Can I calculate beam size if I know length, and weight support requirements?
You can do the math by hand or look for a beam calculator online, but it's easier if I just use engineering software I have to size this beam for you.

Use 3 plies of 10' long 2x6 to support 400# located at the middle of the span.  With 400# that will always be there, it will deflect 0.18", so less than a quarter of an inch. I sized SPF #2, so you'll be fine using any 2x6 stud you can find at your big box store.

Nail the three plies together with 4-1/2" 30d nails in two staggered rows. Technically this detail is for a built-up column, but just use the same nailing pattern.

If your next question is what if I sized it with 200# instead of 400#, the answer is that you can use 2 plies of 2x6. It will deflect 0.14", and nail them together in the same pattern but with 3" 10d nails. Personally I wouldn't do this though, because kids like to hang on these bags and I think the 400# loading and 3-2x6 makes the most sense.