards that cube, every time it slips back in, repeat the process.

In many years time, you'll escape, to kill whichever cruel person trapped you in there!

This does give me an idea:

Spoiler:

Wait, grow nails that extend to outside of the frictionless circle, use them to get some grip and drag yourself out. Drag the gold block with you because you'll probably be legally dead and your assets inherited according to your will. The monetary value of the gold will help you get back on your feet.

[LjSpike]

Well, I've only read through some solutions but...

Spoiler:

We can't create energy alright, so the net movement must be nothing. Now, the problem was very specific in a block of gold, and a rope of 50 meters. So, using our finger nails (were in perfect health! A googleplexian of years should give us time to grow nails, and then do the next step, although the growing of nails would give us momentum if we held our hands and legs in one direction? But, anyway, lets keep that out of the equation).

As I said, using our finger nails, we then cut both the gold block, and rope in half (it doesn't really matter the masslessness of the rope, but, hey ho?). We now have 100M of rope, attached to either hand (we can't break the tether, but can move it round our body!), then we have 2 chunks of gold, and really horrible nails which should be in the guiness book of records.

Currently this is getting us nowhere, we have 100M of rope, the radius of the circle, kind of pointless?
We do however have the distance created by the block itself. Thats not to crucial, but if you wish to factor it in, you can now be infinitely small and solve the problem. I'll factor in the width of me with my arms spread out. Lets say 0.8 Meters for arguements sake, so as not to over-estimate. If we fling one cube in a direction, so that it heads to the edge of the circle, and we fling the other cube the other way to the center, we now should have a cube hit the center, so what, it holds us in place, and the other cube hit the edge, which ISN'T FRICTIONLESS! Woo, now we just slowly clamber towards that cube, every time it slips back in, repeat the process.

In many years time, you'll escape, to kill whichever cruel person trapped you in there!

This does give me an idea:

Spoiler:

Wait, grow nails that extend to outside of the frictionless circle, use them to get some grip and drag yourself out. Drag the gold block with you because you'll probably be legally dead and your assets inherited according to your will. The monetary value of the gold will help you get back on your feet.

True. But with splitting the gold in half, you can use half to get back on your feet, and half to invest in a patent for this frictionless circle, (nobody has to know you didn't make it, and with how long you've been there, unless everyone is trapped in one, I don't think they'll question it).
I mean, it must have some real practical use, like aircraft bodies? Then you could pay a private investigator to find out who actually put you there in the first place.

[Shufflepants]

Solution:

Spoiler:

Spacetime Swimming

Assuming we are in a positively curved space time (which seems like a fair assumption given our scenario has gravity), it ought to be possible to employ space time swimming by repeatedly following the steps listed here:
http://www.science20.com/hammock_physic ... mpty_space

"Imagine a two-dimensional three-legged creature moving frictionlessly over the surface of a sphere.

Let's say this creature is positioned at the equator with one leg pointing east and two legs pointing along the lines of longitude towards the north and south poles. The swim stroke consists of four moves. First the tripodal creature extends its two longitudinal legs, and subsequently it extends the eastern leg. To complete the stroke, it retracts the longitudinal legs and finaly retracts the eastern leg. As a result of each such 'swimstroke' the creature moves a wee bit westward."

For the longitudinal legs we can use our own legs (we've got plenty of time to stretch enough to learn to go into the splits) and for the 3rd leg we can push away and retract the gold block.

Note that these means do not impart any net kinetic energy, so we won't be able to use this to just build up some speed and coast, we'd have to continue to keep swimming the whole way to the edge.

Also of note, the actual amount of translation achieved with each stroke is going to be immeasurably small. I haven't done the math (because I don't know how) but I would guess that the amount of translation you'd get from a single stroke (considering the very small distances and the very nearly flat curvature) would be so small that we would likely still be on our way to the edge of the circle when the heat death of the universe rolls around. But in principle we would eventually escape.

And I'm not 100% sure if this would still work if spacetime were laterally symmetric. In that case, this plan might still work by standing up and sitting down instead of extending our legs to the sides to exploit the vertical asymmetry in spacetime.

[Neil_Boekend]

Solution:

Spoiler:

Spacetime Swimming

Assuming we are in a positively curved space time (which seems like a fair assumption given our scenario has gravity), it ought to be possible to employ space time swimming by repeatedly following the steps listed here:
http://www.science20.com/hammock_physic ... mpty_space

"Imagine a two-dimensional three-legged creature moving frictionlessly over the surface of a sphere.

Let's say this creature is positioned at the equator with one leg pointing east and two legs pointing along the lines of longitude towards the north and south poles. The swim stroke consists of four moves. First the tripodal creature extends its two longitudinal legs, and subsequently it extends the eastern leg. To complete the stroke, it retracts the longitudinal legs and finaly retracts the eastern leg. As a result of each such 'swimstroke' the creature moves a wee bit westward."

For the longitudinal legs we can use our own legs (we've got plenty of time to stretch enough to learn to go into the splits) and for the 3rd leg we can push away and retract the gold block.

Note that these means do not impart any net kinetic energy, so we won't be able to use this to just build up some speed and coast, we'd have to continue to keep swimming the whole way to the edge.

Also of note, the actual amount of translation achieved with each stroke is going to be immeasurably small. I haven't done the math (because I don't know how) but I would guess that the amount of translation you'd get from a single stroke (considering the very small distances and the very nearly flat curvature) would be so small that we would likely still be on our way to the edge of the circle when the heat death of the universe rolls around. But in principle we would eventually escape.

And I'm not 100% sure if this would still work if spacetime were laterally symmetric. In that case, this plan might still work by standing up and sitting down instead of extending our legs to the sides to exploit the vertical asymmetry in spacetime.

It probably saves a couple of million years if you're of above-average height.

[jestingrabbit]

That the first answer that I've liked in this thread Shufflepants. Really nice work.

[Tyndmyr]

First off, pushing won't work. Rolling won't work. Tipping won't work. Ignore the surface, it's utterly unhelpful. You, the block, and the rope are important.

Is there any way you could get the gold block swinging around you, like someone warming up for a hammer throw, but with the block still in contact with the ground? Would/could that help?

Hmm... what if you wrapped the rope horizontally around the block and then pulled on it as though it was a yo-yo? What happens then? Would some of the energy be converted to rotational momentum, and would this help?

Yes. Sliding anything on a frictionless surface would be infinitely easy. Simply grasp the weightless rope while fairly close to the block, and drag it around you in a circular motion by rotating your arm.

Obviously, you will begin sliding around in the opposite fashion, but with a frame of reference centered on you, it'll be pretty easy to get the large gold block orbiting you rapidly. Or, seen from the gold block's perspective, you're spinning around it, and also looking ridiculous. So, yeah, you can do that, it's just a matter of "and then what?"

You could use a length of the rope as a cutting edge, I guess. If it's effectively weightless and indestructible, that shit has to be impressive.

Alternatively, gold is more reflective than your average body. Nobody said it was a closed system. Sail your way out, utilizing the gold block as a sail.

[lordofthesnails]

Sorry for the bump but I wouldn't mind someone telling me what's wrong with this:

Spoiler:

Push the cube on the edge away from your own centre of mass, so that you move away from it, it moves away from you a little, but it's also spinning.

Using the rope you can now zero out your speed relative to the cube (arbitrarily close to the cube by holding the rope not-at-the-end) and you're either both moving in a direction, or stationary (but perhaps spinning)

I would suggest that since it now has angular momentum, you're moving together away from the direction you pushed in. Where else has this momentum come from?

Edit: never mind

[Soupspoon]

I have a problem with imparting (all-body) angular momentum.

Individual components can be given angular momentum of their own. e.g. sitting atop the mass, twisting yourself against that mass, you rotate one way, the mass rotates the other, but with sum-total angular momentum of you both being zero, without any way of leveraging against the static-but-frictionless arena beneath.

Same with turning the mass at (horizontal) arms-length/ropeglength/etc, any twist you apply is 'reflected' in a counter-spin dependant upon how you apply that spin but ultimately the 'barycentre' translates not and whole-system rotation is illusionary.

By wrapping round the rope (around yourself and/or the mass) you add localised rotation to one element and counterpart (counter-rotating) rotation to the other. Upon 'unwinding' the rope, you reverse this (faster or slower, but reverse the reverse too) and then you either go full-unravel and then 'reravel' until you hit the brakes or you slip the chord and (assuming the attachment allows zero torsion) merrily mutually oppositely spin to your heart's content forever, but the combined system still isn't spinning in totality.

I think.

Ultimately, there's no advantage, then, in any situation in which disconnection of any part of the system can be used. And if you can, then throwing chunks of block away (or the whole block) for LW near escape seems more useful than rotational escape...

[Theres]

A little bit of necromancy. I am pretty sure that my solution is overlooking something obvious and is thus wrong. However, physics was years ago and I don't understand what is wrong, so maybe someone can explain.

Spoiler:

If I lie down next to the block of gold (or fall over, without breaking something) and then push against it (legs, preferrably, because they are sort of built for that kind of thing) I'd apply equal force to both the block and myself. In my worldview that would mean momentum equal to mass times velocity applied to both.

m(person)*v(person) = m(block'o'gold)*v(block'o'gold)

Now the goldblocky thing is twice as heavy as the average person, so the net result should be half the speed once the legs are away!!!

And at the end of the tether, things happen. Namely the person gets ripped in half or has their back broken or some such. However at that point, isn't it two kinetic energies "pulling" in different directions, with the kinetic energy of the person being 4 times that of the block'o'gold.

E (person) = m(person)/2 * v(person)^2
E (block'o'gold) = - m(block'o'gold)/2 * v(block'o'gold)^2 = -m(person) * (v(person)/2)^2 = - 1/4 m(person)*v(person)^2

And in that case, why won't there be a net "leftover" speed for the person (or possibly dead body at this point) once the kinetic energy of the gold is subtracted? That "massless rope that cannot be broken" has a lot of explaining to do.

[Soupspoon]

Unless I've misunderstood what you're saying, if you've converted unequal masses of initially zero relative (and absolute) momentum to inversely unequal velocities by an opposite-but-equal impulse at separation, then at the limit of the rope the unequal velocities get converted by the opposite opposite-but-equal impulse back to zero. Assuming inelasticity at the terminus. (If you get a spring-back, the lighter part again advances faster backwards, by suitable factors to cause them to rendezvous back at the centre.)

Is that it?

[BigNose]

Initial thoughts are:

Push the block of gold away from you, but apply a spin to it.

Spoiler:

Assume that your weight is 100kg and the gold block is 150kg and that the spin is sufficient that the rope will wrap around the gold block as it moves away from you, to the extent that 10m of the rope ends up wrapped around the gold block. You are now 40m away from the block and ignoring any 'stretch' value of the rope, any angular spin given to you can be negated by rotating your body in the rope loop at your end or hand to hand passing if holding the rope end.
You will now be at 60% of the rope length from centre (24m) and the gold block will be 40%/16m with 10m/20% of the rope coiled around the gold block.

The gold still has momentum as it continues, by unraveling the coiled rope, so will apply a smaller force, but in effect, will now start to drag you towards it. The torque of the spin, captured in the rope, is now being used to pull you and that same spin does not apply to you, having negated it on the way out.

The rope is of 0 mass, therefore has no impact and as soon as the initial outward momentum is complete and the gold starts pulling you back towards itself with the uncoiling of the rope, so there is no longer any opposing forces and you and the gold block would slowly drift to the edge.

[elliptic]

I'm not sure I've completely followed all of that. But whatever you're trying to do, it blatantly breaks conservation of momentum so it can't work, sorry.

[gmalivuk]

Yeah, I think the issue is that you can't just negate your spin unless you pull on the rope hard enough to also stop the block from spinning.

[BigNose]

More Info:

Spoiler:

You negate the spin by turning yourself. You keep turning on the spot, but the gold continues to unravel the rope.
You have to keep passing the rope around you as you slowly drift, following the gold block.
Momentum still exists in you turning on the spot, that momentum is simply NOT applied back to the rope.
It would, if you did not keep feeding the rope around you and you would be right that you end up going nowhere, but its the fact that you maintain your momentum as a spin, right until you hit the edge.

[gmalivuk]

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.

[SDK]

Working the gold block by hand into a 100 meter wire is still my favored solution. Pretty sure this would be possible, but I don't have 8 kg of gold to try it with.

[HES]

That's what kickstarter is for!

[Soupspoon]

Or someone with easy access to gold.

And egress, easy or otherwise. Just make sure to check with 'em if it's been laundered first!

[elliptic]

More Info:

Spoiler:

You negate the spin by turning yourself. You keep turning on the spot, but the gold continues to unravel the rope.
You have to keep passing the rope around you as you slowly drift, following the gold block.
Momentum still exists in you turning on the spot, that momentum is simply NOT applied back to the rope.
It would, if you did not keep feeding the rope around you and you would be right that you end up going nowhere, but its the fact that you maintain your momentum as a spin, right until you hit the edge.

In addition to what gmalivuk said (which is the bottom line) you seem to be treating angular momentum (spinning) and linear momentum (sliding) as interchangeable, so that one can be traded off to gain the other.

They're not and you can't - each of them has to be conserved separately. You can manipulate the spins of yourself and the block to send both of you orbiting round the common centre of mass (which is what your scenario will probably achieve) but you can't use them to *move* the centre of mass.

[speising]

Any solution to the problem presented here would also work as a reactionless space drive.

[RCT Bob]

Spoiler:

1. Push the gold away from you. You'll move in the opposite direction.
2. Eventually the rope pulls you and the gold back, you'll meet again in the middle.
3. Jump over the block of gold, bounce off of it and push it in its direction of motion with your feet to accelerate it (and yourself in opposite direction).
4. Repeat over and over again, increasing the energy of the system. The potential energy in the rope is maximum when you and the block are both at maximum distance from the centre. The rope will stretch further and further due to the force.
5. After a long time, the rope should stretch sufficiently for you to reach the edge (should stretch roughly 3 times its original length if the gold has twice your mass).

Of course it won't be an exact harmonic motion you have, as the tether force isn't proportional to your distance from the centre, but to the change in length of the tether, but I don't the exact kind of motion really matters, the point is eventually you will be pulled back and no energy is lost to the surroundings in any way (vacuum and frictionless situation). Conservation of momentum guarantees that you will always meet that gold cube again every time in the centre of the circle, until you can interact with the ground outside the circle.

The only thing is that the rope is idealised in some way as it's massless and unbreakable. The problem doesn't specify it but this solution assumes it has finite stiffness, as in, it will experience non-zero strain under tension. Of course most regular materials fail before they reach a strain value of 2, but this rope is unbreakable as specified in the problem. Also, I think this involves ridiculous speeds and frequencies after a while, depending on the stiffness of the tether, and you need to have excellent jumping timing to pull this off (not to mention the tether force and accelerations that your body experiences will most likely kill you long before you escape)

[Mutex]

The OP doesn't state but the general assumption seems to be that the circle is on the surface of a planet, possibly a perfectly uniform sphere of uniform density.

Spoiler:

If there is light from a sun above, and the planet is rotating, you could angle your body so there's more light on one side of your body than the other throughout the day, and use momentum from light to push you in one direction. I think Tyndmyr's last suggestion was along similar lines.

[gmalivuk]

Any solution to the problem presented here would also work as a reactionless space drive.

Except the ones that rely on momentum input from or output to outside the system (i.e. from photons) or that involve stretching something (like the rope or the gold) so it reaches beyond the frictionless area.

[BigNose]

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:

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.

[speising]

Any solution to the problem presented here would also work as a reactionless space drive.

Except the ones that rely on momentum input from or output to outside the system (i.e. from photons) or that involve stretching something (like the rope or the gold) so it reaches beyond the frictionless area.

The first of those violates the premise (only forces excerpted by your limbs), the second seems somewhat cheating to me. (rule lawyering)