Terry Blanton wrote:
No. The rotational kinetic energy was provided as a result of the ball "falling" to a point of lower potential in the (conservative) B-field. The angular momentum was stolen from the Earth, however.From: "Stephen A. Lawrence" <[EMAIL PROTECTED]>
In other words, the Earth itself provides an essentially infinite source/sink for L, which is one reason why it's not always apparent that L is really conserved in real-world situations.
Hmmm. But, it is the magnetic field which imparts the balls momentum and the ball exits the field retaining that momentum.
The point you are striving for here is that the ball rolled up the ramp, gained kinetic energy, fell off the ramp, and retained the kinetic energy. There are several things wrong with this picture.
First, try to plot the direcition of the force acting on the ball due to the magnet throughout the aparatus. I'm no applied E&M whizz, but we don't need precision here: all we need is the fact that the field we're going to plot must be curl-free: we know that metal objects in a static field don't simply start spinning, faster and ever faster. So right away we know there are a lot of field patterns that we CAN'T produce. Here's one (note that the "field" plotted is the force field showing how the B-field would act on a steel ball -- it's _NOT_ the B-field itself):
http://physicsinsights.net/images/B-field-with-curl.png
A closed loop that passes to the left up high, straight down, and back to the right down low would gain energy: it's not curl-free.
Here's a more plausible possible field (again, showing the force field as it would act on a steel ball). This field is radial (though I have only drawn a few of the arrows -- you must picture the rest of them):
http://physicsinsights.net/images/Plausible-field.png
Now the point to notice about the "plausible" field is that, when the ball falls to ground level directly beneath the magnet, it is at a location where the magnetic field is still stronger than it was at the starting point. Furthermore, since the field is radial, for the ball to roll from "Ball Finish" to "Ball Start" at ground level, it will need to roll _against_ the force of the B field which is, of course, not actually zero at "ground level". Consequence: The net energy given back to the B field in the second half of a closed path will equal the energy gained from the B field on the first half of the path.
I'm not claiming that the fields used by Greg matched the ones in my "plausible" diagram. I'm merely observing that (a) it's hard to apply intuition in a case like this without making serious errors, unless you really try to work through all the book-keeping, and (b) you need to take account of the actual value of the B field along the _entire_ path the ball must follow to get back to its starting point. Longer return path => weaker field and it's very easy to mentally substitute "zero field" for "weaker field", which is not correct.
Finally, consider what happens in the "Plausible-field" arrangement if we take away the ramp. Put the ball at "Ball Start". The field points diagonally upward ... and the ball will roll to the left, gaining energy until it arrives at "Ball Finish". It doesn't matter which path you choose through the field; the ball finish point is at a lower potential location in the field, and the ball has more kinetic energy there than it does when it's at "ball start".
(A real layout which actually caused the ball to roll up the ramp and fall down through the hole, rather than just leaping up to the magnet at that point, would require putting the magnet farther away than I have shown it, and would probably require a shallower angle on the ramp.)
