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BigBad

Thu Dec 03 2015, 01:10AM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
Location:
Posts: 600

Slava wrote ...

It will spin forever at the same amplitude... the only way for it to drop to a lower orbit is if it loses energy.

No, that's not correct, it also depends on the eccentricity. The eccentricity can change without any variation in energy at all. It's a completely unstable system.

Dr. Slack

Thu Dec 03 2015, 07:38AM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

In physics, the conservation laws are your friends.

For this example, conservation of energy tells you all you need to know.

The energy of the ball consists of potential energy against gravity, and kinetic energy.

If you have friction, then your total system energy will fall monotonically. At some point, the altitude of the ball will have fallen below the rim of the hole.

If you have a theoretical system that does not lose energy, a Durium ball on an Undraggium funnel in a vacuum, then you have to deduce from the shape of the funnel, or observing a few orbits, whether any eccentricity is stable, to see whether its perigree will be below the hole rim.

If you have a single coin in a cylindrically symmetric funnel, then there is no mechanism to change the degree of eccentricity, if it survives one orbit, it will survive indefinitely.

If the funnel does not have cylindrical symmetry, a few magnets hidden behind one side, or a shape variation, then that will change the local shape of the position to potential energy curve, but I struggle to believe that could cause an increase in eccentricity either.

If you have multiple coins with some sort of force between them, then, like the planets, they could change orbits and exchange energy in difficult to predict chaotic ways, and send one of their cohort down the hole.

Signification

Thu Dec 03 2015, 11:12AM

Registered Member #54278 Joined: Sat Jan 17 2015, 04:42AM
Location: Amite, La.
Posts: 367

I like quality physics college textbooks, esp E&M. Slava's posted video on the funnel and spiraling balls, gravity model, reminded me of this...I have one book here written in the 50's I think that has a very interesting claim--that is, if a planet moves, earth, for example, then the g-force on the moon is INSTANTANEOUS!!! (no light speed delay!) Is this claimed due to ignorance of "gravity waves"?
ALSO: I recall these 'funnel' models being more "elastic". I think that was important.

PS:
The book "PHYSICAL OPTICS" by Jenkins and White led me far. Another thing--it was written in the 30's. Well before the laser of (1960) or hologram (1948), but you can see here, thinking back as you read appropriate sections, that these concepts were well on their way to reality!

Uspring

Thu Dec 03 2015, 11:53AM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

Slava wrote:

argument #1) if there was no energy loss from friction and sound, it would orbit forever (like the orbit of planets around the sun)

No, for some funnel shapes there are spiralling trajectories which will come arbitrarily close to the center, i.e. will eventually drop through the inner rim. This is also true for cylindrically symmetric funnels.

argument #2) gravity will eventually pull the coin/ball into the center regardless of energy loss

No, for some funnel shapes there will stable orbits if there is no friction.

BigBad

Thu Dec 03 2015, 06:09PM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
Location:
Posts: 600

Dr. Slack wrote ...

If the funnel does not have cylindrical symmetry, a few magnets hidden behind one side, or a shape variation, then that will change the local shape of the position to potential energy curve, but I struggle to believe that could cause an increase in eccentricity either.

It can cause increases or decreases in eccentricity. Note that eccentricity and energy are independent variables of orbits.

Consider an inverse square law with energy conservation e.g. gravity

An orbit around a spherical shape is (to an excellent approximation) obeying conservation of energy.

But as a practical observation low orbits around the moon are very unstable! How come?

The reason is that the moon's gravity is 'lumpy'

The energy is constant; going past a mass concentration, temporarily raises the speed, but the body loses it again, with virtually no energy loss at all; but it can change the direction of the orbit; going one side or other of the mass concentration rotates the velocity vector. This changes the eccentricity and orbital plane, and after being bumped around by these mass concentrations enough, the orbit now intersects the surface.

The same thing can happen with a funnel. If a coin goes past (say) a magnet, just above the magnet, it will rotate the velocity vector down. If the magnet is very rigidly fixed and the magnetic losses, eddy current etc. are negligible, then the kinetic energy and potential energy are, overall, unaffected, but the eccentricity changes. The same thing can happen if there's a small depression or hill on the surface; if it's not perfectly axially symmetric.

If the shape is infinitely big, and/or there's no hole in the centre and the energy is less than that required to reach the rim, then it will carry on moving, but it will usually move chaotically forever, provided energy is conserved.

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