Diagnose grinding sound from single rear cog I've got a '70s-era ten-speed road bike with (what I think is) a Huret Eco rear derailleur which shifts well, but for some reason a loud grinding sound emits from the rear of the bike when using the second-smallest cog and riding using this cog feels much less smooth than the others. However, the cog itself doesn't look worn or damaged, and the derailleur tracks quite well.

Of course, since this bike has down tube shifters, some grinding can occur in any gear. Nevertheless, it is very easy for me to trim the derailleur with any other cog in order to eliminate this grinding... except for the second-smallest one: No matter how much I try adjusting the derailleur's trim using the shifter, I can't eliminate this grinding on this one cog and only this one cog. Even more strangely, there is no significant difference in feeling/sound when running the chain on the small chainring as opposed to on the big one.

What is going on here? Is some part somehow worn/damaged or simply needs adjustment?
This could conceivably be a cross-chain issue. Hypothesis: when the chain is on the second cog, it extends toward the chain ring at such an angle that there is interference from the teeth of the larger, third cog.

This kind of thing can't be diagnosed remotely without detailed pictures at suitable angles showing the chain line, derailleur and cassette details and so forth.

Like user zipzit commented: get it in a stand and try to pinpoint the source of the noise. When you find it, slow down to a crawl and try to find the exact location of the metal-on-metal action. If it needs load to reproduce, have someone apply the brake to the wheel.