Imagine a solid sphere with practically zero mass (say made of cork). Embedded randomly within it are two point like masses of mass M1 and M2 (imagine two mahogany pellets). Now it is easy to find the combined centre of mass. But what methods are there of determining the positions of the two point masses? The only knowledge that we have is that there are 2 mass points inside the sphere (invisible to the external observer.) We are allowed to do anything to the sphere, move it, shake it whatever. Is it possible to determine the positions (and mass ratios) of the two mass points?
What are your ideas? I'm thinking maybe if you spin the sphere (with a some-how consistent torque, for a consistent time-interval) and note the angular speed, you could end up calculating the rotational inertia about each diameter of the sphere (within certain limits, because there are technically infinite diameters) This would at least give you some information. It's hard for me to generalize it, but, I see only two possibilities. Either the two points lie along a diameter, or they don't. You should be able to determine not only if they are on a diameter or not, but, (if they are on a diameter,) which diameter they're on. Because, if the rotational inertia for a given diameter is the same as it would be if the sphere was empty, then you know the point-masses lie along that diameter. You could theoretically find more information than that (I think) but I don't really know how to concisely generalize it. I think it would be possible to find the point masses. (Could I find them? Maybe not but I certainly think it's possible that it's possible to find them) Maybe a more clever person on this forum can think of a way to prove or disprove the possibility of finding them with this method? (Maybe it's possible to find them if they lie on a diameter, but not if they don't? I'm just throwing out ideas) Edit: It very well may be impossible though (with this method at least) The fact that you don't know the mass of each particle (only their combined mass) might make this method impossible.
I personally don't know the answer, or if there is an answer. If the centre of mass was exactly in the centre then I suppose if you spun the sphere you would expect it to rotate on an axis through the two mass points. So spinning the sphere should give some information. We are looking for 8 pieces of information in total the coordinates (x1,y1,z1) and (x2,y2,z2) of the two mass points and their masses m1 and m2. So far we have about 6 pieces: The total mass M, the centre of mass, (X,Y,Z) and a spin axis through the centre of mass given by two angles (a1,a2). So we just need to get the two more pieces of information? But is this possible? (My personal interest in this is I am thinking about novel ways of doing body scans). One idea I thought would be to try and use a torque balance to probe the gravitational force of the pellets but this might be too weak practically. At first I thought that gravitational forces only act towards the centre of gravity of a spherical body, but this is only if the sphere is made of shells of constant density. So perhaps there is a way by accelerating the sphere to determine it's inner structure to some degree.
The spin axis being one about which the sphere's moment of inertia is zero, of course. One can recover a seventh piece of information quite easily. Attempt to spin the sphere about an axis that runs through the center of mass at right angles to the spin axis. Measure its moment of inertia about this axis. Unfortunately, this does not distinguish, for instance, between two masses of 1kg and 9kg for instance at 9 meters and 1 meter respectively from the CM (moment of inertia 90 kg m^{2}) and two masses of 5 kg and 5 kg each at about 9.5 meters from the CM (moment of inertia 90 kg m^{2}). Nor can one recover additional information by measuring the moment of inertia about other axes. The parallel axis theorem already tells you the results what those measurements would yield. Theoretically, this would work, yes.
In my mind I had a machine which you strap in a person and it tilts them in all different directions and measures forces on it to try and determine the inner structure. To be honest I'd rather be shook around a bit than be bombarded with X-Rays. An ultrasound is basically shaking (or causing vibrations to ripple though) an object. Or maybe like some Archimedes device which he submerges an object in water to determine it's density. But something more advanced which can determine the density distribution inside an object. If you could somehow weigh sections of an object you could get the density per slice. Then do that from a lot of different directions you should be able to work out the density distribution. Trouble is how can you weigh a slice of an object? (Without actually slicing it?)
This thread has some of the ideas that motor tyre balancing rigs use. That could be a useful area to search on.
MRI doesn't use X-Rays. Shaking and tilting the body during a scan, will change the body's configuration, as soft tissue and bones move relative to each other. So what configuration is the scan supposed to show?
IDK It's just an idea. I think if you had very very sensitive set of scales held over the body at different points you could measure the gravitational field around the body and perhaps determine it's inner structure. They would have do be very sensitive scales though, since the gravitational force between two people standing next to each other for example is about the weight of a grain of sand. So they would have to be able to be about 100 times more sensitive than that. So I think what I'll have to invent is a very sensitive gravitational field sensor! Then incorporate that into an iPhone to make a portable body scanner. I imagine it as a long pole with the sensor at one end which you would scan up and down someone's body to image their innards.
You are right that with a sufficiently sensitive gravitational field sensor you would be able to locate the two point masses within the larger body. - the gravitational field of the two point masses is indeed different from the gravitational field of a single mass located at the center of gravity. However, the sensitivity required is many many orders of magnitude beyond what can be achieved with any currently imaginable technology. Detecting the impact of a mosquito landing on a supertanker would be an easier problem.
I wonder if there's some clever way of doing it though. They said they'd never be able to keep time at sea. But then John Harrison invented the chronometer. It seems like a similar sort of problem. Invent a very accurate device that is impervious to movement. I was thinking of a kind of spring scales which are oscillating. And then you electronically take the average of the oscillations to get the correct result. Instead of trying to eliminate the movement like you do on analogue scales. This is similar to how the torsion balance works. You wouldn't need to get an absolute reading. You only care about relative readings. Basically with a gravitational field reader you are comparing the gravitational field (weight) with the electric field (spring tension). But you could use other forces, such as a magnetic field. You could suspend two magnets and measure the distance between them with a laser. I think there are lots of untried methods.
Yes, other forces instead of gravity. Gravity is a bad choice for this purpose: - Gravity is very weak on this scales - The mass densities of body tissue are very similar - Gravity cannot be shielded to prevent outside influences
No, you can't really figure it out. You get stuck when you narrow it down to a 1D problem (you can find this axis by lots of spinning and help from the Intermediate Axis Theorem). You can then measure the total mass and perpendicular moment of inertia. This leaves 2 equations and 4 unknowns (mass of each and distance to center of mass). Tilting the axis by angle θ scales m.o.i. by cos^2(θ) and you use the Parallel Axis Theorem for a shifted axis. At this point, you are stuck. The best you can do is narrow down the range of possible values. You know there is no negative mass and that the points are between the center of mass and the surface of the sphere. This can give you a solution if the point masses both happen to be on the surface. P.S. Measuring gravity would work if you can get a non-zero measurement. Assuming measuring distance is more than a millimeter and all masses are less than 10 kg, you'd need to measure on the order of 10^-4 m/s^2 or smaller, which is the same order of magnitude as the biggest anomalies in Earth's surface gravity, and also gravity from the Moon felt on Earth.
But magnetism (MMR) and radiation (CAT) have already been used. My method would be to suspend two magnets over each other in a vacuum and measure the distance between them with a laser with sub-micron precision like this one. The distance between the magnets will change depending on the strength of the gravitational field. The magnets might wobble about but you could take the average values using a computer. I think you might also have to shield the magnets from Earth's magnetic field. I'm not sure how you'd move this device around in 3D space but keeping it away from other massive objects. Perhaps you'd swing it on a long string.
https://www.physik.hu-berlin.de/qom/pdfs/Schmidt2011.pdf I think we would have the technology. This research team measured 10^-10 times Earth's surface gravity.
I'm changing my mind again. I think even if you could detect gravitational fields. You couldn't distinguish between two different density arrangements if they both had the same total mass and were both spherically symmetric. I think. It would work for the first case I put but maybe not for a general case. Unless I've missed something. So imagine a spherical organism, it would be impossible to detect the difference even between it having an inner-skeleton or an exoskeleton!
Thinking about this from a more mathematical point of view, I think it is more of just rotating the object with different initial starting conditions. By mapping out the movement and how it reacts to the different initial conditions, you should be able to get enough independent solutions to figure out the placements and weights. It is just a matter of how much information you want to find out about, and how many independent tests you can come up with.
Ohhhh!!! The solution is so simple, right in front of us. Just rotate the mass about an axis with centre dist. r1 and r2 for the masses m1 and m2. I do this along one axis to get x co-ordinates. Then move the axis by dist. d, now the masses are at dist. r1+d and r2-d, and rotate again. We can do this as many times with different d to get a new equation. See the picture for equations: Tell me if there's a catch somewhere or if i did something wrong.
The set of equations you get are not independent. They convey no new information. If the mass and center of mass are already known, the parallel axis theorem tells how each new measurement must relate to the previous measurements.
JBriggs, parallel axis theorem works only if we start at the centre of mass of the system and even then using it would be useless as we will get no new information with respect to changed distances r1 and r2 to some other since the change in moment of inertia will be simply m*d^2. And to use parallel axis theorem from anywhere other then the centre of mass will require the complete expression for I which we dont have. But when we shift the axis like i did, i believe we get new information, as the equations prove.