Well, (no pun intended) I am back. I have replaced the pressure tank and the check valve - both were over 30 years old.
Before I did that there was a dull pulsing in the pipes in the house (not quite a banging) - the pace of the pulses would be about as you can say "on-off", "on-off" over and over at a normal speaking rate. When that happened I went to the well, heard the contacts hitting on and off at the same pace, removed the cover and observed them doing that. I than chatted with a local contractor and was told it could be the pressure tank or the check valve. So, because they were over 30 years old and I did not want things to go South on the wife when I am not here, I bought and replaced the tank and the check valve. The pressure switch was not replaced at this time but it is fairly new - maybe 3-4 months old. The pressure gauge is older.
So, after did all this, I double checked everything, and then turned on a bib to bleed air, then turned on the electricity and pump started up, and I adjusted the pressure switch to 40-60 settings. All seemed to be going well as pump was on and pressure mounted up to 60 and shut off. Then over a period of about 60 seconds the pressure would drop (not smoothly but a bit jerkily) down to 40 when it would turn on again BUT when it went on the same problems resurfaced, i.e., the contacts hit on and off and on and off continually as fast as you can say on-off, on-off. I cut the power off and then on, and it ran smoothly up to 60 lbs, shut off, and then again over about 60 seconds the pressure dropped to 40 and then the on-off, on-off happened again.
So, after a new tank and new check valve the same thing is happening. So, Valveman, or anyone else, can you help me understand what is happening? I am thinking about heading to town to get a new pressure switch but it is sort of new, and I am not certain what could go wrong with that - but really don't know. The pressure gauge itself I think is only a device so you know what the pressure is, not otherwise functional?
So, all that is left is the pump - but it is pumping water - but maybe not as it should? Everything seems to work until it hits the 40 pound cut in, and when it then turns on it is on-off, on-off, on-off until I cut the power.
Certainly hope you guys have some ideas - I have reached the top of my learning curve at the moment, but will start researching.
Thanks for any help you can provide.