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  1. - Top - End - #451
    Surgebinder in the Playground Moderator
     
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    Default Re: Physics In the Playground

    Quote Originally Posted by noparlpf View Post
    1. Say there's a tunnel through the center of the Earth. How long would it take for you to fall to the other side?
    That depends. Do we have to account for air resistance?

    If yes, then you'll never actually reach the other side, as quite a bit of your kinetic and gravitational energy will be lost along the way.

    If no, then I remember solving that in class once, and I think the answer was somewhere around half an hour. I also remember that, provided friction is assumed to be 0 the whole way, it doesn't actually matter whether the tunnel goes through the center or not. A tunnel going only a quarter of the way around the world (or any other fraction, even traveling 1 inch to the side) would have the same transit time, provided only that it is straight and powered only by gravity.

    Quote Originally Posted by noparlpf View Post
    2. How long would you have to yell at a cup of coffee to heat it? (Presumably we're aiming for a drinking temperature of thirty degrees or so.)
    I haven't done any math for this one, but I expect the cup would lose heat to its surroundings faster than any gained from your yelling. So, take your pick between "forever", "undefined", and "no answer".
    Last edited by Douglas; 2013-01-17 at 04:24 PM.
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  2. - Top - End - #452
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    Default Re: Physics In the Playground

    Quote Originally Posted by douglas View Post
    If no, then I remember solving that in class once, and I think the answer was somewhere around half an hour. I also remember that, provided friction is assumed to be 0 the whole way, it doesn't actually matter whether the tunnel goes through the center or not.
    It's 42 minutes:

    http://www.physicscentral.com/explore/poster-earth.cfm

  3. - Top - End - #453
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    Default Re: Physics In the Playground

    Quote Originally Posted by douglas View Post
    If yes, then you'll never actually reach the other side, as quite a bit of your kinetic and gravitational energy will be lost along the way.
    The biggest problem is, that once you go deep enough the air would turn liquid because of the pressure.

  4. - Top - End - #454
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    Default Re: Physics In the Playground

    Quote Originally Posted by douglas View Post
    That depends. Do we have to account for air resistance?

    If yes, then you'll never actually reach the other side, as quite a bit of your kinetic and gravitational energy will be lost along the way.

    If no, then I remember solving that in class once, and I think the answer was somewhere around half an hour. I also remember that, provided friction is assumed to be 0 the whole way, it doesn't actually matter whether the tunnel goes through the center or not. A tunnel going only a quarter of the way around the world (or any other fraction, even traveling 1 inch to the side) would have the same transit time, provided only that it is straight and powered only by gravity.
    I assumed air resistance would stop you from getting there, and that eventually you'd fall back and sort of oscillate around the center of the earth for a while until you starved. So without air resistance (so you have to have a suit or something?) how many gees does the faller experience at most?

    I haven't done any math for this one, but I expect the cup would lose heat to its surroundings faster than any gained from your yelling. So, take your pick between "forever", "undefined", and "no answer".
    That's what I said. Okay, cool.

    Quote Originally Posted by RebelRogue View Post
    The biggest problem is, that once you go deep enough the air would turn liquid because of the pressure.
    I think we're pretending there's no air, because otherwise the answer is "you'll never get there".
    Jude P.

  5. - Top - End - #455
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    Default Re: Physics In the Playground

    Quote Originally Posted by noparlpf View Post
    So without air resistance (so you have to have a suit or something?) how many gees does the faller experience at most?
    One, at the surface. As you fall deeper into the Earth there's less mass underneath you so the gravity actually drops. Don't forget that 1g is actually a pretty hard acceleration--most cars couldn't achieve it; if you were to travel continuously at 1g for 21 minutes, you'd be going at nearly 28,000mph! The actual peak speed at the Earth's core would be less than that because of the reduced acceleration as you go deeper--I think the article I linked to above said you'd hit around 18,000mph.

  6. - Top - End - #456
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    Default Re: Physics In the Playground

    Here is something that isn't about gravity and motion for a change: What are the processes that create fog, and why is it most common in the morning?
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  7. - Top - End - #457
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    Quote Originally Posted by factotum View Post
    One, at the surface. As you fall deeper into the Earth there's less mass underneath you so the gravity actually drops. Don't forget that 1g is actually a pretty hard acceleration--most cars couldn't achieve it; if you were to travel continuously at 1g for 21 minutes, you'd be going at nearly 28,000mph! The actual peak speed at the Earth's core would be less than that because of the reduced acceleration as you go deeper--I think the article I linked to above said you'd hit around 18,000mph.
    Hmm, that makes sense then. I guess there's enough mass being left behind that being closer to the center of mass doesn't quite offset it.

    Quote Originally Posted by Yora View Post
    Here is something that isn't about gravity and motion for a change: What are the processes that create fog, and why is it most common in the morning?
    The Wikipedia page seems to explain it fairly well in simple terms. When it's very humid, and within a couple of degrees of the dew point, you get some tiny water droplets turning liquid around small particles in the air, just like a cloud, but at ground level.
    What you're thinking of would be "radiation fog" partway down the page, and it doesn't happen in the morning, it happens at night; it's just that you only wake up to see it in the morning, I guess. That's just caused by cooling at nighttime bringing the temperature low enough for some water vapor to condense.
    Jude P.

  8. - Top - End - #458
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    Default Re: Physics In the Playground

    Because maximum humidity is lower for colder air and during night it gets cold enough to expell some of the moisture that had evaporated into the air during day?
    Once the sun comes up and warms the air, moisture can evaporate again and "disappear" because the air can contain higher humidity, correct?
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    Quote Originally Posted by Yora View Post
    Because maximum humidity is lower for colder air and during night it gets cold enough to expell some of the moisture that had evaporated into the air during day?
    Once the sun comes up and warms the air, moisture can evaporate again and "disappear" because the air can contain higher humidity, correct?
    Yeah, basically. At higher temperatures more of the water remains gaseous, rather than colloidally suspended in liquid form.
    Jude P.

  10. - Top - End - #460
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    Default Re: Physics In the Playground

    Cool, I feel so smrt now.
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    Or anyway that's what I understand of it. People who know more things about this stuff should probably correct or supplement that.
    Jude P.

  12. - Top - End - #462
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    Default Re: Physics In the Playground

    I know people here have already talked about and answered this question:
    Quote Originally Posted by noparlpf View Post
    1. Say there's a tunnel through the center of the Earth. How long would it take for you to fall to the other side?
    but I wanted to know how the '42 minutes, regardless of whether the tunnel passes through the centre of the Earth or just stays near the edge' answer comes about.

    I wrote up this solution. Does it make any sense if you're not already familiar with Lagrangian mechanics? It could definitely do with some diagrams, but LyX/LaTeX is really not very nice about putting diagrams in sensible places so I haven't included any. (Also if there are any mistakes, or if I've missed a better way to do it, I would be very glad to hear!)

  13. - Top - End - #463
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    Default Re: Physics In the Playground

    You could probably do it simpler starting from the equations for simple harmonic motion (since that's what a journey through this theoretical tunnel would be), but it's a long time since I did applied maths and I have no idea how that would work out.

  14. - Top - End - #464
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    Default Re: Physics In the Playground

    It is SHM (at the end, having applied all constraints, you get the usual quadratic Lagrangian for SHM) -but I'm showing that you get that regardless of the direction of the tunnel, in a way that can be more easily generalized to spinning or non-uniform Earths. I could probably have gotten there easier by working in cylindrical polars from the beginning though.

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    Surgebinder in the Playground Moderator
     
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    Default Re: Physics In the Playground

    The "regardless of tunnel direction" aspect comes from the following facts:
    1) Changing the direction of the tunnel away from directly through the center reduces its length.
    2) Changing the direction of the tunnel causes gravity to pull at an angle, reducing the parallel portion of the gravitational force.
    3) The amount of change for facts 1 and 2 are both the same trigonometric function of the same angle and therefore identical.

    I think it's actually a little more complicated than that, with some extra math to account for what portion of the Earth is actually still below you, but that combined with the angle work out to match the length reduction.
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  16. - Top - End - #466
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    Default Re: Physics In the Playground

    Quote Originally Posted by noparlpf View Post
    Hmm, that makes sense then. I guess there's enough mass being left behind that being closer to the center of mass doesn't quite offset it.
    I have the feeling that you still don't quite get it. As you approach the Earth (or any solid planet), the effect of its gravity increases UNTIL you reach the surface. Once you go below the surface, the effect of gravity decreases again. The strength of the gravity is based on both the distance and the mass between you and the center of gravity. The maximum force is at the surface because that's as close as you can get to the center while keeping all the mass between you and the center.

    As you go deeper into the planet, the gravity from the mass behind you cancels the gravity from the mass on the far side of the planet. The net result is that you only experience gravity from the mass between you and the center of gravity. As you go deeper and deeper, it's like standing on a smaller and smaller planet and gravity gets weaker and weaker.

  17. - Top - End - #467
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    Default Re: Physics In the Playground

    Quote Originally Posted by Xuc Xac View Post
    I have the feeling that you still don't quite get it. As you approach the Earth (or any solid planet), the effect of its gravity increases UNTIL you reach the surface. Once you go below the surface, the effect of gravity decreases again. The strength of the gravity is based on both the distance and the mass between you and the center of gravity. The maximum force is at the surface because that's as close as you can get to the center while keeping all the mass between you and the center.

    As you go deeper into the planet, the gravity from the mass behind you cancels the gravity from the mass on the far side of the planet. The net result is that you only experience gravity from the mass between you and the center of gravity. As you go deeper and deeper, it's like standing on a smaller and smaller planet and gravity gets weaker and weaker.
    Yeah. Oops. I figured that out, I just said something really silly last time.
    Jude P.

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    Default Re: Physics In the Playground

    Say I have a rope affixed to the top of a post, with the other end attached to an effectively immovable object for simplicity. I hang my laundry out on the line. Is heavy laundry further away from the post going to exert more torque on it as the line sags than if I put the heavier laundry closer to the post?
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  19. - Top - End - #469
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    Default Re: Physics In the Playground

    Tricky one. I'm not sure it'll make much of a difference, because the actual weight on the line will be the same and thus the forces transmitted through the pole will be the same--the only thing that *might* be different is if having the heavier washing nearer the pole makes the angle of that force different, and I'm not sure if that would be the case. (This is assuming that the line doesn't sag so much that the washing starts lying on the ground and thus removing some of the force, of course ).

    Actually, thought about it a bit more: I think you'd actually have *less* torque on the post if the heavier washing was closer to it. What I think that would do is cause the line to deflect more sharply close to the post, so the angle of the force acting on it would be steeper and thus the horizontal component (which is where the torque comes from) would be less.
    Last edited by factotum; 2013-02-02 at 02:28 AM.

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    Default Re: Physics In the Playground

    Yeah, the sag will make a difference, and you want to be far from the post.

    Say the line makes an angle to the horizontal of A where it connects to the post, and B where it connects to the immovable. Say the laundry has mass m. Also, say the line is massless. Call the tension in the line to the post Ta, and in the line to the immovable Tb.

    In equilibrium the forces sum to 0, so

    mg = Ta Sin A + Tb Sin B
    T1 Cos A = T2 Cos B

    So T1 = mg (Sin A + Sin B Cos A / Cos B) = mg Cos A (Tan A + Tan B)

    Now, the torque is just Ta Cos A = mg Cos^2 A (Tan A + Tan B)

    A will be smaller if the laundry is by the immovable than by the post. A and B just switch if you consider pairs of positions with equal distances from the post and the immovable, so (Tan A + Tan B) will have the same value. Cos goes up if you decrease A, so torque is higher for small A. So to get high torque on the post, put the laundry near the immovable object.

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    Default Re: Physics In the Playground

    If you had a sealed container with a vacuum that is connected to a pipe that leads to a pool of water, opening the valve would pump the water up the pipe. (Air presses on the pool, squeezing the water up the colum of empty space in the pipe.)

    Would the size of the vacuum container make any difference in regard to the forces at work that push the water up the pipe? Would a 1 liter vacuum suck with the same strength as a 1000 liter vacuum?

    Or is that only a hypothetical question and even best "near vacuum" would change the results?
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    Default Re: Physics In the Playground

    If the pipe was already filled with water or vacuum, then the force would be the same until the water filled your bottle (or when you reached the equilibrium point if the vacuum is trying to pull the water against gravity).

    If the pipe is filled with air, then the force would diminish as the air entered the bottle, so in this case, the size of the bottle is a factor of your force over time, but the initial force would be the same no matter how big it is.
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    Default Re: Physics In the Playground

    Just to add to what Jimor said, the equilibrium point he's talking about is always the same--you can't raise water through use of a vacuum by more than about 30 feet at the earth's surface. Most pumps rely on vacuum to pull more water into their working chambers, so they're limited to 30 feet as well--this was a big problem back when they started using steam engines to pump water out of mines, because they had to stage them every 30 feet up to the surface in order for them to work properly!

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    Default Re: Physics In the Playground

    More practical solution would be be a bucket-coveyor-belt or putting pressure on it at the bottom. But the later has the problem that you would have to move a large machine underground every time you expand the tunnel.

    Another interesting thing I found is that at a certain point, the outside air pushing the water up and the pipe and the gravity pulling it down is pretty much pulling the water apart, potentially causing it to boil.
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    Default Re: Physics In the Playground

    Quote Originally Posted by Yora View Post
    More practical solution would be be a bucket-coveyor-belt or putting pressure on it at the bottom. But the later has the problem that you would have to move a large machine underground every time you expand the tunnel.

    Another interesting thing I found is that at a certain point, the outside air pushing the water up and the pipe and the gravity pulling it down is pretty much pulling the water apart, potentially causing it to boil.
    Well any water at the boundary layer of the vacuum would boil anyway, wouldn't it?


    Edit: I just found an answer for the "shouting at coffee to heat it" question. Supposedly it would take eight and a half years. I guess that's assuming the coffee never loses any heat to its surroundings? Or if you gave it all the heat generated by eight and a half years of shouting all at once?
    Last edited by noparlpf; 2013-02-03 at 10:53 PM.
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    Default Re: Physics In the Playground

    Another question, this one requiring a lot of variables that you could set at anything you consider plausible. It's about getting a general idea about the subject, not any accurate measurements.

    How much energy would you save by using a simple two-wheel cart or a full four-wheel to pull a load instead of carrying it on the back of a person or animal? Let's assume a "decent dirt road" as the surface.
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    Default Re: Physics In the Playground

    Quote Originally Posted by Yora View Post
    Another question, this one requiring a lot of variables that you could set at anything you consider plausible. It's about getting a general idea about the subject, not any accurate measurements.

    How much energy would you save by using a simple two-wheel cart or a full four-wheel to pull a load instead of carrying it on the back of a person or animal? Let's assume a "decent dirt road" as the surface.
    Just conceptually, it should be a lot. A four-wheeled cart supports the entire mass for you, so you're not lifting it; a two-wheeled cart supports some of the mass for you, but still takes a lot of the load off of you. Ideally you should only be doing the horizontal work, and not have to exert any force to hold the mass against gravity. When you carry it, you do both.
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    Default Re: Physics In the Playground

    Quote Originally Posted by Yora View Post
    Another question, this one requiring a lot of variables that you could set at anything you consider plausible. It's about getting a general idea about the subject, not any accurate measurements.

    How much energy would you save by using a simple two-wheel cart or a full four-wheel to pull a load instead of carrying it on the back of a person or animal? Let's assume a "decent dirt road" as the surface.
    This is actually a very complicated situation, because the kind of "clean" physics where me make some basic assumptions to be able to punch in some numbers to our normal equations just doesn't work at all here. The main one is that technically, moving something along a level surface, whether you're carrying it or towing it in a cart, requires zero "work" energy in the vertical plane, and only a small amount of energy to overcome friction in the horizontal plane.

    So while you could work out all the variables of the force vectors and motion of the human body carrying or pulling that burden, the only practical method of finding the real answer is to measure the real metabolic caloric burn rate of somebody doing each method.

    I suspect that if you're talking about an load of weight that somebody could carry, you wouldn't find all that much difference. The advantage of a person-drawn cart isn't that it makes carrying something more efficient, it's that it makes it possible to transport much more than that person could physically lift otherwise.

    I may unpack more of what I mean later, but consider this: there's a REASON why people carrying heavy burdens prop it on their heads, and it has to do with the definition of "work" I mention above.
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    Default Re: Physics In the Playground

    I agree with Jimor...was actually thinking of posting along those lines but really couldn't figure out how to word it.

    I will add, though, that putting the load on the back of a pack animal saves *all* the energy that the human would have used and is thus no doubt preferable from said human's point of view...

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    Default Re: Physics In the Playground

    The whole question is based on the Dragons Hoard problem: You have a practically infinite amount of gold and can take with you as much as you can carry. Based on the gear soldiers tend to carry, let's say you would take 40 kilos of gold to take home with you. You end up pretty exhausted, but damn, it's gold!

    But what if you had a cart? How much more could you take if you are willing to do the same amount of work to get it home?
    Obviously that depends on both the cart and the terrain you are moving on. If the cart collapses under the weight of the gold or the wheels sink into the ground, you are obviously not going anywhere. And if you have to cross very uneven ground, you probably could end up with even less gold than you could carry on your back, since you would have to basically carry the cart over the obstacles as well. And then there's the friction of the axle. If you have roughly cut wood on wood, it would obviously be a lot more than if you had steel ball bearings.

    But any educated guess?
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