What is the effective spring rate of a magnetic spring Consider a magnetic spring as seen on this YouTube video, but ignore gravity. If I wanted to calculate the effective spring rate (Force vs. Deflection) curve for the top magnet, how would I go by doing that?
Consider $N$ permanent magnets made of Neodymium with known geometry (length, diameter etc). Space them equally such as the total span is $L=N\;\Delta x$ such that they can slide along a rod (1 DOF each). Finally apply a unit force  $F$ on one end, while holding the other end fixed.
 A: Lets suppose that all N magnets are identical. Assume that they are quite far from each other so we can replace them with N magnetic dipoles with dipole moment $\overrightarrow{m}$. Also assume that the dipoles are placed along the x axis  in such a way that they repel each other and all N moments   $\overrightarrow{m}$ are parallel. The first step is to find the interaction force between two dipoles. Fortunately, Wikipedia has a formula for the force so that we can save a large amount of work.   Link here 
I write only its x axis component which we need, to save space:  
$$F=\frac{6\mu_0 m^2}{4\pi x^4}$$  
Now, to find "spring rate" between two magnets, it is sufficient to find the differential $dF$:
$$dF=-\frac{6\mu_0 m^2}{\pi x^5}dx$$ 
So for the small displacement the "spring rate": 
$$k(x)= \frac{6\mu_0 m^2}{\pi x^5} $$
Now, since we know $k$ it is easy to find  the effective spring rate for the top magnet. 
Let $F$ be the force on the top magnet, other end magnet fixed. Then the displacement of the top
magnet is $\Delta x=(N-1)\frac{F}{k}$   
So the effective spring rate: 
$$k_e(x)= \frac{6\mu_0 m^2}{(N-1)\pi x^5} $$
