My problem is this: my water pump goes on within 30 seconds of running water. As soon as I turn the water off the pump stops. The pressure on the gauge is normal. I know when the pump goes on because there seems to me a minor drop in the stream of water coming out of the facet. The pressure is there and I know there isn't a leak because it doesn't go on when no one is using the water. I have found no leaks. The pump and well are reatively now, they were both replaced about 5 years ago. Any suggestions?

Sounds like a pressure tank problem. Does you tank have a diaphragm? As new as it is, that is likely.

You should be able to get at least five gallons of water from your system before the pump cuts on. If you can't, and it certainly does not sound like you can get anthing even close to it, then your tank would be the first place to look.

I'm with jlisenbe, It would have helped if you had let us in on what type of tank you have.
Either the membrane in your bladder tank has ruptured or your galvanized tank need recharging. There's no repair for this except to replace the bladder tank or recharge the pressure tank. My bet's on a bad bladder tank. Good luck, Tom