Yes that's correct. String tension keeps the tailpiece upright. It's 'free floating' meaning you can only adjust it up and down using that screw at the bottom. When you go to change strings, if you remove all the strings at once, the tailpiece will fall off the rim of the banjo.
Here's a really great video by John Boulding explaining that type of tailpiece: youtube.com/watch?v=LTkVeL795WI
There is supposed to be a nut on the bottom of the screw. Most often, it is the same as the nuts on the j-hooks.
I came Soooooooo close to losing that little nut off my tailpiece over in Germany on tour. I caught it just before I played in Rodolstadt.
Once, I was changing strings, the nut fell off, the 5th string tuner came loose, the bridge hit the floor and the tailpiece was swingin' in the wind.
Welcome to the Wonderful World of tinkering with banjos.
And to the HangOut.
The good news is:
No matter WHAT the problem, you now have over one hundred and twenty THOUSAND spare brains to help think up a solution.
Free advice from an Old Person:
Do NOT take off all the strings at the same time.
Also: Once you have the bridge set properly, trace around the feet with a pencil, so that IF it gets knocked awry, you can just put it back where it was.
I once had a banjo with a tailpiece attachment with a missing nut. The threading of the screw had enough friction against the endbolt to keep everything in place.
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