A lot of problems arise when players don't use their free arm properly. It's a major problem with a lot of players. Here are two common problems.
First, many players let their free arm just sort of hang down instead of holding it up as a counterbalance to the playing arm. Every time they stroke the ball there is no counterbalancing arm to act as a counterweight, and so they are thrown slightly off balance with each shot. Worse, they become so used to this they don't even realize it is happening. The cure - hold the free arm up for balance at all time, wrist about as high as the elbow, and let it naturally counterbalance your playing arm. Note that this is true on backhands as well as forehands. On the forehand, the counterbalancing is more obvious. On the backhand, as you extend your playing arm out, the free hand needs to counterbalance this by naturally pulling back slightly.
Second, when hitting forehands, players don't use their non-playing side. You should pull with the non-playing arm as you rotate around on the forehand. The non-playing side (the left side for righties) is just as important as the playing side when you hit a forehand. Your body can't rotate properly unless both sides rotate.
Here's a video of the two best players in the world right now, world #1 Ma Long and #2 Zhang Jike of China (15:29), playing the final of the Austrian Open in September. Watch the way they use their free arm on each shot. (Of course, you can learn a lot from this video besides just the use of the free arm!)