Totally normal. Sometimes they need a very thorough clean of the cart connector. I did this by folding some paper around a thin piece of cardboard and pushing it in & out of the cart connector. Move the paper around from time to time as it get black stuff on it.
Yes, totally agree. Very common for dirty cart slots to cause a problem. And the SC-3000 won't do anything unless it has a cartridge inserted as there is no bios on-board.
Edit - the black screen and constant tone is a good thing for a SC-3000. It means that most of the system is probably working
If the screen goes a nice solid black then the video circuitry is probably outputting a reasonable signal and the medium tone means that the sound generator is outputting its default noise from a cold boot. You are then relying on a good connection to the cart to provide some code for the Z80 to execute. If that doesn't happen, you first check the cart contacts, clean the cart slot, try multiple games cartridges to look for different behaviour, then start inserting Basic IIIA and IIIB carts to see if you get some error beeps or not to isolate obvious memory / VRAM errors.
I haven't tried using paper to clean the slot before. I think this is my favourite video on how to clean a cart slot. This is for a Famicon, but the process is basically the same for the SC-3000.http://www.youtube.com/watch?v=bWg2k736KZ4
My usual recommendation is something like a handkerchief around a thin piece of card rather than paper. I found that easier to handle than a T-Shirt as used in the video.
Just one comment. Be careful when cleaning the cart slot. If you bend a pin it is very hard to straighten it out again and you may be looking for a replacement edge connector. The main thing is to make sure that your cardboard + cloth is not too thick (it should be approximately the same thickness as a PCB), or too fluffy (that is why a handkerchief is good). And don't use too much water on the cloth - just slightly damp. And let it dry out afterwards.