## Escape the Frictionless Circle

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### Escape the Frictionless Circle

This puzzle is not so much about logic as it is about physics.

You are standing at the center of a circular surface, with radius 100 meters, that is perfectly frictionless and perfectly flat. There is very high coefficient of friction outside of the circle. You are connected at the waist to a cube of solid gold by a massless rope. The cube has side length 20 cm and is sitting at rest next to your feet. The rope has length 50 meters and is currently coiled neatly on top of the gold block. There is no way to break your tether to the gold block. Assume the atmosphere is a vacuum and you are not allowed to eject anything from your body (but you are magically alive and in perfect health). You are currently at rest.
Using only forces exerted by your limbs, can you escape the circle?
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Znirk
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### Re: Escape the Frictionless Circle

Well, step 1: Fall over. Not helpful per se, but c'mon, it's going to happen on a frictionless surface. Try to avoid hitting the corners of the gold cube (after all we're not allowed to expel blood).

Looking at the situation naively: The cube weighs massifies about 150kg, so I can't lift it. Any cube-pushing or rope-pulling I do will leave the centre of gravity of the (cube+rope+myself) system right where it is, just next to the centre of the circle. If we call the cube's mass at 2 average people, the farthest I get from the centre that way is some 33 meters: nowhere near the edge.

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. But A) would be a partial case of "breaking the tether to the gold block", which is impossible per definitionem, and both ways require tools on top of muscle power. Maybe with extreme patience my teeth could make a dent (hah) in the block, but I'm limited to "forces exerted by my limbs", which excludes the jaw. So, another dead end (short of knocking out one's own teeth and using them as tools; but I assume the intended solution doesn't rely on such a grotesque workaround to what would then be a completely arbitrary restriction).

Hmmm ... is there a way to exploit the rope's masslessness?

Sandor
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### Re: Escape the Frictionless Circle

Znirk wrote:Any cube-pushing or rope-pulling I do will leave the centre of gravity of the (cube+rope+myself) system right where it is, just next to the centre of the circle.

You can move the centre of gravity of the system up and down. You could jump, and a sufficiently strong person could throw the gold cube up into the air (or rather, up into the vacuum).

Depending on where you are and what is meant by "perfectly flat", then maybe centrifugal and Coriolis forces (if you're in a rotating frame of reference, e.g. on the Earth's surface), tidal forces, variations in the gravity field, or light pressure can be exploited. If you're on the Earth's surface, but not at the equator, then centrifugal force on it's own may get you off the surface without you having to do anything. These obviously aren't "forces exerted by your limbs", but you can't exactly choose not to be affected by them.

jaap
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### Re: Escape the Frictionless Circle

I thought for a moment that you could do this:
Spoiler:
Give the gold block some spin. Assuming you are a foot or two away from the gold block rather than directly on top of it, this will push you around the block in the opposite direction, so as soon as you let go of the block you will be moving away in a straight line.

Unfortunately this won't work because:
Spoiler:
You and the block will spin around the combined centre of gravity. So when you let go the block will move along a parallel path to you in the opposite direction. The centre of gravity remains where it is so you will not reach the edge of the circle. Once the rope snaps taught, you will not go any further.

Znirk
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### Re: Escape the Frictionless Circle

Sandor wrote:You could jump

Technically correct, but once again I refer you to the frictionless surface I'm struggling to even stand up on

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### Re: Escape the Frictionless Circle

I think we can agree any solution, if one exists, would be awfully unfeasible. What I'm wondering is if there is any solution at all, short of breaking the laws of physics.

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.
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doogly
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### Re: Escape the Frictionless Circle

What net force are you proposing moves the com?
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Whizbang
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### Re: Escape the Frictionless Circle

Can you phone a friend?

Also, what if you fart and then use your limbs to waft the fart away from you? Does that count? For that matter, just pluck the hairs on your arms and legs and throw them. Repeat until you are free of hairs (on your limbs). Same goes for fingernails/toenails. Does that break "you are not allowed to eject anything from your body"?
Last edited by Whizbang on Fri Nov 06, 2015 5:25 pm UTC, edited 3 times in total.

Chen
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### Re: Escape the Frictionless Circle

Yeah I'm seeing no way to apply a net lateral force to the COM of the system at all here. The only force that can be applied to the surroundings would be normal to the frictionless surface.

Yrgos
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### Re: Escape the Frictionless Circle

This will do the trick:
Spoiler:
Just climb on top of the gold. Any shift in your mass will give you an aceleration. Then sit still and wait

SirGabriel
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### Re: Escape the Frictionless Circle

Not sure whether this would work or not: Balance the cube on an edge or corner, then let it fall. Under ordinary conditions, this would result in the cube's center of gravity shifting sideways as the cube falls over; I don't know if that would also happen on a frictionless surface, or if the edge you're balancing it on would just slide the other way so the center of gravity moves straight down. If the center of gravity would shift sideways even on a frictionless surface, then you've got the beginning of a solution (you just have to figure out how to repeatedly lift the cube and balance it on its edge).

emlightened
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### Re: Escape the Frictionless Circle

The first thing that comes to mind is the answer to a problem that I read at some point:

If a ball is rolling on an infinite, perfectly flat, surface, in a vacuum, will it ever stop?

Spoiler:
yes, because the ball exerts more force on the surface in the direction it is traveling in than the opposite direction, or something. See: https://www.lhup.edu/~dsimanek/scenario/rolling.htm.
So, you should be able to
Spoiler:
repeatedly 'drag' yourself across the floor using my hands like paddles, to exert the required force.

Other ideas are:
Spoiler:
• Your body heats up the surface beneath it, so that surface will expand and rise. Proper positioning should allow motion, like slipping down the world's flattest slope.
• Even less effective: ensure that the majority of reflected light goes in a single direction. It's plausible.
• Perfect health means that your hair continues to grow, so you can cut bits off to make an arbitrarily long rope.
At least you won't be in there as long as the prisoners in the other thread.

The fact that the rope is massless is interesting; it only allows you to either reel in the block directly to you, or actually push the block to move it. It also means that you can throw it, and it won't push you or the block. Can't think of a way to exploit it, though.

Is it relevant whether the surface is conductive?
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### Re: Escape the Frictionless Circle

doogly wrote:What net force are you proposing moves the com?

That's the part that worries me. I'm not sure. The idea is that a torque is applied to my body by the rope. If angular momentum is to be conserved, something has to experience a counter torque. It could be the block or the entire system.

emlightened wrote:Is it relevant whether the surface is conductive?

I guess you are free to speculate about that. I personally imagined it like a special type of metal.
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Chen
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### Re: Escape the Frictionless Circle

Yrgos wrote:This will do the trick:
Spoiler:
Just climb on top of the gold. Any shift in your mass will give you an aceleration. Then sit still and wait

Nah the block will just move in the opposite direction and the CoM of the system won't move.

SirGabriel wrote:Not sure whether this would work or not: Balance the cube on an edge or corner, then let it fall. Under ordinary conditions, this would result in the cube's center of gravity shifting sideways as the cube falls over; I don't know if that would also happen on a frictionless surface, or if the edge you're balancing it on would just slide the other way so the center of gravity moves straight down. If the center of gravity would shift sideways even on a frictionless surface, then you've got the beginning of a solution (you just have to figure out how to repeatedly lift the cube and balance it on its edge)

Not sure how you'd lift the 150 kg block of gold to begin with. The surface can't be used as any type of fulcrum point to tip it either since it will just slide instead. I'm not exactly sure what would happen if you could somehow just pick up the gold block and move it an arm's length away. It seems like that should shift the CoM of the system slightly to that side which means you could eventually work your way towards an edge. But I don't see how that's actually possible looking at the net forces involved. Presumably if you moved the block to your left, you'd still shift in some manner and the CoM would end up in the same place as it started.

emlightened wrote:The first thing that comes to mind is the answer to a problem that I read at some point:

If a ball is rolling on an infinite, perfectly flat, surface, in a vacuum, will it ever stop?

Spoiler:
yes, because the ball exerts more force on the surface in the direction it is traveling in than the opposite direction, or something. See: https://www.lhup.edu/~dsimanek/scenario/rolling.htm.
gold block.png
So, you should be able to
Spoiler:
repeatedly 'drag' yourself across the floor using my hands like paddles, to exert the required force.

I guess it depends on what they mean by perfectly flat. If it's perfectly flat and remains perfectly flat, always, then this doesn't work.

jaap
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### Re: Escape the Frictionless Circle

Chen wrote:
emlightened wrote:The first thing that comes to mind is the answer to a problem that I read at some point:

If a ball is rolling on an infinite, perfectly flat, surface, in a vacuum, will it ever stop?

Spoiler:
yes, because the ball exerts more force on the surface in the direction it is traveling in than the opposite direction, or something. See: https://www.lhup.edu/~dsimanek/scenario/rolling.htm

I guess it depends on what they mean by perfectly flat. If it's perfectly flat and remains perfectly flat, always, then this doesn't work.

Spoiler:
It still works if the ball itself is not perfectly rigid. Rolling friction is due to energy loss from deformation of the ball as well as the ground.

In the OP's puzzle it says that the surface is frictionless, and you might think that this still allows a rolling ball (of gold) to slow down. Well, it would stop rolling, but it would not slow down - its linear momentum is not affected, only its angular momentum. The force from the ground onto the ball is still exactly vertical, just not directly through the ball's c.o.g. due to the deformation happening at the (front edge of) contact patch.

Chen
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### Re: Escape the Frictionless Circle

I wasn't clear there, I meant it wouldn't work for this puzzle's situation, not for a ball situation. If the floor can't distort here and the normal forces are all purely vertical, I don't see how the CoM of the system (gold block + person) can be shifted laterally.

jaap
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### Re: Escape the Frictionless Circle

Chen wrote:I wasn't clear there, I meant it wouldn't work for this puzzle's situation, not for a ball situation. If the floor can't distort here and the normal forces are all purely vertical, I don't see how the CoM of the system (gold block + person) can be shifted laterally.

Sorry, for misunderstanding you. I thought your 'doesn't work' comment referred purely to the ball situation, and so I also didn't intend my second paragraph to refer to you specifically or to anything you wrote, but merely as a further exploration of the OP's situation. I'm glad we agree.

Yrgos
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### Re: Escape the Frictionless Circle

Actually! :
Spoiler:
Any movement of your limbs will shift your COM, thus it will shift the COM of the gold + you system as well. The system will get a net acceleration and then you just wait...

taemyr
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### Re: Escape the Frictionless Circle

Yrgos wrote:Actually! :
Spoiler:
Any movement of your limbs will shift your COM, thus it will shift the COM of the gold + you system as well. The system will get a net acceleration and then you just wait...

No. Any movement of your limbs will also shift the rest of your body so that the COM remains constant. If you move your arms forward your torso moves backwards. When you move your arms back, your torso moves forward again. The COM stays the same.

Yrgos
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### Re: Escape the Frictionless Circle

Try it, stand on ice with your arms down, then thrust them foreward. You'll be accelrated backwards. This is why we don't want objects changing shape in physics problems

Edit: I conceede! I was wrong and you can't. Now I feel stupid

PeteP
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### Re: Escape the Frictionless Circle

I use my extraordinary strength to lift and drop the cube and hope that that slightly damages the surface. Honestly the "work some gold free from the cube" idea might be the most feasible, and its not clear whether that is included in removing the tether.

Yrgos
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### Re: Escape the Frictionless Circle

Can you huff and puff?
Inhale on one side exhale on the other. Problem solved!

speising
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### Re: Escape the Frictionless Circle

Yrgos wrote:Can you huff and puff?

No, you can't.

Anyway, the problem seems easier to imagine if you think of an astronaut in a suit in space.

suriya
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### Re: Escape the Frictionless Circle

Hi,
after thinking about this for 4 days, i still think it is impossible ( except signing to the outside, asking for help and giving the gold to the people who rescue me )

But the question wouldn't be in this forum if there weren't a solution, would it?

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?

SDK
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### Re: Escape the Frictionless Circle

Yrgos wrote:Can you huff and puff?
Inhale on one side exhale on the other. Problem solved!

You're in a vacuum, so I assume your lungs are full of vacuum too.

I asked this question to my wife who came up with a few ideas that she shot down herself immediately. She then resolved to just sit on the block and wait for someone to rescue her. I'm pretty sure that's the correct answer.
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speising
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### Re: Escape the Frictionless Circle

Do i have a microwave at hand? Maybe i can build a reactionless EM drive.

PeteP
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### Re: Escape the Frictionless Circle

Are you wearing clothes? Unraveling them and trying to get a thread onto solid ground should not count as expelling.

Well anyway the surface is in a vacuum somewhere it probably is colder than body temperature. You are a magically healthy human. Lay down and warm up the ground with your body temperature. As it gets warmer it will expand slightly providing you with a slight frictionless decline. Since nothing will slow you any amount of acceleration is enough to eventually escape.

Chen
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### Re: Escape the Frictionless Circle

SDK wrote:You're in a vacuum, so I assume your lungs are full of vacuum too.

I asked this question to my wife who came up with a few ideas that she shot down herself immediately. She then resolved to just sit on the block and wait for someone to rescue her. I'm pretty sure that's the correct answer.

Maybe if your hair continues to grow you could eventually use that to throw towards the outside of the frictional surface and then pull yourself to freedom.

doogly
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### Re: Escape the Frictionless Circle

suriya wrote:But the question wouldn't be in this forum if there weren't a solution, would it?

Some people are just trolls.
speising wrote:Do i have a microwave at hand? Maybe i can build a reactionless EM drive.

This kid right here.
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### Re: Escape the Frictionless Circle

suriya wrote:But the question wouldn't be in this forum if there weren't a solution, would it?

Well technically "no" would answer the question. It's more interesting to think about possible solutions for "yes" though. If someone can prove there exists no solution for "yes", that would be great too!
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jaap
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### Re: Escape the Frictionless Circle

There is a loophole in the puzzle wording:
Spoiler:
A perfectly flat plane does not necessarily have to be perfectly level. If it's a sloping flat plane, you will automatically reach the outside of the circle.

I also thought about what it means to have a perfectly flat surface on a round earth. Unfortunately that just makes it slightly harder, as the gravity near the edges of the circle points slightly towards the centre, pulling you back to the centre. I can probably rule out Coriolis forces because the puzzle does say "Using only forces exerted by your limbs".

So, as there seems to be no way to create a horizontal component to the reaction force against the frictionless ground the only remaining options involve
Spoiler:
expelling mass away. You cannot move the combined centre of mass, so the expelled mass cannot remain attached to you like the gold, because the rope isn't long enough. The only loophole would be if you could reshape the block of gold into a really long bar to move the gold's mass further away, but if you could do that then you could also use that long gold stick to push or pull at the ground outside the circle. Also, if you could reshape the gold that way, then you could also break a piece off and throw that.

Anyway, you need to separate some mass from the system. If it can't be a piece of the gold, then it has to be a piece of you. The "Using only forces exerted by your limbs" phrase suggests you might want to cut off a limb and through that with your remaining limbs, but that would be tricky without a knife (and if you had a knife, you could throw that instead or cut the rope), and has obvious health implications.

In the end, I think I would release my inner monkey and start throwing shit.

CharlieP
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### Re: Escape the Frictionless Circle

If you really are on a magic surface with zero friction, simply push against it and launch yourself into the air. Land with your toes pointed.
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Whizbang
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### Re: Escape the Frictionless Circle

And what will this do (how's and why's appreciated)?

CharlieP
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### Re: Escape the Frictionless Circle

Surely all you have to do is introduce any kind of horizontal movement, however small. If you dropped a stick or rod at an angle onto this surface, the end it lands on isn't going to stay where it is on a frictionless surface, it's going to head outwards as the stick "flattens out". If the surface is frictionless, what is going to bring it to an instantaneous stop the moment the rest of the stick touches the surface?
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PeteP
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### Re: Escape the Frictionless Circle

CharlieP wrote:Surely all you have to do is introduce any kind of horizontal movement, however small. If you dropped a stick or rod at an angle onto this surface, the end it lands on isn't going to stay where it is on a frictionless surface, it's going to head outwards as the stick "flattens out". If the surface is frictionless, what is going to bring it to an instantaneous stop the moment the rest of the stick touches the surface?

The stick can flatten out without it's center of mass moving into any direction beside down. It's other end simply needs to move right if its lower end moves left. And then the two ends stop each other.

The acceleration left or right has to come from somewhere, to conserve momentum something needs to be accelerated into the other direction.

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### Re: Escape the Frictionless Circle

You could get a horizontal component if the stick, upon landing, can deform the surface a bit, so now the normal force is not entirely parallel to gravity. If the perfect flatness of the problem must be maintained throughout the problem, this is not an option.
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### Re: Escape the Frictionless Circle

doogly wrote:You could get a horizontal component if the stick, upon landing, can deform the surface a bit, so now the normal force is not entirely parallel to gravity. If the perfect flatness of the problem must be maintained throughout the problem, this is not an option.

I think frictionlessnes implies perfect rigidity. If the surface can be deformed, the deformerator looses energy.

jestingrabbit
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### Re: Escape the Frictionless Circle

Question: what happens if you fall over? You've lost gravitational potential energy, so where does it go?
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

suriya
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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?

Into hurting yourself?

Whizbang
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### Re: Escape the Frictionless Circle

You speed the rotation of the Earth ever so slightly.

I think.