Ferrite cores ain't cheap when you start talking about the 3 inch ID ones. I can see em for like 40 and 50 bucks....
I don't see why I can't just ground a slinky or something.
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A ferrite core used for radio is specially constructed to conduct magnetic fields while not conducting electrical current, especially eddy currents.
A slinky is (typically) an uninsulated conductive coil of variable spacing, and thus variable inductance. It is nothing like a ferrite core. Typically they are made of steel which his high resistance (for metal), although I have seen copper "slinkies" made specifically for radio use.
Ferrite cores are usually priced by size, which is proportional to the power they can handle. A torioidal ferrite with a 3 inch inner diameter is quite large and likely would be heavy and be able to handle a quite large amount of power. Perhaps you can make use of a much smaller one instead? If you need that size for the power, you should be worried about safety, not cost.
Some people have used slinkies as antennas. In this use, they can act like a broadside helical antenna that you can tune by adjusting the spread of the coil. However, as it is not very stable, the tuning will change randomly as it moves around, so this could be a bit of a problem. Also, steel is generally a poor choice for antenna material due to its magnitivity and resistance (although it will work but with lots of resistive loss).