First thing to check is the circuit breaker or fuse that powers the pump. Since you mentioned sprinklers, I'm guessing things couldn't be frozen solid in ice . If it is tripped, try resetting it. If it trips again, it could be either the circuit breaker (they do go bad, just not that often), or the pump or wiring has a problem.
If the circuit breaker is on, do you ever notice if the pressure rises? Have the lights been dimming for no perceived reason? If the pressure rises, then quickly falls, the check valve may be letting all of the water go back down into the well, pushed by the pressure built up in the pressure tank.
If the pressure quickly rises, then the pump shuts off, then the pressure tank's bladder may be shot - basically what is happening is that since water doesn't compress much, as soon as the pump turns on, the high pressure turn off point is reached, and the pump shuts off. If the bladder (think balloon in the tank) is bad, there is no space to compress and nothing to push the water out of the pipes unless the pump is actually running.
11 pounds SHOULD have the pump running all of the time, since it is below the minimum for any normal system.