How would one analytically calculate the force on a ferromagnetic object, like an iron sphere, from a permanent magnet?
Interestingly, classical physics can't explain the onset of the the magnetic moment induced in the ferromagnetic object by the permanent magnet. The origin of that moment is quantum-mechanical. But you can describe the resulting force classically.
The formula for induced magnetic field is
B = mu0 (H + M),
where B is the flux density created in the responding material, H is the magnetizing field, and M is the magnetization. mu0 is the magnetic permeability of vacuum. B, H, and M are vectors. In diamagnetic and paramagnetic materials there is a simple linear relationship between H and M:
M = x H,
where x (also denoted by Greek letter chi) is the volume magnetic susceptibility. See the Wikipedia table of x/chi for some common materials:
Unfortunately, for ferromagnetic materials, the relationship between M and H is not simple--it depends on magnetic hysteresis. See the hysteresis loop pictured here:
But, since H is given by the permanent magnet in your example (though it depends on distance), if you know M, you can also figure out B in the responding material and then calculate the magnetic force between the permanent magnet and the ferromagnetic material. The induced field is parallel to the applied field from the permanent magnet, so you don't have to do a torque calculation. If the ferromagnetic material has a small hysteresis loop, you may be able to use the approximation M = x H.