The physics of baseball: batting
Since I’ve gotten back to playing ‘ball’ (softball) this summer I’ve been thinking a lot about my hitting. Once I got back to playing second base I fell into my old routine with fielding and have played fairly well (2-3 total errors in six games) even though I’m playing softball this summer and I used to primarily play baseball. Second base seems to come fairly naturally to me. My hitting, however, is another matter.
When I played baseball I was never a great hitter. During my first stint at softball I had a solid on-base percentage (OBP) and usually reached on a hit, though the vast majority were singles. But that was a college ‘beer’ league. The league I’m in this summer is a very competitive co-ed league and my hitting has taken a subsequent turn-for-the-worse. Somewhat concurrently I have been coaching T-ball which is the entry-level game in little league. Despite being average I always prided myself on having ‘good form’ and, as a teacher, have worked hard teaching these kids how to swing (no one keeps score in T-ball since it’s really about learning the fundamentals).
Put all this together and I started to think seriously about batting. While this may be known to those who have been involved with the game for a long time, I realized that the advice I (and countless others) have received (and given) to ‘choke up’ on the bat, is bad advice. Physics – and a little common sense – tells us why.
When a person bats they’re (usually) seeking to maximize the force of contact between the ball and the bat through efficient use of their body. Since the person is swinging, the action is akin to pulling a lever. That is, something is (approximately) rotating about a point or axis. There are a lot of variables that go into this, but roughly the force of contact is related to a torque created by the motion of the arms and bat (as well as a portion of the torso). The strength of this force partly depends on the distance, r, the contact point is from the axis of rotation. Take a look at the picture below that I’ve doctored up a bit (Note: I found this on Flikr via a Google search and it apparently was taken by a guy named Wil C. Fry. Hope he doesn’t mind.).
So the torque at the point of contact, independent of the action of the ball itself, is where is the moment of inertia for the bat/arm combination. Roughly, we can approximate it as a cylinder pivoting around one of its endpoints. This is not a perfect analogy, but it will work for the argument here. is the angular velocity of the bat/arm combination, i.e. how fast the batter swings the bat. So the torque can be determined without even considering the ball. However, the contact itself also creates a torque since torque also happens to be . Note this is a cross product. That means only the components of the force and vector, r, that are orthogonal (perpendicular) to one another multiply. If you’re not sure about that, let’s assume the batter hits the ball dead on (i.e. he/she didn’t swing late or early – no pulling or hitting to the opposite field). In that case it’s just a regular multiplication. So if you know the torque from the other stuff and you know where the bat and ball connect, you know the force of contact. Essentially, this is the same principle behind closing a door. Remember, the batter is trying to get the ‘biggest bang for the buck,’ so-to-speak. When you close a door you need to push harder to get it to close the closer you push to the hinges. Don’t believe me? Try it. This is why you should hold wrenches and hammers as far from the head (as close to the end of the handle) as possible. Hitting a baseball is a lot like hammering a nail.
This summer, despite my team’s plethora of bats, not one is less than (nor greater than) 34 inches in length. Now, I’m not a big guy (5’8″) and was always taught to either use a shorter bat or to choke up. Take a look at our picture above again. Naïvely we might assume that choking up might help the fellow. Of course, for the guy in the picture, the ball’s likely to just hit further down the bat. But for someone like me, who seems to always make contact on the tapered part of the bat, this is seen as an improvement (the ball tends to go more where you want it to, for one). But, this summer, despite choking way up on the bat, contact kept happening along the tapered portion most of the time (not to mention the fact that choking up just doesn’t feel natural). Then someone told me to not choke up, but instead move further back in the batter’s box. Let’s assume the ball always goes right over the plate for now and let’s only consider times I actually make contact. In this case, moving me back has the effect of increasing r! Just like closing a door, that either means I don’t need to swing as hard to get it to do the same thing, or, if I swing just as hard, it should result in a larger force of contact (think about using the same force to close a door by pushing right next to the hinge and likewise by pushing on the handle). In my last at bat (AB) in my most recent game, I did not choke up, but rather moved back further in the box (assuming the umpire would tell me if I was outside the box) and laced a nice single to left-center.
Now, my teammates were also talking about the fact that many batters actually ‘choke down‘ on the bat to get more power. Specifically, the very bottom of the bat is gripped by either the middle or ring finger of their lower hand, leaving one or two fingers gripping at air. Then, right around the time of contact (ideally at the time of contact), they flick their wrists a bit (the grip simply makes this action a bit easier). This applies a secondary torque since, briefly, the bat is also pivoting around its base (in addition to the bat/arm combination pivoting around the axis along the body). Since torques and forces are additive, this increases the force of contact slightly. It’s a nifty little trick of physics – if you can get the timing right. Get it wrong and you’re likely to pull the ball foul.
Now we could also look at all of this from the standpoint of momentum. In the above case, the momentum of both the swing and the flick of the wrist is associated with some angular motion and is thus called angular momentum. Nonetheless, a linear momentum is associated with this at the point of contact and, like forces and torques (and, indeed, all vectors) momentum is additive. This helps explain one more trick batters often use. Frequently batters will, just before their swing, pick their leading foot up and step forward a bit as they swing. This effectively transmits a small amount of linear momentum to the entire body and thus to the contact as well. This gets added to the other two pieces of momentum and, if perfectly executed, is the recipe for a great hit.
Ah, but how does one perfectly execute something like this? Well, it’s all about timing. Take a look at the following analysis based on numbers crunched by Yale physicist Robert Adair. Granted, this is for a 90 mph fastball. In theory it should be easier in slow-pitch softball, but in the latter the ball is not coming directly toward home plate (if it is, don’t swing – it’s a ball since it has to arc to be a strike in slow-pitch). In any case, at least in baseball and fast-pitch softball, timing, which is largely physiologically determined, is what separates a good hitter from a great hitter.
Update: I wanted to clarify the first torque equation (I wrote this when I was exhausted). So, if we approximate the bat/arm combination as a cylinder rotating about one of its ends, then the moment of inertia would approximately be where L is the length. Let’s, for the sake of argument, assume that the ball makes contact with the bat at the very end of the bat every time. As such, we can approximately say that . According to Adair, the bat is moving at about 80 mph when it hits the ball. That’s a linear velocity. Angular velocity is where v is a linear (tangential) velocity. Substituting all that stuff in, the torque becomes . Thus we see that if we choke up, we’re reducing L thus producing less torque and reducing the force of contact!