Modeling The Device

First created: Sat Jun 2 14:22:33 EDT 2001
Last update: Sat Jun 2 14:22:33 EDT 2001

The Equilibrium Position

The equilibrium position of the 2 pendulums is achieved when they are both aligned vertically with the local gravity. In this configuration the pendulums will oscillate around a stable equilibrium under small perturbations.

The force that restores the pendulums to the equilibrium position comes from the coupling magnets. The magnet dipoles on the two strips attract each other.

The potential energy (V) of the float has contributions from 3 sources: the top pendulum, the bottom pendulum and the magnets.
    V(a,b,r) = mgh(1 - cos(a)) + mgh*cos(b) - k/r
(1)
where
    m  measures the mass of a pendulum
    g  is the acceleration from gravity
    h  is the length of a pendulum
    a  is the angular displacement of the top pendulum, in radians
    b  is the angular displacement of the bottom pendulum, in radians
    k  is the magnetic strength of the magnets
    r  the distance between the magnets
Whether the magnetic potential varies as 1/r (monopole) or 1/r2 (dipole), or something in between, is not critical for the arguments that follow.

The potential energy has a minimum at
    a = b = 0
    r = r0
where r0 is the minimum distance between the magnets.

Under the small change
    a -> a + da,
a new equilibrium will be reached at
    r = r0 + dr
    a = b = da
where both dr and da are small and
    2h*cos(da) + r + dr = 2h + r
    dr = 2h(1 - cos(da)) = h(da)2
We can use this to constrain dr and db in terms of da and express the potential as an effective potential in da alone
    V(da) = V0 + k/r02*dr
    V(da) = h*k/r02*da2
After dropping the constant V0. Since a = 0 + da,
    V(a) = h*k/r02*a2
(2)
Around the equilibrium the system behaves like an harmonic oscillator, as most systems do.

The Oscillation Period

The period of oscillation for small perturbations around the equilibrium can be read from (2). The kinetic energy for the system is
    E(d/dt a) = m/2*(h*(d/dt a))2
The equation of motion is
    md2/dt2 (h2*a) = -2h*k/r02*a
A solution is
    a(t) = A*sin(2*pi*t/T)
Where the period T is given by
    mh2(2*pi/T)2 = 2kh/r02
    T = 2*pi*r0*sqrt(h*m/k)
(3)
The period is easy to observe and measure. A period of 3 seconds is not difficult to achieve.

Sensitivity To Changes In Gravity

If we introduce a small horizontal shift in gravity
    g = g + dg
with g vertical and dg horizontal, the restoring force from the top pendulum is cancelled by the destabilizing force of the bottom pendulum and a dx will develop. The force that will restore equilibrium comes from the dr that develops as the coupling magnets separate. Equating the forces
    h*m*dg = k/r02*dr
    dr = h*m*r02/k*dg
The distinguishing feature of pendulums is the square root relation between the horizontal (dx) and vertical displacements (dr).
    dx = h*da
       = sqrt(h*dr)
       = h*r0*sqrt(m/k*dg)
In terms of the period,
    dx = T/(2*pi)*sqrt(h*dg)
(4)
And here is why this device in interesting: dx is proportional to the square root of dg. This greatly increases the instrument's sensitivity. Specifically, with a T of 3 sec and an h of 25 cm, the response to a dg of 10-7g will be a displacement dx = h*da
    dx = 3/(2*pi)sqrt(0.25*9.8*10-7)
       = 4.8*10-4  meters
Between 1 and 0.1 millimeter. This should be easily measurable.

Sensitivity To Changes In Temperature

A higher temperature will cause the coupling magnets to weaken and the device to expand. However the frame expands at the same rate so that effect is muted. So there will be a lower k and maybe a smaller r0. This will amplify an existing displacement, but it should not cause a displacement to appear when none existed.

Sensitivity To Changes In Air Pressure

Buoyancy shifts may vary the weight of the components (mg) and thus the amplitude of a displacement but they should not create a displacement when none existed.

Conclusion

Prospects for measuring horizontal components of tidal forces with this apparatus appear to be good.