## Escape the Frictionless Circle

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doogly
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### Re: Escape the Frictionless Circle

If the surface can't absorb it by deforming, you bounce or break or something fun like that.
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curiosityspoon
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### Re: Escape the Frictionless Circle

Stand on the gold brick. Wrap the cord around your waist repeatedly to get rid of the slack until it is taut, then lunge up and in some direction. If you can jump with enough force to nudge the brick up off the ground by any amount (let's say it outweighs the average person by 2:1 so you'd be lifting three times your body weight), it will follow your momentum and start sliding.

PeteP
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### Re: Escape the Frictionless Circle

curiosityspoon wrote:Stand on the gold brick. Wrap the cord around your waist repeatedly to get rid of the slack until it is taut, then lunge up and in some direction. If you can jump with enough force to nudge the brick up off the ground by any amount (let's say it outweighs the average person by 2:1 so you'd be lifting three times your body weight), it will follow your momentum and start sliding.

When you jump in some direction you push the gold block in the other direction and since it is on a frictionless surface it will begin sliding in that direction. And then your horizontal movement and its horizontal movement cancel each other out.

Sandor
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### Re: Escape the Frictionless Circle

jaap wrote:I can probably rule out Coriolis forces because the puzzle does say "Using only forces exerted by your limbs".

A lot of people seem happy to try and use gravity, which also isn't exerted by your limbs. I don't see why gravity would be OK, but Coriolis or centrifugal forces wouldn't.

doogly
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### Re: Escape the Frictionless Circle

Gravity totally comes from your limbs. F = m_limb m_other thing / r^2.
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Chen
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### Re: Escape the Frictionless Circle

doogly wrote:Gravity totally comes from your limbs. F = m_limb m_other thing / r^2.

Hmm if the surface is perfectly flat wouldn't some of the net gravitational force at some points on the circle not be directly normal to it? This would mean there'd be a net gravitational force acting on you and the gold block's CoM with a lateral component. Means you'd just eventually "fall" out of the circle anyways.

douglasm
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### Re: Escape the Frictionless Circle

suriya wrote:When I sit on a swing, i can put myself in motion by moving my legs forwards and backwards to shift my center of mass, without pushing myself off the ground. Would this help in any way here?

When you do that, you're not actually shifting your center of mass. What you're really doing is shifting the point where the swing is attached to you. Without a swing, this technique isn't applicable.

Sandor
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### Re: Escape the Frictionless Circle

doogly wrote:Gravity totally comes from your limbs. F = m_limb m_other thing / r^2.

There is no "m_other thing" in this scenario that has mass. If there where it would be exerting a force (gravity) on you, and that's specifically disallowed.

The minutiae of the set-up is crucial, and the OP hasn't provided it. Is there gravity or not, and if so from what? Is the surface in a rotating frame of reference (e.g. on the Earth's surface)? What does flat mean (mathematically flat, or perpendicular to local gravity)? And so on...

doogly
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### Re: Escape the Frictionless Circle

I feel like it was safe to assume it was meant to take place on the earth, but yeah, it's a weird setup.
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tomandlu
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### Re: Escape the Frictionless Circle

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?
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Chen
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### Re: Escape the Frictionless Circle

tomandlu wrote: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?

The block being in contact with the ground or not here is not terribly relevant. Neither you nor the block can exert any lateral force on the ground since it's frictionless and flat.

As I mentioned earlier the only lateral net force I can think of would be gravity if, whatever the perfectly flat circular area is sitting on, is not perfectly uniform. Even then, if the CoM of the system is at the dead center of the circle, you'd need there to be more mass in one direction compared to the opposite direction to get you moving at all. Consider this flat surface on the top of a ball. The gravitational force as you move towards the outside of the circle would actually have a component that pulled you BACK towards the center of the circle assuming the contact point of the flat surface was the exact center of the circle and the ball was perfectly spherical. Now if it's on earth we know the "ball" is not perfectly spherical so there could be a chance that the gravity vector wouldn't pass directly through the center of the circle, in which case you'd "fall" out of the circle eventually, if the vector was pointing outwards. I suppose something like having a tree on one side of the circle and nothing on the other could also lead to the very slight gravitational force of the tree acting on you to pull you out of the circle too.

mward
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### Re: Escape the Frictionless Circle

Spoiler:
Stretch the tether (without breaking it, of course) to a suitable length. Push the cube away from you. Wait until you reach the edge of the circle.
Or even simpler:
Spoiler:
Untie the rope (without breaking it, of course). Push the cube away from you.

Sandor
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### Re: Escape the Frictionless Circle

Chen wrote:Consider this flat surface on the top of a ball. The gravitational force as you move towards the outside of the circle would actually have a component that pulled you BACK towards the center of the circle assuming the contact point of the flat surface was the exact center of the circle and the ball was perfectly spherical.

In this setup, if you pushed the gold block slow enough that both you and it started sliding back towards the centre before the limit of the rope was reached, and if from then on you jumped over the block to avoid hitting it, then you and the block would both oscillate around the centre forever.

However the force pulling you (and the gold) back towards the centre would not be proportional to the distance from the centre, so this wouldn't be simple harmonic motion. This implies (I think) that it will take a different amount of time for you and the gold to reach the centre, so the crossing over point won't be at the centre. If you grab hold of the gold (instead of jumping over it), you would find you and the gold off-centre and with some residual velocity. It's possible that repeating the procedure enough times will allow you to reach the edge (although I'm very unsure about this bit).

This is all based on my intuition - I haven't done any calculations. So it may all be rubbish

PeteP
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### Re: Escape the Frictionless Circle

So another question. You are standing outside the circle with the tether and the gold block. Is there any reliable way to get into the initial position in this setup? I mean you could drop down from the air and if you have no horizontal component to your movement you will stay there, but no horizontal component at all is not exactly simple.

Sandor
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### Re: Escape the Frictionless Circle

PeteP wrote:So another question. You are standing outside the circle with the tether and the gold block. Is there any reliable way to get into the initial position in this setup?

In this kind of idealised setup, the physics are reversible. If you can somehow get out from the centre, you can reverse the procedure to get into the initial position instead.

SirGabriel
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### Re: Escape the Frictionless Circle

Random thought: is there any way you could create static electricity in this environment and use that to exert some net force?

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### Re: Escape the Frictionless Circle

jestingrabbit wrote:Question: what happens if you fall over? You've lost gravitational potential energy, so where does it go?

Heat. If you had some way to eject heat (as black body infrared realisation) in a particular direction, you could use that, but that may be forbidden by the "you can't eject anything" clause.

Jeff_UK
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### Re: Escape the Frictionless Circle

If the rope is massless, and we're in a vacuum.. does that mean I can throw the rope at the speed of light?

I don't think it helps, but it would be lots of fun...
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### Re: Escape the Frictionless Circle

As far as I can tell, you either need to alter the surface in some way, release energy in some direction, or exploit the minute acceleration of the surface as the Earth rotates and the moon orbits Earth, and centrifugal force.. I think the last one is automatic, if the circle isn't so big as for the gravity to be enough to pull you towards the centre, as you slowly come away from the centre of earth.

On the other hand, as I know nothing about how wormholes work, I want to use Jeff's idea to make one, and hence displace the entire system slightly.
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mward
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### Re: Escape the Frictionless Circle

Push the block away from you in the current direction of the moon: the differential tidal force should create a net acceleration.

BedderDanu
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### Re: Escape the Frictionless Circle

Is it possible to use the friction that exists between you and the gold block to accomplish anything?

Say I climbed on top of the block and jumped off, but my shoes slipped while jumping, causing the gold block to squeak and heat up. Wouldn't that cost it some of the momentum I'd be transferring it? Or would symmetry still cause that to be lost equally by both me and the gold?

qetzal
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### Re: Escape the Frictionless Circle

Or, what if you rub one side of the gold block, making that side get warmer. Could that do anything? Maybe the higher temp of that side would slightly increase the (infinitesimal) evaporation rate of gold atoms, creating a *tiny* net force on that side? 😀

Edit: More seriously, would the following work? [Answer: no.]
Spoiler:

I was thinking the extra angular momentum would come from the energy expended by your muscles, and that it would cause you to hit the block faster than when you pushed off, resulting in a net directional movement. Started posting before I thought it through. I'll leave this here, though, in case it inspires anyone else.

DrZiro
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### Re: Escape the Frictionless Circle

Cradarc wrote:Here's my idea, but I'm not sure if it will work:
Spoiler:
You wrap the rope around yourself so you become something like a top. You then push the gold block directly away from you. As the rope spools out, you should start to spin about your (not the system's) center of gravity. To conserve angular momentum, the block must also rotate about your center of gravity, albeit, very slightly. This would then shift the center of mass of the system. Eventually the rope will run out, and you use your angular momentum to start winding it in the other direction. At some point, this will cause you to stop spinning, and therefore the block must also stop rotating about your center of mass. You then reel in the rope in a linear fashion and repeat the process.

Spoiler:
I don't think angular momentum is conserved here, since the person-gold system is not a rigid body.

I thought for a while that that might be the key to the answer. By spinning your arms appropriately, you would be able to rotate yourself, and thus you could also sit on the gold and rotate that. Presumably it should also be possible to rotate the whole system, going into orbit around the cube. It seems like that might be helpful, but now I'm getting more convinced that it's not.

Ignoring outside-the-box solutions and wordplay, there is no way to move the person more than 50 m from the CM. There is also nothing that can exert a force on the system. Although angular momentum isn't conserved, linear (horizontal) momentum is, so the CM can't move. That should be sufficient evidence that there is no solution.

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### Re: Escape the Frictionless Circle

Spoiler:
I think you can totally melty-brain this. I mean, if you don't literally melt your brain attempting it anyway.

just instead of modulating power to a drive wheel, you throw your arms out or do something else to change the CoM of the rotating system at a particular point on each rotation to set the whole system drifting
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### Re: Escape the Frictionless Circle

For those talking about rolling, you can't roll without friction.

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### Re: Escape the Frictionless Circle

EdgarJPublius wrote:instead of modulating power to a drive wheel, you throw your arms out or do something else to change the CoM of the rotating system at a particular point on each rotation to set the whole system drifting

Throwing your arms out with nothing to anchor you or push against doesn't change your CoM. It just throws the rest of your body the other way to compensate.

DenisL704
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### Re: Escape the Frictionless Circle

Will this work? (assuming you're very strong, being in perfect health and all)

1. Pick up block, drop it on opposite side from where you picked it up from.
2. Climb over block.
3. Repeat 1 and 2.
4. Win.

Would picking up the block and moving it through the vacuum to the other side cause your feet to slide back to the original COM?

jestingrabbit
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### Re: Escape the Frictionless Circle

DenisL704 wrote:Would picking up the block and moving it through the vacuum to the other side cause your feet to slide back to the original COM?

Yeah.
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Demki
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### Re: Escape the Frictionless Circle

DenisL704 wrote:Will this work? (assuming you're very strong, being in perfect health and all)

1. Pick up block, drop it on opposite side from where you picked it up from.
2. Climb over block.
3. Repeat 1 and 2.
4. Win.

Would picking up the block and moving it through the vacuum to the other side cause your feet to slide back to the original COM?

You would have no hold on the ground, so as you attempt to move the block you will move in the opposite direction, you'll probably fall over before you even pick it up.

PsiSquared
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### Re: Escape the Frictionless Circle

I think Znirk got the general gist of a solution in his first post:

Znirk wrote:Now presumably there's a reason why the cube is specified to be gold. Gold is relatively soft and ductile, which suggests approaches like A) working free a bit of gold, throwing it, and waiting a few weeks (a long time to hold in all bodily ejections, but that's a biological rather than a physical problem); or B) changing the shape of the cube into something long enough to reach the high-friction outside of the circle.

(A) may be against the rules of the problem, but (B) looks like it could work.

Just how soft is 24-carat gold? Can you flatten 20-cm cube of gold by repeatedly jumping on it?

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### Re: Escape the Frictionless Circle

PsiSquared wrote:Just how soft is 24-carat gold? Can you flatten 20-cm cube of gold by repeatedly jumping on it?

I'm thinking soft enough that you could yank the chain off the block, if you put some effort into it, but not quite soft enough that it would be practical to treat it like a goomba. At least not in a frictionless vacuum.

But since "there is no way to break your tether to the gold block", presumably no amount of gold can be separated from the block. Otherwise, if you could carve pieces off, there would come a point where what the rope was connected to did not qualify as "the block" which it can't be separated from.

There's certainly more than enough gold there to make a rod the length of the diameter of the circle, given enough time and body heat (unless that counts as ejecting something, in which case, well, don't touch anything, especially not the gold or the vacuum).
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SDK
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### Re: Escape the Frictionless Circle

Lead is pretty easy to shape by hand. Gold is softer than lead. Therefore, this'll work.
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### Re: Escape the Frictionless Circle

SDK wrote:Gold is softer than lead.

I don't think it is...

I might be reading my hardness charts upside-down, but I'm pretty sure lead is much softer[1].
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### Re: Escape the Frictionless Circle

tomandlu wrote: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?

Since there's no friction to slow you down... Start swinging the gold block around you, like someone warming up for a hammer throw. Keep swinging. More swinging. More angular velocity. With enough time you can basically create a centrifuge of arbitrary power rotating around the center of mass. The gold-rope force will start to deform the cube into an elongated shape. (Hopefully the cube is big enough to result in a 50m length of gold... The other option is that the (presumably indestructible) rope cuts it in half and two chunks go flying out of the circle at extreme velocities, which I think would break the rule about tethering.) Once the gold is long enough, start swinging the rope the opposite direction relative to your body to slow down. Use the gold to push yourself to safety.

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Znirk
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### Re: Escape the Frictionless Circle

Magnanimous wrote:Start swinging the gold block around you, like someone warming up for a hammer throw. Keep swinging. More swinging. More angular velocity.

How?

douglasm
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### Re: Escape the Frictionless Circle

Znirk wrote:
Magnanimous wrote:Start swinging the gold block around you, like someone warming up for a hammer throw. Keep swinging. More swinging. More angular velocity.

How?

As described, it simply doesn't work. The attempt at swinging results in your feet sliding ineffectually on the ground without giving you any spin at all.

SDK
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### Re: Escape the Frictionless Circle

eviloatmeal wrote:
SDK wrote:Gold is softer than lead.

I don't think it is...

I might be reading my hardness charts upside-down, but I'm pretty sure lead is much softer[1].

Whoa. Don't know what I'm smoking, but it probably had some lead in it. In reality, gold is somewhere in between lead and copper. Might be possible? I wouldn't want to be trying this with copper, that's for sure.
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tomtom2357
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### Re: Escape the Frictionless Circle

douglasm wrote:
Znirk wrote:
Magnanimous wrote:Start swinging the gold block around you, like someone warming up for a hammer throw. Keep swinging. More swinging. More angular velocity.

How?

As described, it simply doesn't work. The attempt at swinging results in your feet sliding ineffectually on the ground without giving you any spin at all.

Spoiler:
Can I assume I we are allowed to alter the place the rope is attached to me (as long as the rope is always connected at some point)? If so:

Grab the rope with your right hand, and swing it round to behind your back, while at the same time moving your left hand behind your back, now your left arm has balanced some of the momentum of the gold block and your right arm. Now grab the rope behind your back with your left hand, and rotate it in front of you, while moving your right hand in front of you as well. Repeat. This should generate some rotational energy. The rotational energy is balanced by the arm that is not holding the rope, effectively creating two rotating systems.

Alternatively, if you're right next to the gold block, spin the gold block. You would then start spinning in the opposite direction. When you're going fast enough, grab the rope right where it connects to the gold block. Since you are spinning, it will force the (also spinning) gold block (this may work better if the gold block was spherical, can I mold it to a sphere), to also revolve around you.

I have no idea what to do at this point, I'm simply pointing out that you can generate rotational energy, you just have to have the equal and opposite rotational energy as well. Also, the second alternative involves you and the gold revolving in different directions, so it could conceivably smash into you. This is ok, if it is about to do that, simply jump over it.

Actually, you can also climb on top of the gold block and turn clockwise (I assume that the gold has friction), then after you have gained sufficient speed, jump off. I'm not exactly sure what happens next though.
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### Re: Escape the Frictionless Circle

tomtom2357 wrote:Actually, you can also climb on top of the gold block and turn clockwise (I assume that the gold has friction), then after you have gained sufficient speed, jump off. I'm not exactly sure what happens next though.

You continue spinning and the block continues spinning (in the opposite direction).
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mfb
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### Re: Escape the Frictionless Circle

With actual Earth: perturbations from Moon, Sun and other sources are sufficient to give some non-zero acceleration. Just wait.

With an idealized, rotating sphere and no other objects in the universe, if you are not very close to one of the poles:
Spoiler:
Jump up repeatedly. Coriolis force will eventually get you to one side.

If the surface is perfectly flat in a mathematical way and does not follow Earth's curvature and the central point is exactly orthogonal to the line of gravity, there is a perfectly reasonable way to escape:
Spoiler:
We are effectively in a potential well. Its dominant shape is a parabola, but we have higher orders (4th power of distance). We will exploit those tiny deviations from a parabolic shape.
Push the block of gold away in one direction, slow it down gently to stop it once the rope is fully extended. Assuming the block of gold has twice your body mass, it is now 50m/3 away from the center, while you are 2*50m/3 away from the center. In a perfect parabola, you and the block of gold would experience exactly the same force but in opposite direction (as force is proportional to mass*distance), not giving a net force. But we do not have a perfect parabola: we have a tiny term of 4th order. This makes the force on you (towards the center) stronger than the opposing force on the block of gold. The effect is tiny compared to your individual accelerations. To fully exploit it, you'll have to collect the block of gold and push it out again frequently, so it spends most of the time ~50 m away from you. After a few days, your net (center of mass) motion by a few nanometers was sufficient to have the parabolic component cancel the effect from the 4th order correction. Collect the block of gold again, but now push it in the opposite direction. That way, you can slowly increase the amplitude of your oscillations. You can escape, but I guess the timescale are several years.