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September 26, 2011 3:12 PM   Subscribe

What happens when you drop a slinky?
posted by empath (83 comments total) 39 users marked this as a favorite

 
Here's the (hidden) link to the answer
posted by Llama-Lime at 3:15 PM on September 26, 2011 [1 favorite]


well I was wrong.
posted by hypersloth at 3:16 PM on September 26, 2011 [4 favorites]


Cool!
posted by Specklet at 3:19 PM on September 26, 2011


I GUESSED RIGHT I GUESSED RIGHT! I'm a fucking physists or a pyhsosist. I'm smart.
posted by nola at 3:21 PM on September 26, 2011 [5 favorites]


Well, that doesn't make sense. Is this something to do with neutrinos?
posted by seanyboy at 3:22 PM on September 26, 2011 [4 favorites]


The plane takes off, obviously.
posted by Horace Rumpole at 3:25 PM on September 26, 2011 [18 favorites]


This also blows my mind because it comes with some intelligent youtube comments. Who'd have thought?
posted by hypersloth at 3:26 PM on September 26, 2011 [6 favorites]


I was wrong and that man is therefore a witch. Burn him at once.
posted by The Bellman at 3:26 PM on September 26, 2011 [12 favorites]


I heard that certain slinkies can travel faster than the speed of light, but old physicists really want to be proven wrong.
posted by entropone at 3:26 PM on September 26, 2011


His use of the word "information" is intriguing, but from a pedagogical perspective it opens up a whole can of worms. In the STEM field, "information" is overloaded with meanings, ranging from conceptual abstractions to advanced mathematical formalisms. In the video it seems to just mean causality, in that it takes time for the event of releasing the top end to cause something to happen at the other end of spring, starting from a state of equilibrium. Handwavy is bad. Always.
posted by polymodus at 3:28 PM on September 26, 2011 [5 favorites]


When other people complain about Radiolab treating them like they're stupid, is this how it sounds to them? Like this guy? Because if so, now I get it.
posted by penduluum at 3:31 PM on September 26, 2011 [3 favorites]


Well, of course it works that way, south of the equator. Up here, in North America, it goes counter-clockwise.
posted by crunchland at 3:32 PM on September 26, 2011 [17 favorites]


Slinky's Slunkle but the don't fall down.
posted by Cerulean at 3:32 PM on September 26, 2011 [1 favorite]


Interesting. It's pretty boring at regular speed but gets wild in slo-mo. Or as wild as a Slinky gets.
posted by tommasz at 3:33 PM on September 26, 2011


CONTROLLED DEMOLITION, SHEEPLE
posted by dhartung at 3:35 PM on September 26, 2011 [5 favorites]


Apparently you *can* pull yourself up by your own bootstraps.
posted by Tell Me No Lies at 3:39 PM on September 26, 2011 [3 favorites]


A slinky is a spring under tension. Gravity is pulling the weight and body of the spring downward. When you release the top that spring is going to contract because the entire thing is momentarily weightless and in freefall, allowing the spring to relax to its natural closed/contracted state. It contracts downward because of the potential energy in the stretched spring causing the contraction while the entire thing is still being pulled by gravity.

That's all fine and good in this controlled demonstration but a totally lame representation of the real world.

Everyone knows that if you drop a Slinky in the real world it will immediately tangle into a useless pile of steel coils that you'll never untangle without putting permanent kinks and bends into to the coil. QED.
posted by loquacious at 3:41 PM on September 26, 2011 [38 favorites]


What usually happens, in my experience, is either Slinky Stretch, a toy affliction in which one coil is forever bent out a little causing the thing to never again coil neatly in a stack, or Slinky Sprain, in which two coils interleave out of order, the fixing of which proving either impossible or resulting in Slinky Stretch. Both are, to my experience, grievous maiming ruining the prospects a bright young slinky has in life. As happened to my poor slinky, Gretchen. (None of my friends will own up to breaking her. It's a matter of some strife with me.)
posted by JHarris at 3:44 PM on September 26, 2011 [5 favorites]


Or what loquacious said.
posted by JHarris at 3:45 PM on September 26, 2011 [1 favorite]


In the STEM field, "information" is overloaded with meanings, ranging from conceptual abstractions to advanced mathematical formalisms.

If you're ever bad I will have a human geographer come round and tell you about what quantum means in their field.
posted by biffa at 3:45 PM on September 26, 2011 [3 favorites]


It falls.
posted by Edogy at 3:46 PM on September 26, 2011


I predicted that also. Both ends meet in the middle, but the whole thing is falling at the same time, negating the bottom going up, and doubling the speed of the top going down--exactly like fucking on a trampoline, if you've ever done that.
posted by weapons-grade pandemonium at 3:46 PM on September 26, 2011 [4 favorites]


Meanwhile, what's up with that scientist's trousers? He looks like he dropped a slinky or two in there before coming out to the field.
posted by chavenet at 3:47 PM on September 26, 2011


Metafilter: exactly like fucking on a trampoline
posted by chavenet at 3:47 PM on September 26, 2011 [10 favorites]


It gets the information and knows to start falling? Any physicists care to comment on that? Are those technical terms I just don't understand?

It is like the diagram I once saw on a tide bottle, it was called "how tide works", I expected some cool shit about emulsions, but all it said was "Tide gets deep down and attacks dirt" at least they had a cool pic of a blob of tide attacking a bit of dirt. Tide really hates dirt.
posted by Ad hominem at 3:51 PM on September 26, 2011 [7 favorites]


So I get that the top end shoots down because it was under tension, but why does the bottom end remain stationary? Is is (a) actually accelerating (and just seems to be stationary relative to the top end, which is accelerating faster than g--thus the need for a high speed cam), or is it (b) actually stationary due to the tension release not having yet propagated to the bottom end? Or does (c) the upward force on the bottom end just happen to perfectly cancel g, until the top end impacts?

If it's (b) then that would imply this would work with any object, like a weighted piece of string?
posted by danny the boy at 3:54 PM on September 26, 2011


I really like that more than I should have.
posted by Mental Wimp at 3:56 PM on September 26, 2011


Like if you have a string weighted on two ends:

[]-------------[]

holding one end, and then letting go, would it behave the same way? If so, I'm gonna need someone with a high speed cam to upload this to youtube, now.
posted by danny the boy at 3:56 PM on September 26, 2011


You people clearly own shitty slinkies.

/me has done this experiment dozens of times this afternoon, with no ill effects on her slink.
posted by jacquilynne at 3:59 PM on September 26, 2011


danny the boy: I don't think so. I think the crucial part is the tension generated by the spring that is counteracting gravity for a sec. If you just had a two weights, I think both weights would just start to fall at the constant because there is no spring tension being exerted.
posted by lazaruslong at 4:04 PM on September 26, 2011


I WANT TO BELIEVE.
posted by iamabot at 4:06 PM on September 26, 2011 [1 favorite]


Yeah, the bottom is falling and being pulled up at the same time, thus remaining in the same place. The top is falling and being pulled down so it is moving twice as fast.
posted by Ad hominem at 4:07 PM on September 26, 2011


Wait, I'm not physics-inclined, but there's no magic here, and this isn't about the "propagation of information", as if the slinky is waiting to be told it can fall. Gravity is pulling it down the whole time!

As soon as he lets go, the entire slinky object starts falling and the slinky bottom and top are contracting at each other due to tension. Since the tension is equal to that provided by gravity when he's just dangling it, and tends to be in the top part of the slinky, the contraction energy will pull the ends at each other as fast as g will pull down the whole object... right?

So all we're really seeing is what weapons-grade pandemonium describes, where the top of the slinky appears to fall twice as fast due to gravity, while the bottom seems motionless- but rather, it's that the slinky as a whole is falling but contracting at the same speed, so it only looks like the bottom isn't moving, and the top is moving at g + contraction speed.

This is no different than jumping off a chair, and pulling up your legs at the last moment to make you stay in the air longer. Or am missing something?
posted by hincandenza at 4:09 PM on September 26, 2011 [6 favorites]


The tennis ball's natural bounce keeps it suspended in the air.

Okay. What about other objects. Would a 10lb weight hang there just the same?
posted by run"monty at 4:09 PM on September 26, 2011


Everyone loves a Slinky, you gotta get a Slinky, Slinky, Slinky, go Slinky go
posted by growabrain at 4:10 PM on September 26, 2011


If you could some how see the slinky's center of mass, you would see it fall as normal, exactly like a tennis ball.
posted by 2bucksplus at 4:12 PM on September 26, 2011 [7 favorites]


So we have to make the hoverbikes out of the bottoms of falling Slinkies, I get it now. 2012 will be the year we finally get our hoverbikes.
posted by not_on_display at 4:12 PM on September 26, 2011 [4 favorites]


Some people are like Slinkies. They'll usually just sit there and do nothing but if you push them down a flight of stairs, it brings a smile to your face.
posted by Renoroc at 4:12 PM on September 26, 2011 [14 favorites]


I downloaded the videos and blew them up to max size on my 30" monitor, and then put my cursor on the exact bottom of the slinky, and in the 3rd video, on the tennis ball, so I could see if the bottom of the spring moved or not.

YES, the bottom of the slinky does move down slightly, even before the spring reaches maximum compression. The bottom of the spring starts falling the moment the top is released. Yes, gravity has an effect on the whole apparatus, independent of the spring compression.

SO, the claims made these videos are not totally true, at least, not with this crude experimental apparatus. And that explanation about "information" is the most unscientific thing I ever heard.
posted by charlie don't surf at 4:16 PM on September 26, 2011


It gets the information and knows to start falling? Any physicists care to comment on that?

The main point, without having looked at the math, is that the centre of mass of the slinky is falling at 9.8 m/s^2 the whole time.

I'd argue that the in the frame of reference of the slinky, both ends are contracting towards the centre. The fact that the whole system is falling just makes it appear that the bottom is holding in the same place. The bottom of the slinky is getting pulled up by the spring force at the same rate it's being pulled down by gravity.

If you watch the slo-mo with the tennis ball attached to the bottom of the slinky, you can see the whole system is actually falling at a pretty slow rate. I'd bet that the heavier the weight you put on the bottom of the slinky, and the longer the spring you used, the more obvious it would be that the whole system is falling while it contracts.
posted by auto-correct at 4:16 PM on September 26, 2011 [2 favorites]


Or, what hincandenza said.
posted by auto-correct at 4:19 PM on September 26, 2011


crap
posted by mrgrimm at 4:19 PM on September 26, 2011 [1 favorite]


I'd argue that the in the frame of reference of the slinky, both ends are contracting towards the centre. The fact that the whole system is falling just makes it appear that the bottom is holding in the same place. The bottom of the slinky is getting pulled up by the spring force at the same rate it's being pulled down by gravity.

Yes, that is precisely how the Slinky is designed. If the spring had any more contractive force, it would not be able to "walk" down the stairs. The spring would pull back up rather than pull the rest of the spring down. It's a fine balance between force and mass.
posted by charlie don't surf at 4:19 PM on September 26, 2011 [2 favorites]


I'd bet that the heavier the weight you put on the bottom of the slinky, and the longer the spring you used, the more obvious it would be that the whole system is falling while it contracts.

[facepalm]

This effect would most easily be demonstrated by taking a slinky, and a slinky with a tennis ball attached to the bottom, and dropping them simultaneously from the top of the Leaning Tower of Pisa.
posted by charlie don't surf at 4:27 PM on September 26, 2011 [4 favorites]


Would this work the same way underwater?
posted by DaddyNewt at 4:37 PM on September 26, 2011


You can't explain that.
posted by punkfloyd at 4:41 PM on September 26, 2011 [2 favorites]


Notice that the slinky is not uniformly contracting toward the center - each turn of the slinky is staying roughly in the same place. The only forces on each turn of the slinky is from gravity and from the tension from the section immediately above it. The only section that is initially affected by letting go is the very top of the slinky - it takes time for the effects of letting go to propagate to the bottom of the slinky. That is why he is talking about it in terms of transfer of information - the travel of a wavefront, instead of just saying the slinky contracting at the same rate at it is falling. You miss a key piece of understanding with the latter description.
posted by Zalzidrax at 4:46 PM on September 26, 2011 [7 favorites]


Man, Einstein is really taking a pasting this week.
posted by metaxa at 5:03 PM on September 26, 2011 [5 favorites]


Or does (c) the upward force on the bottom end just happen to perfectly cancel g, until the top end impacts?

To simplify matters by moving all the mass to the end points, imagine one small weight of unitary mass hanging motionless below another, suspended by a stretched spring of negligible mass. The upward force on the bottom weight must be g, since that's the downward force and it isn't moving. The downward force on the upper weight must be 2g, since that's the weight of whole apparatus.

When the upper weight is released, it starts to accelerate downward at 2g; but while it's accelerating, it still anchors the top end of the spring, which is still pulling the bottom weight upward, and that force is counteracting gravity and holding the lower weight in place.

Per Hooke's law, a (imaginary, perfect) coil spring under tension pulls back in linear proportion to the length it has been stretched (up to the limit of its elasticity). So as the top weight falls, accelerating at 2g downward, the upward force on the lower weight shrinks at the same rate as the distance between them.

In reality, though, a Slinky's rest position isn't when it's completely closed up (e.g. see this plastic one in zero gravity), so the bottom weight starts falling before the Slinky is fully compacted.
posted by nicwolff at 5:04 PM on September 26, 2011 [3 favorites]



I had to watch it six times, but I finally got it right on the first try.
posted by Pogo_Fuzzybutt at 5:21 PM on September 26, 2011 [1 favorite]


See also: The Abyss.
posted by odinsdream at 5:43 PM on September 26, 2011


Is it Ghostbusters II?
posted by kcds at 6:12 PM on September 26, 2011


They have a lot of other videos.

Don't miss the 2.5 ton granite sphere Earth globe floating on a film of water!

I didn't guess the correct answer for dropping a weight attached to a chain compared to a free weight.

"If the Earth were the size of a basketball and the moon a tennis ball, how far apart would they be?" "about that much?"
posted by jjj606 at 6:12 PM on September 26, 2011 [3 favorites]


If Adam Savage sees this post, he's gonna be pissed! These guys have leaked the entire premise of the 2012-2013 season of Mythbusters to the internet.
posted by crunchland at 6:15 PM on September 26, 2011


This effect would most easily be demonstrated by taking a slinky, and a slinky with a tennis ball attached to the bottom, and dropping them simultaneously from the top of the Leaning Tower of Pisa.

Clever. I wasn't saying the whole system would fall faster. I meant that if the slinky where stretched out further by a heavier weight it would be more obvious that the whole system is falling while the slinky contracts.
posted by auto-correct at 6:16 PM on September 26, 2011 [1 favorite]


What happens when you drop a slinky?

Storm: The same thing that happens to anything else.
posted by Joey Michaels at 6:22 PM on September 26, 2011 [2 favorites]


This is why Republicans want to unfound science. Your money is going to figure out worthless shit like this. No slinky never cured no cancer!
posted by cjorgensen at 6:24 PM on September 26, 2011


The law of gravity isn’t as indiscriminate as people often think. You learn things like that when you’re a bird.
posted by Devonian at 6:32 PM on September 26, 2011 [1 favorite]


I wasn't saying the whole system would fall faster. I meant that if the slinky where stretched out further by a heavier weight it would be more obvious that the whole system is falling while the slinky contracts.

yup. the explanation in the video is misleading (at best) and the tennis ball most certainly does start to move instantly.
posted by DavidandConquer at 6:35 PM on September 26, 2011


What happens when you put a slinky on an escalator? This doesn't help.
posted by twoleftfeet at 6:36 PM on September 26, 2011


Here is my question, physics people:

Suppose I am tied to the bottom of a very large, sturdy slinky (a la the tennis ball video). The top of the slinky is held by a helicopter hovering 2 miles above the earth's surface. I, at the bottom end, am suspended only 2 feet off of the ground. Now the person in the helicopter lets go of the top of the slinky.

Ignoring wind effects screwing with the slinky's movement, will I be suspended 2 feet off the ground until the top of the slinky meets the bottom?
posted by 3FLryan at 7:23 PM on September 26, 2011


Man, Einstein is really taking a pasting this week.

True, but the OP is an example of Australian science.
posted by charlie don't surf at 7:49 PM on September 26, 2011


Not physics, but the University of Nottingham Periodic Table of Videos is fucking awesome.
posted by Ad hominem at 8:12 PM on September 26, 2011 [1 favorite]



Like if you have a string weighted on two ends:

[]-------------[]

holding one end, and then letting go, would it behave the same way? If so, I'm gonna need someone with a high speed cam to upload this to youtube, now.


This would behave the same way, to a lesser extent. A wave of reduced tension would propagate down the string. And indeed, when it is in free fall the string will no longer appear to be taut, but will be slack, as the top object will have caught up with the bottom one somewhat, while the bottom waits for the wave of relaxed tension to arrive.

Speaking about this in terms of information is a legitimate way to discuss it, and it is just like the obvious speed of light analogies. If instead of using electromagnetism, you're using a mechanical system, nothing you do can have an effect until there's time for the wave of "information" to propagate there. The speed of sound in steel is 6km/s. If I dangle a 6 km steel rod, an observer at the bottom will not be able to tell if I've hit the top with a hammer or have let go for a full second - during which time the rod is relaxing or compressing. The more rigid the material, the faster the tension or release of tension (information) will propagate.

This happens in all kinds of media - air compression can't travel faster than the speed of sound, water waves can't travel faster than their speed in their media. This even happens with traffic - when they clear a broken down car away, that doesn't mean the 3 mile backup suddenly goes back to driving 60mph; that has to propagate backwards, and the effects aren't felt for some time.
posted by aubilenon at 8:44 PM on September 26, 2011


Here is my question, physics people:

Suppose I am tied to the bottom of a very large, sturdy slinky (a la the tennis ball video ). The top of the slinky is held by a helicopter hovering 2 miles above the earth's surface. I, at the bottom end, am suspended only 2 feet off of the ground. Now the person in the helicopter lets go of the top of the slinky.

Ignoring wind effects screwing with the slinky's movement, will I be suspended 2 feet off the ground until the top of the slinky meets the bottom?


Assuming the slinky is not completely stretched out, yes, you will hang there, 2 feet over the ground, until suddenly rather a lot of steel hits you at hundreds of miles per hour.

I do not recommend this experiment.
posted by aubilenon at 8:50 PM on September 26, 2011 [7 favorites]


Fucking slinkies. How do they work?

No, I do not want to talk to a scientist. The entire centre of gravity of that apparatus is moving downwards even as the tension of the slinky gives the bottom the appearance of remaining stable, which is clearly not the case in the slow-motion tennis-ball video. The experiment only appears to yield the absolute results it does on account of its small scale. Motherfucker's lying, getting me pissed.
posted by bicyclefish at 9:19 PM on September 26, 2011


The law of gravity isn’t as indiscriminate as people often think. You learn things like that when you’re a bird.

Especially if you wear a slinky dress.
posted by chavenet at 10:35 PM on September 26, 2011 [1 favorite]


Now if it were a neutrino slinky, it would f
posted by not_on_display at 10:52 PM on September 26, 2011 [2 favorites]


This makes great sense if you think about it from gravity's perspective--the earth has pre-pulled as much of the Slinky toward it as it can, and the potential energy stored in the spring's contractile tension simply turns into velocity as the upper portion falls.

So it might actually fall faster than normal, at least at first.

(And yes, the plane takes off.)
posted by Camofrog at 11:32 PM on September 26, 2011 [1 favorite]


Camofrog:

The top of the slinky unquestionably accelerates faster than 9.8 m/s/s; it's got the combined forces of gravity and spring tension pulling it downwards. My thinking is that this is exactly enough faster than 9.8 m/s/s to make the total average acceleration of the slinky exactly 9.8 m/s/s. As long as there's no outside force acting on it, any body's center of mass will trace the same path, regardless of how it shifts around that center.
posted by aubilenon at 12:35 AM on September 27, 2011


At the moment before he releases the top of the slinky, every part of the system is in equilibrium, with the acceleration due to gravity balanced by the tension exerted by the part above. At the instant he releases it, the only thing that changes is that the top of the slinky is no longer supported by his hand, and it proceeds downwards under gravity and the compressional force that it experiences from the part of the slinky immediately below it. Beneath _that_, each part of the slinky will only start to fall once the upward force stops acting on it, and that only happens when the part above it has started to fall.

If the tennis ball was on the end of a broomstick, the whole lot would apparently fall at the same time (although there would be a delay due to the elasticity of the wood, it's tiny), but with a slinky, the tension is released effectively as a noticeable wave that moves down.

That's what he means by 'information', and it's not as daft as it sounds - if you think of pushing a stick that's, say, a light year long, the end would only start to move a year afterwards. The effect would look like a wave at c moving down the stick - same's true of an ordinary broomstick, but good luck in observing that - and that's the sort of thing physicists are comfortable talking about in informational terms. The ball will only fall once it 'knows' that the system above has changed (the fact it does start to fall slowly as soon as the top is let go is perhaps due to the incompressibility of the slinky along the line of the metal; it's not a 'perfect spring', but that's wild-eyed raving on my part).

A lot of physics is like this; it's weird when you look at it closely and you can come up with a number of equally valid ways of modelling what's actually happening. Pick the one that's most useful to the task in hand. But you are only modelling it: what's _actually_ happening is a physical system behaving in its own way; the rest is analogy and maths.

It's less common for this to be as obvious as this in classical physics as it is in quantum, but there it is nonetheless.
posted by Devonian at 1:40 AM on September 27, 2011 [4 favorites]


aubilenon writes "Assuming the slinky is not completely stretched out, yes, you will hang there, 2 feet over the ground, until suddenly rather a lot of steel hits you at hundreds of miles per hour."

And then you fall the two feet to the ground and the steel falls on top of you.
posted by Mitheral at 1:55 AM on September 27, 2011


His "golf flurry" was actually the end of a batsman's swing in cricket. I'm surprised he ballsed that up so bad, coz I'm pretty sure he claimed he was some sort of sports scientist in one of the vids.
posted by uncanny hengeman at 2:25 AM on September 27, 2011


"Yes, that is precisely how the Slinky is designed. If the spring had any more contractive force, it would not be able to "walk" down the stairs. The spring would pull back up rather than pull the rest of the spring down. It's a fine balance between force and mass."
Read what Devonian (for example) says in his first paragraph. It's not any design feature of the slinky that makes it have that amount of tension - it's the fact it's in equilibrium and therefore the tension must be exactly equal to gravity. It'd be the same for any spring. The slinky merely makes it more obvious by stretching more for a given force.
posted by edd at 2:25 AM on September 27, 2011


Now do it with an anvil!
posted by silkyd at 4:21 AM on September 27, 2011


My physics knowledge is still there! I predicted exactly what would happen. Damn, that feels good!
posted by Decani at 7:02 AM on September 27, 2011


Assuming the slinky is not completely stretched out, yes, you will hang there, 2 feet over the ground, until suddenly rather a lot of steel hits you at hundreds of miles per hour.

I do not recommend this experiment.

And then you fall the two feet to the ground and the steel falls on top of you.


Oh, no no no. My slinky, when compressed, turns to spaghetti. Yum.

I think it would be rather nice to hang two feet in the air for a time, only to land in a heap of spaghetti.
posted by 3FLryan at 7:15 AM on September 27, 2011


Now I understand how Wile E. Coyote could run off the cliff and hang in the air - until the information caught up with him.
posted by bitmage at 8:03 AM on September 27, 2011 [2 favorites]


You people just have to suck the magic out of everything, don't you.
posted by slogger at 8:57 AM on September 27, 2011


There's a reason they call it the center of gravity. I'm betting that if you calculated the acceleration of the center of mass of the slinky, Newton would be pleased with the result. What the conformation of the slinky is, constrained by that framework, is pretty much a matter of indifference to gravity.

A vertically spinning baton falling to earth has the analogous counterintuitive motion, as one end is actually going up at times as the other end falls faster than the center of mass.
posted by Mental Wimp at 9:11 AM on September 27, 2011 [2 favorites]


"Drop a Slinky" is my new favourite euphemism for defecation.
posted by Grangousier at 9:57 AM on September 27, 2011 [1 favorite]


JHarris: "Slinky Stretch, a toy affliction in which one coil is forever bent out a little causing the thing to never again coil neatly in a stack"

In my house we used to call that a slernia.
posted by Riki tiki at 8:17 PM on September 27, 2011 [1 favorite]


His "golf flurry" was actually the end of a batsman's swing in cricket. I'm surprised he ballsed that up so bad, coz I'm pretty sure he claimed he was some sort of sports scientist in one of the vids.

Also, he was MOCKING golfers who have a nice follow thru. The mind boggles how this clown can come to call himself a sports science expert. I'm always a bit dubious about Professor-Dudes who like being in front of a camera, and this guy has done nothing but re enforced that suspicion.

[Great experiment BTW. Everyone else has discussed the physics, so I thought I'd concentrate on the h8ing.]
posted by uncanny hengeman at 10:12 PM on September 27, 2011


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