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Brainteaser ...sort of

Discussion in 'Other Off Topic Forum' started by Turb0flat4, Oct 13, 2004.

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  1. Turb0flat4

    Turb0flat4 Formula 3

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    #1 Turb0flat4, Oct 13, 2004
    I said sort of, because this question apparently occurred in a local major school exam for 11 to 12 year olds. Think about it, give your answer, and then, in all honesty, tell me if you'd have been able to answer the question when you were 12. Oh, and if you have kids around that age give them the question too, it's fun. :D

    OK, here goes :

    You have a thin rod, composed partly of metal and partly of wood. The rod is balanced on a small pivot and adjusted so that it balances neutrally. You then take a lit candle and heat up the metal part of the rod uniformly. Would you expect the rod to a) remain balanced b) move down on the wood side, or c) move down on the metal side ? Explain.
     
  2. 134282

    134282 Four Time F1 World Champ
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    #2 134282, Oct 13, 2004
    A) Remain balanced...

    When heated, the mass would not increase or decrease, therefore, it would just get hot.
     
  3. MarkPDX

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    #3 MarkPDX, Oct 13, 2004
    what if the wood bit catches fire?
     
  4. darth550

    darth550 Six Time F1 World Champ
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    #4 darth550, Oct 13, 2004
    Then it stands to reason that there would be smoke!

    DL
     
  5. TcpSec

    TcpSec Formula Junior

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    #5 TcpSec, Oct 13, 2004
    If the rod is balanced vertically, nothing will happen. :)

    Otherwise, the rod will move down on the metal side. The reason is that metal has a higer coefficient of expansion than wood, causing the center of gravity to move outwards on the metal side. This will increase the torque on the metal side and the rod will move down on that side.
     
  6. SRT Mike

    SRT Mike Two Time F1 World Champ

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    #6 SRT Mike, Oct 13, 2004

    I disagree.

    Metal has a higher coefficient of expansion than wood does. Therefore, the metal part will expand, whereas the wood part will not. The two sides are balanced because of the distance they are from the center point, as well as their weight. Metal would be heavier, so I would expect the balance point to be somewhere on the metal part, but it would certainly be some amount towards the wood side, because the metal alone would be halfway down it's length for a balance point, and the wood would move it SOME amount (no matter how small) towards one side.

    So, in other words there is equal mass on both sides, but the coefficient of expansion would be different for both sides. The metal part heats up, and more weight is a further distance from the center on the metal side, giving it a higher torque (since torque is force times distance) which would rotate it about the centerpoint towards the metal side.

    Or I could be completely wrong :)

    The metal would expand both ways, pushing the wood out further too - although the metal should expand evenly meaning there would be more expansion on the metal side than the wood side of the balance point - so it should always go towards the metal side.

    What did I win (or lose)???
     
  7. jordan747_400

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    #7 jordan747_400, Oct 13, 2004
    I say the metal side gets lowered.
     
  8. hwyengr

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    #8 hwyengr, Oct 13, 2004
    I say its balanced. Even though the metal is expanding, the mass doesn't change. Therefore, the density of the expanding metal is less than the cold metal, so the center of mass should stay in the same spot.
     
  9. 96impalaSS

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    #9 96impalaSS, Oct 13, 2004
    Well I say chaos will ensue.

    Or I expect the thing to remain balenced. To me it depends on one question.

    Is the wood on one side and the metal on the other or are they layed on top of one another?
     
  10. zach_300zxTT

    zach_300zxTT Formula Junior

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    #10 zach_300zxTT, Oct 13, 2004
    But what if the metal was in a spiral with the wood? If that was the case, I'd have to say rabid kittens would take over the world.
     
  11. tbakowsky

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    #11 tbakowsky, Oct 13, 2004
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  12. Mike360

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    #12 Mike360, Oct 14, 2004
    The scales will tip to the Metal side.
    The increase in size is not due to an increase in the size of the particles that make up the metal rod, but rather to an increase in the average distance between the particles. When an object is heated, its particles vibrate faster, collide more violently, and consequently move farther apart, thereby increasing the volume of the object.

    When the object is cooled, the opposite change occurs and the volume of the object decreases. This decrease in volume is called contraction.

    Thefore the Metal will expand with the heat, making the metal side longer, therefore offsetting the point of equilibrium off the balance, tipping it to the metal side.

    Theres your answer!
     
  13. Turb0flat4

    Turb0flat4 Formula 3

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    #13 Turb0flat4, Oct 14, 2004
    Wow, 11 replies while I went out for lunch ! Excellent :D

    BTW, the rod is composed of a length of wood joined collinearly (lengthwise, in a straight line) with a length of metal (not vertically on top, or in a spiral ;)).

    I don't know what the "expected" answer is, but my answer is also that it would lean towards the metal side. The metal expands on heating (due to a lower density). Even though the mass on the metal portion remains the same, its center of gravity would shift further away from the fulcrum towards the metal end, causing an increased moment (or torque) about the fulcrum, leading to a lowering of the metal end.

    Initially I considered 3 cases : 1) when the rod is balanced right between the metal and wood parts. (this would mean that the wood part is longer than the metal part) 2) when the rod is balanced over the wood part (this means that the wood part is really "a lot" longer than the metal part) and 3) when the rod is balanced over the metal part (in this case, the moment of the wood part and the short part of the metal portion counterbalances the moment of the remaining long part of the metal portion). The first two cases are very easy to "see". The last case is the most involved, but the result is the same because proportionate expansion of the metal [which can be assumed to expand isotropically] would guarantee that the longer portion of metal component would gain more in absolute length than the short part, leading to the same conclusion.

    Fun wasn't it ? Give this problem to your kids ! :D
     
  14. 96impalaSS

    96impalaSS F1 Rookie

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    #14 96impalaSS, Oct 14, 2004
    In that case I vote for the medal side to be weighed down more than the wood side.

    The wood side would either burn up or all the moisture would be evaporated out of the wood causing it to weigh less.
     
  15. Turb0flat4

    Turb0flat4 Formula 3

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    #15 Turb0flat4, Oct 14, 2004
    Actually, now I'm thinking it might go either way in case 3 depending on the exact assumptions. Let me work out the math and repost, but it's looking like it's possible that in the final case I considered (fulcrum over the metal portion), it's possible balance might prevail or even that the wood side would go lower. I'm not entirely sure yet, but I'll work it out and get back...
     
  16. DrStranglove

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    #16 DrStranglove, Oct 14, 2004
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  17. Turb0flat4

    Turb0flat4 Formula 3

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    #17 Turb0flat4, Oct 14, 2004
    Dr Strangelove LOL :D

    OK, fantastic, I'm feeling really satisfied now. The final answer is (drum roll please) :

    The metal end will always go down

    I got a little thrown by Case 3 in my post, because some "back of the napkin" calculations seemed to suggest that the rod could go either way depending on the exact algebraic relationship between the weights of the wood, the short metal part, the long metal part, the lengths of each part and the coefficient of expansion.

    So I knuckled down, wrote up 3 pages of equations, solved the whole thing symbolically. The short version, there is NO WAY the bar could remain balanced or sink on the wood side. It always goes down on the metal side.

    I'm not a sadist (or a masochist) or I'd post the algebra. ;)
     
  18. 400SPYDER

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    #18 400SPYDER, Oct 14, 2004
    Tom

    Still laughing out loud - superb pic.

    Cheers Kevan
    :) See you down the road - with huge grins on our faces :)
     
  19. Turb0flat4

    Turb0flat4 Formula 3

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    #19 Turb0flat4, Oct 14, 2004
    Yeah, I really liked the bunny pic too, I've used it in other places to reply to cryptic posts, it always gets a laugh, at least out of me. :D
     
  20. speedy_sam

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    #20 speedy_sam, Oct 14, 2004
    I got a headache just thinking about this... but I think I would have a better chance of answering this when I was 13 or so than now... I forgotten all my high school physics .... Lord help my kids when they come to me with such questions. At least thankfully we have the internet and enthusiastic f-chatters...

    Homework questions answered --> new topic in the Other Off Topic section LOL!
     
  21. ashsimmonds

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    #21 ashsimmonds, Oct 15, 2004
    i think questions such as this would be right in step for a 12 or 13 year old who is in school learning this stuff daily.

    a couple times a year i have reason to do wierd math problems, i'm 28 and my brain is as fast as ever, but i can't help thinking when i'm doing the problem that before i had pubes i would have been able to spit out the answer in 1/10th of the time.
     
  22. UroTrash

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    #22 UroTrash, Oct 15, 2004

    100% agreement.
     
  23. milstanselnino

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    #23 milstanselnino, Oct 15, 2004
    Wouldn't the rising warm air on the metal cause an upward force on the metal side? Enough to offset the mass changes due to heat?
     
  24. damcgee

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    #24 damcgee, Oct 15, 2004
    I was laughing so loudly my wife had to come into the room to see what I was laughing at :)
     
  25. SRT Mike

    SRT Mike Two Time F1 World Champ

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    #25 SRT Mike, Oct 15, 2004

    You could have saved yourself the math. See what I wrote above...

    "Metal would be heavier, so I would expect the balance point to be somewhere on the metal part, but it would certainly be some amount towards the wood side, because the metal alone would be halfway down it's length for a balance point, and the wood would move it SOME amount (no matter how small) towards one side. "


    Bottom line is that assuming the metal is even in density, it would balance precisely along it's halfway point. Even if the wood part was tiny in comparison, it would move the center of mass SOME distance (no matter how small) towards the wood-side of the metal bar (since more than 50% of the metal would need to be past the balance point in the opposite direction to balance out the wood). Even if it was 50.0001%, it would still be some number greater than 50%.

    So, as the metal expands, it does so evenly, and given that proportionally more is on the all-metal side, and that wood doesn't expand the same way with heat, the metal side will ALWAYS be the one to drop.
     
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