## Modeling The DeviceFirst created: Sun Mar 26 2000Last update: Sun Mar 26 2000 | |||

## The Equilibrium PositionUnder 3 different configurations of the magnet stacks, the minimum of the black curve indicates the equilibrium position of the float. (Click image for full picture).
The two stack design is the one with the broadest, flatest, equilibrium for the float. This ensures that small changes in gravity will make the float oscillate with an unusually large amplitude. The potential energy (V) of the float has contributions from 3 sources: the bottom stack, the top stack and gravity. V = k/r - k/(h - r) + mgrwhere k is the magnetic strength of a stack h the distance between the stacks r the distance of the float above the bottom stack m the mass of the float g the acceleration from gravityWhether the magnetic potential varies as 1/r (monopole) or 1/r ^{2} (dipole),
or something in between,
is not critical for the arguments that follow.
Equilibrium points are reached when the potential energy is a minimum (stable) or maximum (unstable), d/dr V = 0 k/r Provided h is big enough, there are 2 equilibrium points,
a stable one at r ## The Oscillation PeriodThe oscillation period at the stable point is of great interest because oscillations are easily produced and their period is easily measured. The period T at the stable point is T = 2 * pi * sqrt(Q) (2)and it does not depend on g. The quantity Q m 1 Q = -- * ------------------- (3) 2k 1/ris an important characteristic of the probe. It has dimensions of a time squared. This result can be obtained by a formal expansion of the potential around
the stable position r V(r) = V(rC is some constant value. The term linear in dr vanishes at the stable point. For small dr, the cube an higher powers can be neglected and the potential is that of an harmonic oscillator. The expression for the period follows from there. To achieve a long period one must try, according to (3), to get the stable and unstable points close together, on each side of h/2. This is tricky because one also needs to limit the oscillation amplitude to stay at all times on the stable side of the unstable point. The longer the period, the smaller the amplitude allowed and the more care is needed in designing the stopping mechanism. A heavier float and weaker magnets will also help in extending the period. The longest period I have been able to achieve is between 1 and 1.5 seconds. ## Sensitivity To Changes In GravityUnder a small change g -> g + dg the equilibrium position
changes from r k/(rto first order 2k*(1/r If we can achieve a long period (2),
then we also achieve high sensitivity to changes in g (4).
Specifically, if the period is 1 sec, then in response to dg = 10 dr = 1/(2*pi)This is exceedingly small. ## Sensitivity To Changes In TemperatureThe temperature has a significant effect on the magnetic strength k. Under a change k -> k + dk, relation (1) implies (k + dk)/(rto first order dk*mg/k + 2k*(1/rdr and dg have a similar depency on dk, proprortional to Q. This is because (1) really only depends on the combination k/(mg). For the dg effect to dominate, we would need dg >> g*dk/k, but ... this is not the case. dg/g is about 10 ## Sensitivity To Changes In Air PressureChanges in atmosperic pressure will also appear as a small change in g.
Air is mostly N One way to minimize this effect is to seal the experiment in a pressure tight container. ## ConclusionsProspects for measuring Moon and Sun tidal forces with this device are grim.
The first challenge is to be sensitive enough to measure
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