GotMex? said:
Anyone have any experience with this example and can tell me why it wouldn't work?
No experience with this example, but I
can tell you why it will not work (only to subsequently find someone else come up with a much simpler and more elegant explanation).
It will not work because you can not build the system without friction. In the absence of friction, it (or some form of it) would work just fine, and as pointed out by Jeff, wouldn't even need the magnet.
However, the magnet does not "offset" the losses due to friction.
Quick reason: the magnetic field is conservative, unlike friction.
Longer explanation:
Magnetic and gravitational fields are conservative, i.e, the work done by these fields is independent of the path. As a result of this, the work done by these fields on a particle, over the course of a loop is zero. (See this by writing the path integral from A to A as the sum of two path integrals from A to B and again from B to A; these integrals are equal in magnitude and opposite in sign, irrespective of the actual paths from A to B and back, and cancel each other off). So, after each loop, while there's a loss of energy from dissipative forces, there is no gain of energy from the magnetic field.
Still longer, and more rigorous proof:
1. Pick any point (call it A) where the velocity of the ball is v(A;n) during the n'th looping of the loop.
2. Watch the ball do a loop (assuming it can; if the ball can not do a loop there's nothing left to prove) and come back to A. During the course of this loop, the total work done by the conservative forces (gravitational + magnetic) is zero. The work done by the non-conservative forces (friction, air resistance), must hence equal the loos of KE of the ball (ref: Work-Energy Theorem). The ball thus has a smaller velocity when next it arrives at A. i.e, [itex]v(A;n+1) \leq v(A;n)[/itex]
3. One could still argue that as the ball gets slower with each successive loop, the loss of KE also gets smaller, and the velocity asymptotically approaches a terminal value. However, since air resistance is proportional to velocity, a non-zero velocity will imply a non-zero drag and hence a non-zero loss to the KE. So, the only possible asymptotic value, if one exists, is zero. (For the specific case of the ball on the ramp/hoop, the loss of speed is faster than asymptotic. One can show that there's a non-zero lower bound on the work done by rolling friction, and hence, a finite upper limit on the number of loops before the KE reaches zero at A).
4. Now we make use of a specific flaw in this system. Since the magnetic field has a spatial variation and the gravitation field does not (or it has a much smaller spatial variation), we can safely conclude that that the net field is not zero everywhere on the loop. This in turn implies that the net potential energy is not a constant during the motion, and specifically, there must exist a pair of spots where the net PE are respectively a maximum and a minimum for the loop.
5. If the PE at A is smaller than the maximum PE by some P' > 0 (which we can ensure by our choice of A), we need a minimum KE at A given by: KE(A,min) = PE(max) - P', which then imposes a minimum speed required at point A, v(A,req). Now since v(A) at best approaches zero asymptotically (with the number of loops completed, n), we can always find an n=N when v(A;N) is smaller than any chosen number. Specifically, there exists some N, where v(A;N) < v(A,req). At this N, the ball has insufficient KE to reach the point of maximum PE and hence fails to complete the loop.
Since N is finite the motion is not perpetual.
