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Topic: Two identical spheres? (Read 14561 times) |
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Kozo Morimoto
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Two identical spheres?
« on: Oct 8th, 2002, 4:53am » |
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You are given 2 identical looking spheres. They have the same mass and have the same diameter. Physically, they look the same, and have the same surface texture. (ie you can't visually pick them apart) They are both hard, thus they won't bounce and they won't have any 'give'. They both have perfectly smooth surface.
One is made of less dense material and is soild and uniform through out. The other is made of higher density material, but since having the same mass and volume as the other, it is hollow at its centre (assume a spherical cavity with the centre of cavity and centre of the whole sphere at the same point).
With a minimum of instruments, how can you determine which one is hollow and which one is solid?
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| « Last Edit: Sep 20th, 2003, 6:46pm by Icarus » |
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James Fingas
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Re: New Riddle: Two identical spheres?
« Reply #1 on: Oct 8th, 2002, 9:54am » |
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Put them on the floor and spin them. The one that spins for longer is the hollow one.
You could also see which rolls down a ramp faster, but this requires a ramp. The faster-rolling one is the solid one.
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Kozo Morimoto
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Re: New Riddle: Two identical spheres?
« Reply #3 on: Oct 12th, 2002, 6:24am » |
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You didn't explain why one would spin longer nor why one would roll down the ramp faster!
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Chronos
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Re: New Riddle: Two identical spheres?
« Reply #4 on: Oct 14th, 2002, 6:09pm » |
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The spinning solution depends on the details of how you spin them, but assuming that you start them at the same speed, the hollow one will spin longer. The hollow sphere has a higher moment of inertia than the solid one, so when they're spinning at the same rate, it has more angular momentum. Assuming that, at any given speed, they'll both feel the same frictional torque (identical textures, after all), it'll take longer for that torque to get rid of the higher angular momentum.
The ramp solution is perhaps preferable, because the experiment is more easily controlled. Again the solution depends on the moment of inertia, but here, it's easiest to use conservation of energy.
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eviljed
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Re: New Riddle: Two identical spheres?
« Reply #5 on: Apr 7th, 2003, 12:08pm » |
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it doesn't say that you can't damage them, so couldn't you just drill a hole into each?
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NickH
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Re: New Riddle: Two identical spheres?
« Reply #6 on: Apr 7th, 2003, 12:14pm » |
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But using a drill hardly constitutes a "minimum of instruments."
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Beau
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Re: New Riddle: Two identical spheres?
« Reply #7 on: Apr 8th, 2003, 7:10pm » |
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Well, if the spheres are a certain weight, you could put them in a liquid and see which would float. If it's too heavy, weighing them underwater would show the hollow one to be lighter due to the lightness of the air inside the sphere.
The dumb answer would be to hit it with a blunt object (ie. your head) and you could tell by sound alone which one is hollow. <Please note this will not work if head is hollow also>
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aero_guy
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Re: New Riddle: Two identical spheres?
« Reply #8 on: Apr 8th, 2003, 7:15pm » |
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I think you need to examine who's head is hollow. (Sorry, couldn't resist from how your last post was worded.) Same weight + Same volume = Same buoyancy. Hollow don't matter for squat.
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Beau
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Re: New Riddle: Two identical spheres?
« Reply #9 on: Apr 8th, 2003, 7:20pm » |
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You sound awefully sure about that.
Okee, I guess I'll explain. Oh and by the way, I wasn't referring to anyone specific when I said "Your Head" and the subsequent reference to hollowness. It was a non-directive comment.
Anywho, I'd like to create a little experiment to explain to those skeptics "The Glories of Air."
2 plastic cups, store-bought, have similar mass/volume. For the purposes of this experiment they are identical. Weigh the cups in normal atmosphere, one upside down, one right-side up. Then place the two cups 10 feet underwater, one upside down (containing air) and one right-side up (containing water). You see where I'm going with this? The upside down cup will shoot to the surface while the other cup rests at the bottom.
Ok, so this is an extreme of the 2 spheres riddle, but fatten those cups up and leave just a little air in only one and it will weigh less even if it doesn't shoot to the top.
So, it's still my belief that air "does matter squat."
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| « Last Edit: Apr 8th, 2003, 7:34pm by Beau » |
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aero_guy
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Re: New Riddle: Two identical spheres?
« Reply #10 on: Apr 8th, 2003, 7:50pm » |
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What you are missing is that the hollow one is denser so that, "They have the same mass."
So, we take your upside down cup and attach lead weights to it until it weighs the same as a cup full of water.
Yes, I know the analogy breaks down when the air is compressed at different depths, but hey I didn't come up with it originally.
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aero_guy
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Re: New Riddle: Two identical spheres?
« Reply #11 on: Apr 8th, 2003, 7:50pm » |
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And yes, I am awfully sure, so I definitely fall into the no squat camp. I will explain further if you ask.
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| « Last Edit: Apr 8th, 2003, 7:52pm by aero_guy » |
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Beau
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Re: New Riddle: Two identical spheres?
« Reply #12 on: Apr 8th, 2003, 8:06pm » |
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I need to get out more.
I will gladly volunteer my head for all "hollow-sound" tests. Please fax me the spheres and I'll give you results immediately.
Perhaps the hollow portion of the sphere is a vacuum, or, like you said, extra weight was added to offset the air. Ack, whatever.
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towr
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Re: New Riddle: Two identical spheres?
« Reply #13 on: Apr 8th, 2003, 11:25pm » |
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the problem with the cups is that they are not the same volume once they are in the water, the one upright has the volume of just the cup, the other has the volume of the cup+air, or in both cases how much water they displace. With the spheres that not an issue since they are convex shapes.
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budderbear
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Re: New Riddle: Two identical spheres?
« Reply #14 on: Apr 9th, 2003, 3:41pm » |
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i have to say beau's example makes more sense to me. to use the spheres themselves as examples: take the hollow one, make the 'shell' of mass a little denser or thicker to compensate for the holes we are going to drill in it. now we can assume that the one with the holes will be less boyant because of the lack of an air pocket(we can subtract the density of the air trapped in our other sphere to be more accurate). the air, being less dense than the water, is the source of bouyancy. the point about water displacement becomes relevant now in that the sphere with the holes will displace less water than the one with the air pocket if both are submerged. at this point we can return to the original scenario-one hollow and one with consistent density. submerge both(again we can consider the air in the total mass of the hollow sphere) and the hollow one wants to rise more than the other unless the substance the spheres are made of allows for and equal amount of air in the consistent density version, possibly in the form of very small air pockets. but if the substance prevents other elements from penitrating it(such as a water molecules when they are frozen) then there are 'natural' voids in the substance. also in beaus example it doesnt seem that the water should be added to the mass of the rightside up cup. the air in the upside down cup may legitimatly be subtracted from the mass of that cup but the water is simply the medium and the real issue to be resolved in that example is the water displacement issue. i would like to comment that this rant is simply my limited view on the situation and i mean no disrespect to the authors of other posts. after all i find discussion, such as this, to be a great forum for learning and fine tuning our understanding of the world around us and indeed the people we live among.
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Chronos
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Re: New Riddle: Two identical spheres?
« Reply #15 on: Apr 9th, 2003, 7:08pm » |
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We're not going to drill any holes in either of them, because we don't need to. The one made out of the denser material has a hole already in the middle, but it's sealed off so nothing can leak in.
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Icarus
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Re: New Riddle: Two identical spheres?
« Reply #16 on: Apr 9th, 2003, 7:27pm » |
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It is not the displacement of water by air that causes bouyancy. It is the displacement of water - period.
When you submerge an object, it displaces a volume of water equal to the object's volume (in the case of the overturned cup, the object submerged is not just the cup itself, but the cup + the trapped air). The water has only one way to accommodate the extra volume, and that is for the displaced water to raise up the water level - equivalent to lifting the displaced water up to the surface level. Suppose the object has mass M, volume V, and is at an average depth h. Let d be the density of water.
The potential energy gained by the water being displaced raised is given by
Ew = dVgh,
where g is the acceleration due to gravity. The potential energy lost by the object when it is submerged is
Eo = -Mgh.
So the total change in potential energy is
E = Ew + Eo = dVgh - Mgh = (dV - M)g.
Force is the change in Energy with respect to distance. (No calculus is needed here because the Energy changes linearly with distance.)
So the force acting on the object (positive for upward because of how I chose the signs above) is
F = E/h = (dV - M)g.
The bouyant, or perceived, weight is the downward directed force acting on the object: W = -F = Mg - Vdg. Note that d and g are constants.
That is, the weight of each sphere, when measured under water, depends on exactly two things: the Mass of the sphere, and its Volume. More specifically, the volume of water it will displace - that is, its outside volume. The void in the middle of one makes no difference in this, since water can't get into it.
Since the Mass and "outside" volume of the two spheres is the same, their weights are exactly equal under water, just as it is out of the water.
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Clay Black
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Re: New Riddle: Two identical spheres?
« Reply #17 on: Jun 7th, 2003, 9:57pm » |
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Perhaps this is just me being silly, but couldn't you just click them together like pool balls and listen for the hollow one? I guess this is assuming they are small enough to move easily  I believe this solution would require no tools.
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yellowchicken
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Re: New Riddle: Two identical spheres?
« Reply #18 on: Jun 17th, 2003, 10:33am » |
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i really can't get my head round this... i'm so stuck.
here's a solution... you heat them. the one which is denser will get hot on the other side faster than the less dense ball (the one without the air bubble).
Also... to what extent does it work if you put them under water?! i'm assuming that because they both have the same mass, gravity will act on both in exactly the same way. thus they'll both sink at the same speed etc... but i have a friend who insists that the one with air will sink less quickly because of the air bubble... isn't this just ignoring the density problem?
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redPEPPER
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Re: New Riddle: Two identical spheres?
« Reply #19 on: Jun 17th, 2003, 3:25pm » |
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It doesn't work in water. And you summed it up well: the overall density is the same.
Two forces are applied on the immerged spheres:
1. weight, pointing down. Since both spheres are the same mass, the force is the same on both.
2. buoyant force, pointing up. Since both sphere are the same volume, the force is the same on both.
In order to differenciate the spheres you have to use the fact that the density varies inside the sphere. That's when spinning them comes to mind as a possible solution.
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Dummy
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Re: New Riddle: Two identical spheres?
« Reply #20 on: Jul 15th, 2003, 2:53am » |
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In an effort to result perspective... I'd rather knock both spheres and analyze the noise  ...if there's a gas cavity you get a different sound propagation... easy and still quite effective (our ears are quite a good sonar, i guess 
Seems that you deprecate this solution, don't like easy responses uh? ^^
PS: All this does not apply if we have a VOID cavity, wich does not propagate sound hehehe
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Icarus
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Re: New Riddle: Two identical spheres?
« Reply #21 on: Jul 15th, 2003, 6:04pm » |
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While the sound method will most likely work, it is not certain. Since the two spheres are necessarily made of materials with different properties (in particular, different densities), it is possible that the material properties also differ in such a way that they produce the same tone when knocked, despite one being hollow and the other solid.
The spinning method is guaranteed to work, however.
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Kozo Morimoto
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Re: New Riddle: Two identical spheres?
« Reply #22 on: Jul 16th, 2003, 5:42pm » |
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Would the spinning/rolling method still apply if the cavity was not centralized? i.e. if the material was spreadout evenly, like a sponge with tiny cavities everywhere?
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James Fingas
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Re: New Riddle: Two identical spheres?
« Reply #23 on: Jul 17th, 2003, 9:12am » |
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Kozo,
No, the rolling method only works if the sphere is less dense in the middle than farther out. If you distribute the hollow parts around, you're effectively decreasing the average density so it's that same as that of the hollow ball.
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Kozo Morimoto
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Re: New Riddle: Two identical spheres?
« Reply #24 on: Jul 20th, 2003, 11:37pm » |
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How about if the mass was concentrated in the centre like a bicycle wheel hub (but in 3D of course) ?
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