I have never worked with woodwork and I am very interested in starting to create my first pieces. Anyway, there is one piece that I have no idea how to do this circular groove. What are the tools for doing this type of circular groove?
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Some more context would be helpful. How large is the piece for example? What is it for? How accurate does it need to be? What tools do you have?
I would cut the main curve with a coping saw or bandsaw depending on the size of the piece and then finish it with a spokeshave, spindle sander, or router depending on how accurate it needed to be.
I would then re-saw it into two thin pieces on the bandsaw (or tablesaw) and carve half of the groove in each piece with a router with a ball nose bit. Then I would glue the two pieces back together (after smoothing the mating surfaces).
EDIT: I had a quick look and you can get a 'radius groover' router bit that would allow you to skip the re-saw and re-glue operations. Either way you would need to find a router bit the correct size. What will the radius of the groove be?
EDIT: adding a picture of the radius groover bit.
I'd suspect you can cut the circular profile with whatever you like: hole saw if you're lucky, band saw, jig saw, etc. With that done, look for a router bit in a half or full circular shape. You can either edge-route the groove or try to gouge it out with a hand tool. (Plenty of lathe gouges have a round profile.)