Originally posted by sonhouse This seems to boil down to exactly what level of inverse squareness is involved. Maybe it's an inverse cube law. Not sure.
How about doing it just assuming various inverse laws, square, square and a half, cube and so forth, just seeing what the velocity would be under each assumption.
Originally posted by joe shmo Well, I don't think that is a very straightforward scenario, but lets start with some definitions.
The Electromagnet will be body "A"
The Flat Plate will be body "B"
The distance between the bodys "A" and "B" will be "L"
The electromagnetic pushing force will simply be "F(L)"
Examining body "A" (the electromagnetic hoverboard) using Newtons Second ...[text shortened]... someone can take it from here or restate the problem in terms of energy somehow to simplify it.
The main problem with this is that the electromagnet would attract the plate and not repel it. You could use two electromagnets but they would be unstable against rotating so they attracted.
Your maths is correct, except that the differential equation is a linear one. The quantity m1m2/(m1 + m2) is known as the reduced mass. Suppose m1 were very large, then the reduced mass is very close to m2 and only the smaller object moves noticeably. If the two masses are equal then the reduced mass is m/2, with m=m1=m2.
The problem with electromagnets is that they have the field one associates with a dipole, so it goes as 1/r^3 and comes with a turning moment when applied to another dipole. So with two electromagnets you'd need powerful gyros to stop the magnets flipping over under small perturbations, when they'd start attracting each other. So the problem with the proposal as a practical engine is stopping the thing going into reverse! A solar sail or a rocket seem more manageable.
Originally posted by DeepThought The main problem with this is that the electromagnet would attract the plate and not repel it. You could use two electromagnets but they would be unstable against rotating so they attracted.
Your maths is correct, except that the differential equation is a linear one. The quantity m1m2/(m1 + m2) is known as the reduced mass. Suppose m1 were very la ...[text shortened]... engine is stopping the thing going into reverse! A solar sail or a rocket seem more manageable.
We are talking magnetic hoverboards, they are electromagnetic AND thrust away from the metal board it rides on. It won't work on a wooden board, non-conductor, it needs a conductor, maybe just iron or magnetic steel, not sure.
Originally posted by sonhouse This seems to boil down to exactly what level of inverse squareness is involved. Maybe it's an inverse cube law. Not sure.
I checked in Wikipedia as I wasn't entirely confident in my statement of an inverse cubed law. Wikipedia gives an inverse square law for the force between poles, in other words the force between poles as if they were magnetic monopoles. There are four of these in the problem, two pairs for each magnet. We can get the field strength at a distance r from a magnet by doing the calculation for a pair of monopoles whose separation l is much smaller than r, the field around a monopole goes as 1/r^2, so the field due to two monopoles of opposite charge goes as 1/(r - l)^2 - 1/r^2 and when one expands the first term to leading order we have 1/r^2 * (1 + 2l/r) so the difference is 2l/r^3. So I'd expect the force between two magnets where r >> l to go as 1/r^3.
Originally posted by DeepThought I checked in Wikipedia as I wasn't entirely confident in my statement of an inverse cubed law. Wikipedia gives an inverse square law for the force between poles, in other words the force between poles as if they were magnetic monopoles. There are four of these in the problem, two pairs for each magnet. We can get the field strength at a distanc ...[text shortened]... difference is 2l/r^3. So I'd expect the force between two magnets where r >> l to go as 1/r^3.
Is that enough information now to solve for velocity at X time or X distance separating the steel structure from the hoverboard system with say, mass of 100Kg? On Earth it would provide a thrust of 100Kg exactly to keep the system with rider hovering.
Originally posted by sonhouse Is that enough information now to solve for velocity at X time or X distance separating the steel structure from the hoverboard system with say, mass of 100Kg? On Earth it would provide a thrust of 100Kg exactly to keep the system with rider hovering.
No, my calculation is for the field around one magnet. Incidentally, my statement was based on the well known tendency of iron to be attracted to magnets. This is because it is magnetiseable, your metal plate needs to not have this property. The problem is that even with a superconductor it won't generate thrust, the reason maglev works is that the plate has currents induced to resist changes in the magnetic field. It's more like a magnetic trap than a drive, so you'll get acceleration to cancel out relative motion, the effect is more like drag in that sense.
Originally posted by DeepThought No, my calculation is for the field around one magnet. Incidentally, my statement was based on the well known tendency of iron to be attracted to magnets. This is because it is magnetiseable, your metal plate needs to not have this property. The problem is that even with a superconductor it won't generate thrust, the reason maglev works is that the plat ...[text shortened]... u'll get acceleration to cancel out relative motion, the effect is more like drag in that sense.
Not sure about this but it seems the board uses highish frequency coils and there might be magnetization and demag shortly after to leave the metal in the same state as it was without the fields.
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Originally posted by DeepThought The main problem with this is that the electromagnet would attract the plate and not repel it. You could use two electromagnets but they would be unstable against rotating so they attracted.
Your maths is correct, except that the differential equation is a linear one. The quantity m1m2/(m1 + m2) is known as the reduced mass. Suppose m1 were very la ...[text shortened]... engine is stopping the thing going into reverse! A solar sail or a rocket seem more manageable.
"The main problem with this is that the electromagnet would attract the plate and not repel it. You could use two electromagnets but they would be unstable against rotating so they attracted."
Yes, I certainly was unaware of that. What about a coil wrapped around a steel bar? It wouldn't levitate, but it would certainly provide the launch sonhouse was looking for. These equations could with (minor modification) properly apply to that situation, and it would be a nice uniform magnetic field providing the force.
"Your maths is correct, except that the differential equation is a linear one."
Are you saying this as a result of your derivation for the proportionality of the driving Force? F(l) ≈ ( 2 / r^3 )*l
Originally posted by joe shmo "The main problem with this is that the electromagnet would attract the plate and not repel it. You could use two electromagnets but they would be unstable against rotating so they attracted."
Yes, I certainly was unaware of that. What about a coil wrapped around a steel bar? It wouldn't levitate, but it would certainly provide the launch sonhouse was ...[text shortened]... a result of your derivation for the proportionality of the driving Force? F(l) ≈ ( 2 / r^3 )*l
Thinking about it you are right, it is non-linear, but not very strongly so. I tend to only think it's non-linear if one of the derivatives is included in the non-linearity.