The magnetic pendulum model
I'm reading about this thing called the magnetic pendulum experiment, and I got curious about the equations of motion governing it. Essentially, imagine a pendulum whose bob is made of a magnetic metal and placed above some magnets. See this video for the actual experiment. I've seen this equation for the bob's acceleration when looking it up:
$\begin{align} m\ddot{\mathbf{x}} = \sum_j \frac{\mathbf{r}_j}{(h^2 + \mathbf{r}_j \cdot \mathbf{r}_j)^{3/2}} - g\mathbf{x} - \mu\dot{\mathbf{x}} \end{align}$
In this equation:
* $\mathbf{x}$ is the position of the pendulum bob in the x-y coordinate plane. (The bob is assumed to be locked to a plane some fixed height above the plane the magnets lie on.)
* $\mathbf{r}_j$ is the distance between the bob's position and that of the $j$th magnet, i.e. $\mathbf{r}_j := \mathbf{x}_j - \mathbf{x}$, where $\mathbf{x}_j$ is the center of the $j$th magnet.
* $h$ is the height of the bob above the magnets.
* $g$ is the kickback force of the pendulum (think <PERSON>'s law).
* $\mu$ is friction/air resistance.
* $m$ is the bob's mass.
* We assume the magnets are of equal strength.
I'm trying to understand this equation and I understand most of it. From what I understand, the $-g\mathbf{x}$ term comes from <PERSON>'s law, and describes the force restoring the pendulum back to its state of rest. 782
The $\mu\dot{\mathbf{x}}$ term just damps the velocity. We put it all together using <PERSON>'s second law.
However, I don't understand why the denominator in the summation is raised to the power of $\frac{3}{2}$. This source says it's to eliminate the vertical component from what is obviously a distance formula in the equation, but I don't see why that is. From what I understand, magnetic force obeys <PERSON>'s law, which is essentially a type of inverse square law. I've read it has something to do with dipoles and an inverse-cube law. Is this true?
I asked this on Math Stack Exchange and didn't really get an answer, and someone suggested that I post it here. Please try to keep in mind that I don't know too much about electromagnetics. Thanks for your help. 483