The hard drive where the music files are is not plugged in before starting Amarok. Then when I start Amarok, and then mount the hard drive, the music files in Amarok are all faded and do not work.

Well, how about plugging in the drive before you start Amarok? It needs to access the files stored in the database, plugin in a drive referred to in the database after the launch is not the correct way to do things.
Mind you, a full collection rescan can solve the issue with the grayed out tracks, but still, this is a workaround, plugin in and mounting the sources before using them is the recommended way