I need to make a swinging cat-door in a piece of furniture but I can't figure out how to hinge it
If there is such a hinge it would leave a big gap and the door might be too heavy for your cat to comfortably go through it. I looked into this myself and decided it would be best to just use a cat flap like this one.



http://store.petsafe.net/2-way-locking-cat-door

Alternately you can do something similar like round over the top of your wood and use 2 metal pins on the side but you still have the weight issue. Not sure if they make spring pins big enough but that would make it easy to mount like a toilet paper roll.

Update: I had a little time today to draw out what I was talking about with the rounded over top... I still think the cat flap is the best option but if you want to give this a go...

Couldn't find spring pins big enough to handle this so just routing out a channel to accept the pins then covering with a straight metal bracket or strip of wood might be the only option.

Top needs to be rounded over. Radius should be 1/2 the width of the stock and rounded over on both sides to make a half circle top.

Drill holes and insert metal pins. The metal shelf pins should work.

Insert the door in the slots you routed out for the pins and cover with a small metal bracket attached with screws or something like that to prevent the pins from coming out.