Hi thePirateRooster!
See this Apple Article re "Cannot see or control camera":
http://support.apple.com/kb/TS1059#3
Here is an excerpt from that Article relevant to your camera:
+"If your USB camcorder is not recognized, follow these recommendations:+
+1. For best results, use a qualified device. iMovie '09 works with the DVD, HDD, and Flash memory-based camcorders listed in iMovie '09: Camcorder Support. iMovie '08 also works with the DVD, HDD, and Flash memory-based camcorders listed in iMovie '08: Camcorder Support.+
+2. An Intel processor is required for AVCHD video support. AVCHD DVD camcorders do not work with iMovie.+
+3. If your camera is connected via USB and the camera turned on, follow one or more of these tips to reestablish the connection to the camera.+
+Turn the camera off and back on again.+
+Disconnect the USB cable from your computer and reconnect it. Do the same with the camera as well. If the camera is connected through a USB hub, try connecting the camera directly to the computer.+
+Quit and reopen iMovie.+
+Restart the computer.+
+Try another USB cable or try the camera with another computer running iMovie '09 or iMovie '08.+
+4. Open Apple System Profiler (from the Utilities folder) to see if your camera appears in the USB section. If after following the tips in Step 3, you do not see your camera in Apple System Profiler, try resetting the USB connection as described in What to do if your computer won't recognize a FireWire or USB device , and check again.+
+5. If you are using a flash-, disc- or hard-drive-based camera and the camera and its clips are recognized in iMovie, but certain clips will not import, or import without audio -- try another flash card or disc in the camera. Alternately, back up the footage from the camera and reformat the camera's flash card or the drive from the camera. Record some new footage and see if this works properly. (Consult your camera's documentation for more information on how to back up and reformat the camera's contents.)+
+6. Make sure you are recording your footage at the normal or 'high quality' camera setting. Some cameras allow you to set a higher compression rate to fit more footage on the camera's storage medium. Footage recorded in this format may not be recognized by iMovie."+
Your camcorder, Sony DCR-SR47, is not listed in this Apple Article re iMovie '08 supported camcorders:
http://support.apple.com/kb/HT1014
BUT - is listed in this Article re camcorders supported by iMovie '09:
http://support.apple.com/kb/HT3290#2
Your camcorder records in MPEG-2 format. Please also see this Article re iMovie and muxed MPEG-1 and MPEG-2 clips, as this may be relevant to your situation:
http://support.apple.com/kb/HT1372
Hope this helps!
John