At places like kid museums, you may have seen large coin funnels where you drop a coin into it and spirals around the funnel until it finally drops through a hole on the bottom. They look like this. The funnel is stationary and the coin spins round and round.
But what would happen if the funnel itself was spinning rather than stationary and you dropped marbles into it? Would the marbles be able to find stable orbits around the hole in the center? Since the funnel itself is spinning, it would provide some outward force on the marbles to counteract gravity. If the funnel was spinning really fast, the marbles would be thrown off. If it was spinning very slowly, the marbles would spiral down the hole. But it seems like if the funnel was spinning just right, a marble couuld find a sweet spot where the outward centrifugal force matches gravity and would end up in a stable orbit around the hole.
I tried looking for such an experiment, but I couldn’t find anything. There’s plenty of videos of marbles spiraling around funnels, but the funnels are always stationary.
That would probably work. The coins fall into the center – as do marbles used in some museum versions of this – because friction quickly eats up the energy the coin or marble starts out with. But if the funnel itself is rotating, it’s essentially providing the energy needed to keep the coin or marble “up”, so that even in the presence of friction you’ll get a stable situation.
I’ve often thought of something more ambitious – construct a “double funnel” model that duplicates the gravitational potential of a two-body system, with one body being significantly larger than the other. Now rotate THAT model, and see if a marble placed at the L4 or L5 Lagrange points maintain equilibrium. Even cooler, disturb them slightly from that equilibrium and watch them execute their kidney-shaped orbits around the Lagrange points. (You could also demonstrate how the other Lagrange points are un stable.)
As an alternative, consider those circus acts where one or more motorcyclists are going in circles on the interior of a big mesh ball (cage). What would happen if the motorcyclist sat still and the cage spun under him? He would not upside down hang in mid-air.
The motion of marbles, pennies, or motorbikes depend on their movement. The coin spirals (and would work the same with marbles or anything that moved smoothly) rely on the motion of the object - it gains momentum then wants to go in a straight line. the surface of the spiral chamber forces it to curve, and that is the apparent centrifugal force that keeps the coin spiralling.
Of course, the small amount of friction could begin to drag the marble along until it is spinning with the funnel container. The closest real-world analogy would be a roulette ball; by hitting the spinning wheel surface and being impacted by the irregular surface, it gains momentum that causes it to spin with the wheel, until it has more momentum that the wheel (less friction between rolling metal ball and surface than between wheel and table.) At that point it continues rolling while the table under it slows down.
You may be able to find a stable point with marbles or coins if you could impart just the right forces for a little while. It would be non-trivial though and when the marble’s orbit would actually decay faster than in the typical example.
Centrifugal force is a “fictitious force” which always proportional to the mass of the object on which it acts. In other words a feather and an elephant will see the same “acceleration”. The issue is that under General Relativity the force we typically call “Gravity” is also a “fictitious force”. Orbiting bodies under GR are traveling in a straight path with zero acceleration, but spacetime is curved in just the right way that results in an orbit.
So the important thing to remember is that if you built this contraption it wouldn’t be an actual model of gravity but just another visualization tool that approximates the behavior.
The math to figure it out involves inertia, rotational energy, angular momentum, gyroscopic motion, and rotational stability so it is a bit complicated to figure out exactly. But getting it to slip in just the right way would be the trick and may guess is that precession and nutation would make it unstable.
The constant torque and angular momentum vectors in the marbles frame precess about the angular velocity vector. As a rolling marble will act like a rolling gyro and if the speed was just right with some slip it would orbit but in the reference frame of the spinning funnel the marble would still slow down leading to nutations to cause a stall or the rotation of the marble slowing down and dropping in he hole.
Remember that the coins stay up because they are gaining forward speed as they drop due to the conservation of momentum (remember to include angular). An orbiting body need to have a faster sidereel speed to have a closer orbit. So when that marble slows down the orbit is going to degrade.
Consider that Wall of Death motorcycles have a throttle lock or you have to manually close the throttle. You don’t really gain anything from spinning the funnel as the relative speed will still drop due to air resistance and rolling friction so a motorized unicycle would probably be easier to accomplish your goal. While the rotating funnel would have a centrifugal/centripetal component that force is going to work to maintain the distance from the origin and not forward momentum.
IANAGambler, but it’s my recollection that the ball in a roulette wheel is directed opposite to the direction that the wheel itself spins. Also most of the time the ball is revolving it is not actually touching a spinning portion of the wheel but is above that. Friction slows the ball then it falls into the spinning portion, but at that time it’s motion is mostly chaotic.
When you roll a marble (or a coin) into a stationary funnel you will impart an initial velocity which can be broken down into a vector towards the center of the funnel and one tangential to the circumference of the funnel at the point where the marble first hits the funnel. The path of the marble through the funnel will be determined by this initial velocity.
If the funnel is spinning, the tangential part of the initial velocity is increased or decreased by the rotation of the funnel, but once the marble is rolling the path should be the same as if it were launched into as stationary funnel with an increased (or decreased) initial tangential velocity. If you launch the marble opposite the direction of spin with a tangential velocity exactly the same as the rotational velocity, the marble should fall straight down the hole.
Some variance will occur from different wind resistance. Also, friction between the marble and the funnel will be different initially, at least until the marble transitions from skidding to rolling.
If you mounted the funnel slightly eccentric, you could use the motion to pump the marble. You wouldn’t have to depend on rolling friction to drive the marble.
The problem, as I see it, with using rolling friction (imagine a very slightly rough surface), is that the marble would spin/roll backwards in it’s stable (forwards) orbit. If it dropped down slightly, it is spinning too fast for it’s forward motion, but instead of the spin kicking it faster (and up), it kicks it slower (and down).
(Intuitive science, not engineering: I haven’t done an analysis).
Cal, I strongly suspect that your Lagrange funnel wouldn’t work. Stability of the Trojan points is actually a fairly subtle effect, and the funnel is only an imperfect analogue for gravity. But I’d nonetheless love to see someone try.
I have posted this video before, but seem to always credit MIT and not University of Toronto. Despite being from 1960 it is the best example of frames of reference impact this.
I tried to link to the relevant time period of 17:04 to 18:10, but it will clarify a very common misunderstanding on the nature of forces in a spinning frame.
Correct. The ball runs around the track because the dealer started it spinning, not because it’s spun up by the wheel. The track is slightly angled such that when the bacll drops below its ‘capture velocity’ it will fall into the rotating part of the wheel, which as you say rotates in the opposite direction. There it bangs into ‘randomizers’, or little metal plates that stick up in various places on the spinning surface to make sure the ball doesn’t move predictably.
That said, they aren’t perfect randomizers, and if you know the exact position and velocity of the ball in the track and the velocity and position of the spinning wheel, you can beat roulette by a pretty good margin. This is why computers are illegal at the roulette table.
As for the marble dropped into a spinning funnel, the most important factor is friction. The ball is going to want to go straight down the hole, but will be slowly accelerated by the spinning wheel under it. If the spin is too low or there isn’t enough friction, the ball will just follow an arc down into the hole. If the table is spinning fast enough, the ball would actually trace a hyperbolic trajectory and be thrown completely clear of the rotating cylinder. If you get the speed just right you could get the ball to stay, but it would be tricky.
Best bet - set up a stepper motor and a camera, so that if the ball moves towards the hole the funnel speeds up, and if it moves away the funnel slows down. You should be able to set up a nice feedback circuit that keeps the ball just about where you want it in the funnel. That would be a fun project to try.
Basically these funnels kind-of work for circular orbits but don’t obey Kepler’s third law, which should be the inverse law or:
T[sup]2[/sup]∝R[sup]3[/sup]
On the funnel is:
T[sup]2[/sup]∝R
Basically no cylindrical symmetric surface at this scale exists that would be capable of reproducing the orbits of an object subjected to uniform gravity.
While circular orbits can get close they are unstable if you have a “closed” orbit and are only stable for these devices because none of the “orbits” are closed. Looking at other papers spandex or other elastic surfaces and mimic an orbit more accurately but a ridged surface would have to be shaped in a way where the situation really resembles spinning a bucket of water over your head than an orbit.
Which part? About the paper, the link or my math conclusions? I do agree it didn’t say “Ring-ding-ding-ding-dingeringeding! Gering-ding-ding-ding-dingeringeding!Gering-ding-ding-ding-dingeringeding!”
I provided the cite that will let you find other papers, and the math is there but that paper I provided did error on the side of finding pedagogical value which is appropriate for their main use case.
And here is a random other paper who agrees with my math, which was based on the above cite, on the Third law, look under this text to find that.
If you look on page 9 at Figure 1, note how the “κ = −1 surface” at 1 is approaching a cylinder at r =1, exactly as I found based on the original paper.
“The square of the period of a planet in a circular orbit is proportional to the cube of the orbital radius”
But I think you will find that this part of the original paper:
Remember those papers are considering if it is close enough to use as a teaching tool, and it is, I was trying to use those same assumptions to see if one could figure out a way to maintain stable orbits per the OP. Any torque which would cause the “marble” to climb is problematic in the case of this thread but not in the case of a physics demo.
As my math related to Kepler’s 3rd law, related to T and the previous paper didn’t solve for that solution obviously the paper and my math wasn’t written in the text but if you look at the math in section “III. A SURFACE PROPORTIONAL TO r[sup]-1[/sup]” it is trivial to come to the same conclusions using high school level mathematics.
As the period is just T = 2πR/v it is all there. As I stated above the effects of climb and added torque are the easiest place to see the issue.
