That piece of wood immobilizes the tremolo. It's good as hard tail.
(you may not need to add springs as most guys here suggested. Just tighten the existing springs will do.)
don't let the bridge float. By tightening the springs the bridge will lay flat to the body. Then you won't have problem getting out of tune because of spring and string pull. In case you break strings, the rest of the strings will stay in tune.
