It is common to see a batter drop his bat in fair territory in the area in front of home plate. Usually this results in no further complication of the play. But when the dropped bat touches a batted ball, every umpire, coach and player should know the rule.
Live batted ball
A batter is responsible for his bat. If a batter drops his bat and it touches a live batted ball in fair territory, the batter is out for interference, whether he does it intentionally or not. The umpire should call, “interference, time, batter is out.”
In the case of a broken bat, the batter is not penalized if a piece of the broken bat touches a fairly batted ball or interferes with a fielder making a play on the ball.
Ball rolls into dropped bat (fair)
Another fairly common situation is when a batter drops his bat and it comes to rest in front of home plate (fair territory) and the ball rolls into the bat. If the ball rolls into the bat and stays in fair territory, the ball is live and in play with no penalty for the batter. If the ball rolls into the stationary bat and the goes into foul territory, it is a foul ball.
Ball rolls into dropped bat (foul)
If the bat is dropped in foul territory and the ball rolls into the bat and then goes into fair territory, the ball is ruled foul. Upon touching the bat in foul territory, the ball is foul and is a dead ball.
Batter Unintentionally Drops His Bat and it Touches a Live Ball
Ted Barrett Explains Different Dropped Bat Scenarios