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        riddles >> easy >> Two identical spheres?
        
(Message started by: Kozo Morimoto on Oct 8^th, 2002, 4:53am)

Title: Two identical spheres?
Post by Kozo Morimoto on Oct 8^th, 2002, 4:53am

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?

Title: Re: New Riddle: Two identical spheres?
Post by James Fingas on Oct 8^th, 2002, 9:54am

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.

Title: Re: New Riddle: Two identical spheres?
Post by NickH on Oct 8^th, 2002, 4:15pm

Here is an interesting link on another (almost) spherical body, the Earth:

http://www.geol.binghamton.edu/faculty/barker/demos/demo10.html

Title: Re: New Riddle: Two identical spheres?
Post by Kozo Morimoto on Oct 12^th, 2002, 6:24am

You didn't explain why one would spin longer nor why one would roll down the ramp faster!

Title: Re: New Riddle: Two identical spheres?
Post by Chronos on Oct 14^th, 2002, 6:09pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by eviljed on Apr 7^th, 2003, 12:08pm

it doesn't say that you can't damage them, so couldn't you just drill a hole into each?

Title: Re: New Riddle: Two identical spheres?
Post by NickH on Apr 7^th, 2003, 12:14pm

But using a drill hardly constitutes a "minimum of instruments."

Title: Re: New Riddle: Two identical spheres?
Post by Beau on Apr 8^th, 2003, 7:10pm

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>

Title: Re: New Riddle: Two identical spheres?
Post by aero_guy on Apr 8^th, 2003, 7:15pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by Beau on Apr 8^th, 2003, 7:20pm

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."

Title: Re: New Riddle: Two identical spheres?
Post by aero_guy on Apr 8^th, 2003, 7:50pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by aero_guy on Apr 8^th, 2003, 7:50pm

And yes, I am awfully sure, so I definitely fall into the no squat camp. I will explain further if you ask.

Title: Re: New Riddle: Two identical spheres?
Post by Beau on Apr 8^th, 2003, 8:06pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by towr on Apr 8^th, 2003, 11:25pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by budderbear on Apr 9^th, 2003, 3:41pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by Chronos on Apr 9^th, 2003, 7:08pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by Icarus on Apr 9^th, 2003, 7:27pm

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

E_w = dVgh,

where g is the acceleration due to gravity. The potential energy lost by the object when it is submerged is

E_o = -Mgh.

So the total change in potential energy is

E = E_w + E_o = 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.

Title: Re: New Riddle: Two identical spheres?
Post by Clay Black on Jun 7^th, 2003, 9:57pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by yellowchicken on Jun 17^th, 2003, 10:33am

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?

Title: Re: New Riddle: Two identical spheres?
Post by redPEPPER on Jun 17^th, 2003, 3:25pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by Dummy on Jul 15^th, 2003, 2:53am

In an effort to result perspective... I'd rather knock both spheres and analyze the noise ;D ...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

Title: Re: New Riddle: Two identical spheres?
Post by Icarus on Jul 15^th, 2003, 6:04pm

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.

Title: Re: New Riddle: Two identical spheres?
Post by Kozo Morimoto on Jul 16^th, 2003, 5:42pm

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?

Title: Re: New Riddle: Two identical spheres?
Post by James Fingas on Jul 17^th, 2003, 9:12am

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.

Title: Re: New Riddle: Two identical spheres?
Post by Kozo Morimoto on Jul 20^th, 2003, 11:37pm

How about if the mass was concentrated in the centre like a bicycle wheel hub (but in 3D of course) ?

Title: Re: New Riddle: Two identical spheres?
Post by Icarus on Jul 21^st, 2003, 4:52pm

What is necessary for the rolling/spinning method to work is a difference in the inertias of the two spheres. The inertia of a point mass is given by the product of the mass times the square of the moment-arm (the moment-arm is the distance from the mass to the axis of rotation). For a distributed mass, it is the integral of mass density times the moment-arm squared.

By concentrating the mass at the outside of the sphere in the original problem, you increase the moment-arm, and in so doing, you increase the inertia. This causes the sphere to be less responsive to rotational forces - so it will rotate slower than the homogenous sphere.

If you concentrate the mass in the center instead, you decrease the moment-arm, and decrease the inertia. This causes the sphere to be more responsive to rotational forces - so it will rotate faster than the homogenous sphere.

When you suggested spreading the mass out evenly - you made it a second homogenous sphere, so of course it will be indistinguishable from the first. Which is why James said it would rotate at the same speed.

You could also vary the density - having it higher in the center, then decreasing, but finally picking up again at the outer rim - in such a way that the overall inertia will match that of the homogenous sphere. In this case, the rolling/spinning method will not work either.

Note also that the rolling/spinning method is only useful if you know that the non-homogenous ball has most of its mass at the outside, or if you know that it has most of its mass at the center. If you only know that one or the other is true, but not which, when you spin the two balls you will see a speed difference, but will not know whether the faster ball has its mass in the center or the slower ball has its mass concentrated at the outside.

Title: Re: New Riddle: Two identical spheres?
Post by tmj0nes99 on Jul 29^th, 2003, 10:06am

Cut them in half.

Title: Re: New Riddle: Two identical spheres?
Post by Dalv on Jul 29^th, 2003, 4:24pm

Couldn't you just hit them? The hollow one should have a higher pitch since it is denser (if I remember my physics right)

Title: Re: New Riddle: Two identical spheres?
Post by James Fingas on Jul 30^th, 2003, 12:52pm

My guess is that the solid one would be higher in pitch. For identical materials, the hole in the middle should lower the pitch, and for the same sort of material, higher density should also lower the pitch. But the hollow one would probably be louder...

Title: Re: Two identical spheres?
Post by Jimmy Hatt on Feb 11^th, 2004, 10:06am

Geez, lots of math going on. All you have to do is knock them together. The solid one will "thud" and the hollow one will "ring". Try it with hammers the next time you're at the hardware store. A ringing hammer is hollow and a bad investment. ;D

Title: Re: Two identical spheres?
Post by nonothing on Feb 16^th, 2004, 8:09am

James Fingas was right on in the 2nd post on this thread. From there it went crazy. I'm sure the intent of this problem had nothing to do with air in the middle. You know if you spin around with your arms close in you can go fast, but if you put your arms further out you go slower. This is what this problem was trying to get at. (spinning the ball and going down a ramp are the same thing, they are both spinning about its central axis)

Title: Re: Two identical spheres?
Post by Icarus on Feb 16^th, 2004, 5:02pm

The only craziness were those who do not understand buoyancy well enough to know that the spheres will be equally buoyant. The rest was merely a series of questions into alternative situations.

For the original problem, the spinning or rolling technique is guaranteed to work.

The hollow sound technique is almost, but not completely, guaranteed. It is possible that the structure & material of the hollow sphere is such that it's sound when struck mimics that of the solid sphere. Note that in the hammer example, both hammers are made of essentially the same material, steel, whose density does not vary greatly.

But we are not guaranteed this for the spheres. Indeed we know there must be a significant difference in the material structure of the two spheres or else one could not be hollow and the other solid, yet still have the same mass and exterior volume.

If, as Kozo asked in posts #23 & #25, we do not know that the center of one is hollow, only that hollows exist within it, then there is no non-invasive technique that is guaranteed to show the difference. And even if we do find a difference, without additional information concerning the materials the spheres are constructed of, there is no way of telling from that difference which sphere is which. In this case, drilling or cutting or x-raying, or otherwise probing the interior is the only way to uncover which is which.

Title: Re: Two identical spheres?
Post by nonothing on Feb 19^th, 2004, 2:01pm

And another thing. If the balls were of identical mass and diameter, and the ball with the hole in the middle had a vacuum, water would have the same effect on both of them. They would both sink at the same rate, or float, or whatever. But of the hollow ball had air in, then you are increasing its mass, and since its volume is staying it actually becomes more dense overall, and would therefore sink faster than the other ball (less bouyant).

Title: Re: Two identical spheres?
Post by aero_guy on Feb 19^th, 2004, 2:06pm

I think the problem intends that the two weigh the same including whatever mass may or may not be included from air in the center.

Title: Re: Two identical spheres?
Post by Icarus on Feb 19^th, 2004, 4:56pm

Yes, if the "identical mass" meant only of the ball itself, and not the air in the cavity (assuming it did contain air), then a sufficiently fine balance would be able to tell them apart. (Yes, there are balances that sensitive - it is even possible to build one yourself from easily obtained parts.)

Title: some complications with spinning and hitting.
Post by bodo on Mar 13^th, 2004, 10:45am

problems with spinning:

The problem states "They both have perfectly smooth surface." If it was indeed perfectly smooth, you would likely not be able to spin it: the suface would be frictionless - how would you 'grip' it to spin it?

problems with hitting:

The problem further states that "They are both hard, thus they won't bounce and they won't have any 'give'" If you substance has no give, it will not distort or vibrate in any way. It will not conduct sound.

Title: Bitten off more than we can chew?
Post by Mythras on Mar 13^th, 2004, 2:40pm

uhh -
I think it's pretty safe to say, that if we take the puzzle verbatim, it's practically impossible.

Bodo is right, in that 'perfectly smooth' spheres would A) be impossible to spin. and B) even if you did get them spinning, it would be impossible to tell how fast they spun, or even, when they stopped.

Now, I really hope, I'm not the only one to notice that - perfectly 'hard' spheres DO bounce - and they bounce perfectly elastically.

Imagine if you will - the two balls on a collision course. What happens when they collide? Well, they gotta bounce off of eachother - if they both stop dead - energy would not be conserved! (in real life - objects that don't bounce, are ones who had their energy dissipated 'breaking' or 'squishing')

not only that - a perfectly rigid body, won't allow its constituent atoms to vibrate. Hence, they would carry no waves what so ever - no sound - no heat.

These balls are at ABSOLUTE ZERO (the atoms are not moving!) Only, they won't interact thermally with anything. So - fat lot of good that does us.

If they are at absolute zero, are stuck there, are perfectly hard, and perfectly smooth, how are we going to drill into these?? Uhhh we aren't.

No lasers either kids - do these things even interact with photons? (Ofcourse a lightsaber would work! - but lets stick to realit...ummm)

So - I think it's pretty clear, that some of the constraints have to be lessened, in order to allow us to come to the simple solution of "spinning the balls" to find the answer.

But, given these constraints - I've thought up a little solution...

We could get a neutron gun - and shoot at the balls for a long time (with a stream of neutrons).
Quantum mechanics says that - there is a probability that on any given collision, that a neutron will tunnel through the barrier of the walls.

This tunneling effect, will cause a pile of neutrons to collect inside of our hollow sphere - thus, making it heavier than our solid sphere... over time.

Since we are in a magical world of 'super rigid balls, with smooth surfaces, that violate energy conservation' - I'm going to have a equally magical scale - that allows me to measure the mass difference caused by a single neutron. So we dont have to wait long - for it to DING!

Mythras

Title: Re: Two identical spheres?
Post by rmsgrey on Mar 13^th, 2004, 3:39pm

1) Even at absolute zero, there's still some residual heat energy

2) Since normal matter is actually mostly "empty space" with relatively small specks of concentrated (charged/coloured) mass a sphere which is "solid and uniform throughout" must be some exotic form of matter or a known elementary particle (if any of them are spherical and uniform) - which also explains most of its other properties. The second sphere then is either something the size of a known elementary particle but with a central spherical cavity (note that the puzzle doesn't specify how the remaining mass outside the cavity is distributed - there could be additional cavities or other local density variations), or is also of some exotic form of matter - a "sphere" of normal matter has an uneven "surface" even if you ignore the vast empty spaces.

If both spheres are elementary particles, then they presumably have different properties that can be tested.

Title: Re: Two identical spheres?
Post by John_Gaughan on Mar 14^th, 2004, 8:35pm

on 03/13/04 at 15:39:29, rmsgrey wrote:

I think rather than saying the matter is uniforms throughout, a more accurate description would be that the molecules are evenly distributed.

Title: Re: Two identical spheres?
Post by Mythras on Mar 16^th, 2004, 10:25am

on 03/13/04 at 15:39:29, rmsgrey wrote:

1) Even at absolute zero, there's still some residual heat energy

I was not aware of this... could you explain how "we know this?" - perhaps there is some advanced theories, that I dont know about? -- but, to let you know where I stand. I do know about bose einstein condensation - and that is the closest we have gotten to absolute zero. I also know, that the uncertainty principle makes it impossible to reach absolute zero. Along with that, i know - whatever you'd expect someone with a degree in physics would know.

on 03/13/04 at 15:39:29, rmsgrey wrote:

1) Even at absolute zero, there's still some residual heat energy

2) Since normal matter is actually mostly "empty space" with relatively small specks of concentrated (charged/coloured) mass a sphere which is "solid and uniform throughout" must be some exotic form of matter or a known elementary particle (if any of them are spherical and uniform) - which also explains most of its other properties. The second sphere then is either something the size of a known elementary particle but with a central spherical cavity (note that the puzzle doesn't specify how the remaining mass outside the cavity is distributed - there could be additional cavities or other local density variations), or is also of some exotic form of matter - a "sphere" of normal matter has an uneven "surface" even if you ignore the vast empty spaces.

If both spheres are elementary particles, then they presumably have different properties that can be tested.[/quote]

well, I am assuming the spheres are 'exotic material' because of their exotic properties of perfection... and I am guessing your comment about 'empty space' - is claiming the probability of tunneling to be equal between the spheres? hmm, well, I can't definitely say you are wrong (or right!) off the top of my head... without looking at the math and experiments, and the overall point that the question needs to be rephrased still stands.

Now, I'm pretty sure sure that having a non-uniform density would cause variations in the object's local gravitational field (on its surface.) Much like the gravity on earth varies across the surface - according to the density of the ground directly beneath it. (I went to a lecture on this subject, so I'm sure that guy has research to back up that claim!)

if it is non-uniform, we just get a gravity meter and measure around its surface! :)

Title: Re: Two identical spheres?
Post by rmsgrey on Mar 17^th, 2004, 9:03am

If you have an object with spherical symmetry, then (under Newtonian physics, not sure how relativity affects it) the only things affecting its gravity are the mass and your radial distance - the actual distribution of the mass is irrelevant, provided it satisfies the symmetry condition.

Back to absolute zero: my recollection is that absolute zero is defined as the lowest achievable temperature - rather than as the temperature with 0 energy. Uncertainty means that you can't ever have 0 energy, because then you'd have no accuracy in time, so the coldest you can get has to be a bit more than 0 energy.

[e]Just noticed that this post got me my fourth star... perhaps not the best start for a new senior riddler[/e]

Title: Re: Two identical spheres?
Post by Noke Lieu on Mar 17^th, 2004, 3:14pm

1st: quickly admit that I don't know much about absolute zero.
2nd: Giggle at that statement.
3rd: It seems that obtaining a state of 0 energy would be incredibly hard- the whole mc² might be something to do with it. Another is mgh- unless its a solitary atom. Even then, I would stab a guess at it still having gravitional potential energy.
4th: Get myself prepared to learn something.

Title: Re: Two identical spheres?
Post by Icarus on Mar 17^th, 2004, 7:56pm

1) The third law of thermo-dynamics says that you can never reach absolute zero, so that is not the definition! (Interesting, it does not rule out going below absolute zero, and this in fact has been accomplished in a very limited circumstance.)

2) Absolute zero is by definition simply the zero of the Kelvin temperature scale. The definition of this scale is based on general scientific principles (other than an arbitrary constant which is set by demanding that the triple point of water be at 273.16 K). Here is how my old Thermo book, "Heat and Thermodynamics" by Zemansky & Dittman, 6th Edition, McGraw-Hill, describes absolute zero: "If a system undergoes a reversible isothermal process between two reversible adiabatic surfaces without transfer of heat, the temperature at which this process takes place is called absolute zero."

Trying to define all of that is more then even I want to tackle at this time.

3) The same textbook also has this to say: "When it is necessary in statistical mechanics to correlate termperature to molecular activity, it is found that classical statistical mechanics must be modified with the aid of quantum mechanics and that, when this modification is carried out, the molecules of a substance at absolute zero have a finite amount of kinetic energy, known as the zero-point energy."

Title: Poorly worded riddle.
Post by bodo on Mar 24^th, 2004, 8:41am

I think Mythrias' post really makes the point that this riddle is in need of re-wording. I seriously doubt the 'easy riddle' category should contain a riddle that wanders so easily into college level physics.

I hadn't even thought about Mythrias' point that the spheres would bounce - especially if you consider they must bounce of something. For instance, you could imagine bouncing them off a rubber mat - the rubber doesn't care about their unphysically hard property - it'll still bounce them. Good points on the heat conduction as well.

But as long as we are playing this game with our surreal balls I propose this solution.

Imagine if the balls were travelling at relativistic speeds towards you, the would deform into ellipsoid - ruining their spherical symmetry that they have enjoyed thus far. Now, when we measure photons that have either been bounced or emmited from the minor axis of both we will observe they are different wavelengths. This will be because the hollow one will 'pull' less on the photon than will the solid one therefor the hollow one's photons will be more energetic i.e. 'bluer' and the solid one's will be redder.

Now since surreality is abounding. I propose that we should be able to detect this change a non-relativistic speeds and with our eyeballs. So, simply throw the balls towars you (or away from you) and the redder one is solid.

Perhaps we should turn our attention to rewording this riddle...

Title: Re: Two identical spheres?
Post by Logix128 on Nov 16^th, 2004, 6:52pm

heat them up and measure how much they expand, whichever one changes more in volume is likely the one with the more evenly distributed density.. 8)

Title: Re: Two identical spheres?
Post by Logix128 on Nov 16^th, 2004, 6:57pm

on 11/16/04 at 18:52:07, Logix128 wrote:

heat them up and measure how much they expand, whichever one changes more in volume is likely the one with the more evenly distributed density.. 8)

SORRRY!! i posted too early, forgot to add that you could also add the same amount to heat to each of them and measure how fast they give off heat ;; the one with the air or a vaccuum inside of it will give off heat slower than the other

Title: Re: Two identical spheres?
Post by Icarus on Nov 17^th, 2004, 3:48pm

No - this is no more dependable than the sonic test. It depends upon the properties of the materials used to construct the two spheres.

If the two spheres are both made of materials with the same coefficient of thermal expansion, then they will expand by the same amount when heated, regardless of any internal voids. And if they do not expand by the same amount, this just tells you which material has a higher coefficient of thermal expansion. Sadly, the coefficient of thermal expansion is not related to density, so this does not give you any information about which sphere has a void.

Similarly, thermal resistance is also not particularly related to density, so knowing which sphere cools faster does not tell you anything either.

Title: Re: Two identical spheres?
Post by Lewis Temple on Dec 2^nd, 2004, 7:59pm

Could you not put a piece of tape around them with the ends barely touching and put them into a vacuum tube, the one with the hollow space should expand as the air inside is put under vacuum, thus making the tape pull apart at the ends?

Title: Re: Two identical spheres?
Post by Icarus on Dec 2^nd, 2004, 8:28pm

The vacuum outside does not magically reach through the sphere to influence the cavity inside. (By the way, there is nothing that says the cavity is filled with air. It could be a vacuum as well.)

Placing the spheres in a vacuum will relieve the pressure on their outer surfaces, which will generally cause them to expand. However:
(1) This expansion is very small for solid materials for differences of only 1 atmosphere.
(2) Both spheres will expand, so expansion alone will not help.
(3) The presence of the hollow will not change the expansion of that sphere: a solid sphere with the same composition would expand by the same amount.

Because the two balls are made of different materials, it is possible that they would expand by differing amounts. But this would be difficult to detect. Tape would almost certainly be insufficient for the task.

Title: Re: Two identical spheres?
Post by Lewis Temple on Dec 3^rd, 2004, 2:31am

OK, then what if you soak them in water, and immediately measure them after you take them out, the less dense one should weigh more because it has soaked up more water than the dense one. I know it sounds crazy, but at 4:30 in the morning, it sounds like a good idea.

Title: Re: Two identical spheres?
Post by rmsgrey on Dec 3^rd, 2004, 8:22am

That only works if they are porous, the less dense material is more porous, and the more dense material doesn't let water leak through to the hollow...

Since the riddle specifies that the less dense material sphere is "solid and uniform throughout", it's unlikely to be porous. Since both have the same perfectly smooth surface texture, they're both unlikely to take on water anyway.

Title: Re: Two identical spheres?
Post by SWF on Dec 3^rd, 2004, 8:22pm

The question states that the solid ball is uniform, but does not say so about the hollow ball. It is therefore allowable for the hollow ball to have density which varies with position. That allows the possibility of the hollow ball to have the same moment of intertia as the solid ball, nullifying the moment of inertia methods.

Given ideal accuracy of measurement, variations on some of the other proposed methods are feasible. For example, the thermal methods are more likely to be foolproof than the moment of inertia method. It is is easier to design a hollow ball with the same moment of inertia, than to make one thermally indistinguishible. Good luck making a hollow ball with the same steady state and transient temperature response for all possible boundary conditions over a range of temperatures. Don't forget to account for radiation heat transfer across the cavity.

Of course, rolling down a ramp (I have doubts about spinning working well) is easy and effective most of the time, and there is always X-ray.

Title: Re: Two identical spheres?
Post by alien on Dec 4^th, 2004, 6:59am

Use some kind of a sophsiticated scanner. ;D

Title: Re: Two identical spheres?
Post by Lewis Temple on Dec 9^th, 2004, 3:38am

OK, How about checking continuity with a voltmeter. The less dense one should have a higher continuity than the dense one with the hollow center.

Title: Re: Two identical spheres?
Post by Icarus on Dec 9^th, 2004, 9:10am

By "continuity", I assume you actually mean "conductance", since you are using a voltmeter to check it (conductance is the inverse of resistance).

Alas, but this depends on the properties of the materials the spheres were made out of. One of them may have higher resistance than the other. So you wouldn't know if the differences you are seeing are due to materials or to geometry.

The same failing applies to SWF's comments. One does not need to try and match physical characteristics of the two spheres to render these tests ineffective. They are only effective if you know something about the materials used. Otherwise the differences measured still tell you nothing, as you have no way of knowing which response is appropriate for the hollow ball.

Basically, the problem is unsolvable without additional information than what was given, and what can be learned with non-invasive experiments.

Title: Re: Two identical spheres?
Post by SWF on Dec 9^th, 2004, 6:59pm

on 12/09/04 at 09:10:39, Icarus wrote:

The same failing applies to SWF's comments. One does not need to try and match physical characteristics of the two spheres to render these tests ineffective.

Measurements from the two spheres are not compared to each other, they are compared to other measurements on the same sphere.

First, apply and measure boundary conditions on the surface so you know distribution of (voltage and current) or (temperature and heat flux). Assuming steady state case and ideal measurements, from mathematics you can then figure out the conductivity (electrical or thermal) of each sphere under the assumption of solid homogeneous spheres. Conductivity is the material property that determines the steady state field.

Next, you can apply different boundary conditions and compute conductivity again. You are looking for results inconsistent with the sphere being solid and homogenous. I claim that with the standard basic laws of physics that it impossible to have a hollow sphere and a solid homogeneous sphere give identical response for all possible boundary conditions applied to the surface- assuming finite, non-zero conductivity.

Title: Re: Two identical spheres?
Post by Icarus on Dec 10^th, 2004, 10:25am

Ingenious idea. But I am less sure than you that is impossible to match the characteristics of the solid ball with an appropriate build of the hollow one. It might be impractical, however.

Title: Re: Two identical spheres?
Post by OnlyAnEgg on Jan 12^th, 2005, 9:16am

If the mass were concentrated to the centre, then you'd have the situation reversed: the uniform sphere would have MORE angular momentum than the one with centralised mass. Then the uniform sphere would spin down more slowly than the other, or roll down the ramp more quickly.
I'm not sure how one would construct such an "inverse hollow", though. An intrigueing idea, isn't it?

Title: Re: Two identical spheres?
Post by Icarus on Jan 12^th, 2005, 4:10pm

It isn't that hard: Use a very dense substance as your core, then a less dense mantle. Done right, the mass still totals the same as a uniform substance with medium density.