I'd like to point out to you that a Jacobs chuck mounted by Morse Taper in the tailstock can assuredly come loose......and sometimes create a dangerous situation for the lathe turner, as Ken Grunke can confirm. I've had this happen when I've failed to back off soon enough, and accumulated drill shavings in the flutes of the drill bit become binding......and dislodges the Jacobs Chuck from the Morse taper grip.
I purchased this hard rubber (?) #2 Morse Taper cleaner more than 20yrs ago, and at the moment, I don't remember just where I got it. If you can make a wooden one on your own, that's great, but I'd like to point out just how good this one is. it has little spiral blades molded into the tapered end, and they do a great job of dislodging and removing dust that has accumulated in your female Morse taper......sort of like a squeegee. They're cheap, too.....I can't recall the exact price, but I think it was under ten bucks.
Also, if you are not already doing this......get a regular ol' cork to put into your Morse taper when not in use. That will eliminate 90% of the dust that would otherwise end up in there.
otis of cologne