quote:You have a ten centimeter by ten centimeter square with a flap on the top edge. Unfolding this flap reveals a second flap, and unfolding this second flap reveals a third, as shown. If there are an infinite number of these flaps, what is the area of the entire figure (in square centimeters) when they are all unfolded?
What is a "flap" exactly? I mean, I understand it's a flap of folded-over paper, but how wide is it?
"as shown" where?
This is an infinite series problem, but I think I need more specifics to answer it . . . at least, I'm too lazy to think about it much until I know more.
posted
Okay, I seem to have found the song in its entirety.
Of course, it's a two or three minute download, and I'm not sure it's totally legal, althought the site I found it on had a link to the band's own page, so (I'll delete it if you think I should.)
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posted
On the flap one ... if there are an infinite number of flaps, and you are unfolding them one at a time, then there is no such thing as a time when you've got them all unfolded.
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posted
We have to have a diagram for the "flap." Is it the same size as the original 10 cm square? Half the size?
Assuming the flap is at all smaller than the original, and each succeding flap is smaller than the previous one by any amount, the total area will be finite. The exact number will vary with the ratio of each flap to the preceeding one.
posted
The weight may stay still, go down, or go up, depending on how the monkey climbs. Assuming that the rope is taut, if the monkey climbs in a way such that he keeps the rope taut (he maintains the original distance from the pulley) the weight will not move. If he climbs in such a way that the rope is pulled (he increases his distance from the pulley), he is imparting energy on the sysem, and the weight will move the exact same distance closer to the pulley. Vice versa if the monkey moves in some manner towards the pulley (he is implicitly reducing he Potential Energy with respect to the monkey-pulley-weight system). This is all because this situation is a closed system.
Of course, if the monkey is frictionless as well (which it would have to be or else you'd have to take that into account as he climbed), then the monkey couldn't climb at all.
So I claim the weight won't move, no matter how hard the monkey tries to climb.
posted
I assumed the flap was the same size because he didn't specify how much the flap is compared to the previous. if each one gets a fraction of the length of the previous then the solution is: 100 / (1 - x) where x is the fraction of the length of one flap / the length of the previous flap.
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posted
MEC - I just ran this out to 63,000+ terms (Excel is a wonderful tool) and it's not even close to the formula you gave. And all numbers in at least the last 2000 are 0 to 12 decimal places.
Here are the results
code:
RATIO TOTAL AREA 1/10 101.0101010101010 1/5 104.1666666666670 1/4 106.6666666666670 1/3 112.5000000000000 1/2 133.3333333333330 2/3 180.0000000000000 3/4 228.5714285714290 4/5 277.7777777777780 9/10 526.3157894736850 .99 5025.1256281406800 .999 50025.0125016523000
The relationship between .9 and .99 is particularly intriguing.
posted
Um, are you sure your figures are right? with 1/10 it would be 100 + 10 + 1 + .1 escentally 111.111 with 1 repedeated forever, the same thing my equation gives.
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posted
My assumption, as an uneducated sort of person, is that under normal circumstances (ie, barring all weird methods of climbing), the weight will rise together with the ape.
Reason is, Newton's law. When you apply force to the rope, you will experience an equal and opposite force yourself. Whenever the ape pulls on the rope, both he and the weight experience the same exact force.
If the ape were heavier than the weight, and standing on the ground, pulling down on the rope would cause the weight to rise while the ape stayed put, because the pulling force necessary to lift the weight would not be enough to lift the ape.
If the ape were lighter than the weight, and if the weight were sitting on the ground, then the ape would rise and the weight would stay put, because the force necessary to lift the ape would not be enough to lift the weight.
Since both are balanced midair, when the force of the ape's pull on the rope is applied to both, neither will weigh enough to resist the force and support the other's independent motion, so both will rise together at half the rate.
I know it lacks a physicist's flair, but it makes sense to me.
Posts: 1907 | Registered: Feb 2000
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posted
I think the problem for me is that it's very difficult for me to imagine a pulley with no friction. All the real-world examples you can pull from your head utilize the friction of the pulley.
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posted
Sun, the pulley is really just a distraction. Imagine an ape and a weight suspended in a gravityless vacuum, connected by a rope. If the ape "climbs" the rope, it will move forward along the rope, but the weight will simultaneously get pulled toward the ape.
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posted
I understand that, my first choice was that the weight goes up.
but I still contend that part of the reason it's difficult to me is that I'm used to friction pulleys.
Posts: 1892 | Registered: Mar 2002
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posted
Yeah. Both questions, as stated, are nonsense, and require interpretation. A pulley with no friction makes no sense; my interpretation of "frictionless" was that they meant to say a pulley that had 100% efficiency, where a certain amount of force wasn't required simply to put it in motion. In other words, with a real pulley, two objects that don't weight exactly the same could remain in equilibrium if the differential wasn't enough to overcome the pulley's own inertia. I vaguely remember there was a word for this, but not what.
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if the ape and weigth are perfectly balanced at rest. no matter what the internal friction of the pulley, the ape must experience an *increase* in the upward force applied by the rope if he is to move upwards. This increase willl be transmitted throught the rope to the weight and make it move upward as well. If the pulley has friction, it will interfere with (eat up some of) that force, but it will not prevent some of it from being passed on to the weight.
Make more sense that way?
Posts: 3846 | Registered: Apr 2004
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posted
hmmm... though there's a chance that the ape goes downwards, actually. if you take two equal weights on a pulley (frictionless, at that) at two different heights, then the lower one would go down wouldn't it?
so as soon as the ape shortens the rope and the other side goes up a little, the ape would start going downwards.
Either way, i'm heading out the door so you will have to respond without me to respond.
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posted
For ever action there is an equal and opposite reaction. If the monkey pulls down on the rope, trying to climb up, the weight will rise.
I also made a little picture in MSPaint. It doesn't help at all, but it was entertaining to draw for a couple minutes. I was trying to see if I could do alright with my little trackpad, as I don't have my mouse with me. http://s88345032.onlinehome.us/images/pulley.gif
posted
I'd just like to point out that jehovoid's analogy on the previous page with the two rockets is exactly correct. All the pulley does is change the direction of the tension in the rope, so if you replace gravity with thrust from rocket engines (assuming you could keep the thrust going forever, and neglecting change in mass from the exhaust) you get exactly the same situation. Except for the fact that a rocket is not an orangutan.
From this point it should be pretty clear that the rocket doing the pulling is pulling itself and other other rocket closer to the system's center of gravity at an equal speed, which is half the relative speed of the pulling rocket and the rope.
This isn't rocket science, folks.
Edited to add: imagine you and a friend are playing tug of war. In a pond. In innertubes. You pull the rope. You and your friend both float closer to the midpoint between you. Now, same situation, but you're both kicking away from each other equally. Voila (or viola as some would say).
[ May 01, 2004, 11:13 PM: Message edited by: Mike ]
Posts: 1810 | Registered: Jan 1999
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posted
For more exciting arguments and theories based one these example of questions, quickly go to www.highiqsociety.org . This is where I went to to find out my iq. To join the high iq society you need at least an iq of 120 to qualify, and even then its not free...so I didn't join. But the free iq tests are fun and the puzzles are brain rattling. Have fun and post your IQ's. Let the "Member in Hatrack with the Highest Intelligence of 2004" begin... huff huff...
Pericles p.s (I qualified for the "special" division. I feel "special")
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posted
I was wondering, if the rope is massless and the ape exertes energy to climb, wouldn't it then burn calories climbing, effectivly reducing it's weight, and causeing the mass to fall?
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posted
But just because it's 100% efficent doesn't mean it doesn't use energy. It just means that none of it goes to waste, right?
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posted
Just because the ape used up energy doesn't mean that his mass will have decreased. It's just that his chemical composition will have changed, but no mass.
Of course, he has to breathe, and will constantly be losing H20, and therefore mass. Unless he's wearing a space suit. Posts: 16551 | Registered: Feb 2003
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posted
And then he'd eventually run out of air in the suit, die, and fall of the rope.
Pericles, I took the "Ultimate IQ" test on their site. And a lot of the questions didn't provide enough information. Anyway, I scored a 150 on the test, I'm in the highly gifted catagory.
posted
There's no place like my carefully maintained mental construct. There's no place like my carefully maintained mental construct. There's no place like my carefully maintained mental construct.
Posts: 26071 | Registered: Oct 2003
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posted
You guys are so intelligent. Way out of my league. Maybe by "special" you think I meant super intelligent. By special I meant "special-education" or "special olympics" "special". I only scored 129. I mean I'm sixteen and I ONLY SCORED 129!!!
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