Last week, Sammy Sosa got into trouble for accidentally using in a baseball game a bat that had been partially cored and filled with cork. Big-time baseball sluggers like Sosa think that a corked bat helps them make bigger hits--Sosa uses his corked bat to show off during batting practice.
There are a couple of prevailing theories about why a corked bat might improve a slugger's abilities.
Theory 1: A corked bat has more flex.
This one, it's hard to state in a way that doesn't make it immediately obvious that it's wrong. The theory is that a corked bat is less stiff than a solid wood bat and hence has more spring when it hits the ball. That sounds sort of plausible--after all, a springy rubber ball bounces higher than a baseball. The problem is that a steel ball bounces even higher, and we wouldn't think of it as springy at all.
The underlying phenomenon is that the stiffer and harder an object is, the less energy is lost in a collision. The softer an object is, the more energy is lost in the deformation of the object. A beanbag doesn't bounce at all--all of the falling energy goes into changing the shape of the beanbag. A baseball is also pretty soft--it squashes quite a bit when you drop it, and although it springs back into shape, it does so without much energy, so it doesn't bounce very high. A rubber ball doesn't deform very much, so most of its energy is left for it to rebound. A steel ball hardly deforms at all, so nearly all of its falling energy is available for its rebound.
In the case of a corked bat, to the extent that it's less stiff than a solid bat, it will deform more, mostly by bending, when it impacts the ball. That deformation eats up more of the energy in the swing, leaving less for the ball's rebound off the bat. Note that metal bats are known to be slugger's bats--they're much stiffer than wooden bats, lose less energy in the hit, and sending the ball flying further.
It seems plausible that a chunk of cork out of the center would tend to make the bat more flexible; to the extent that it does so, corking your bat won't help you hit homers.
Theory 2: A lighter bat is swung faster, and therefore hits the ball harder and further.
Let's start by saying that a batter swings with a constant torque on his bat, and that his swing goes through the same angle. With a lighter bat, he swings it around faster.
If you assume a bat is a perfect cylinder, corked wood is 30% lighter than solid wood, and 1/3rd of the bat is corked, you get a "moment of inertia" that's 20% less than a solid bat. That's simplified from reality, but it's a ballpark number. Given that, our slugger is going to be swinging his corked bat about 12% faster than his solid bat.
When the bat hits the ball, momentum is conserved. Momentum is the product of mass times velocity. A corked bat has about 90% as much mass as a solid bat and is 12% faster, so its total momentum is about 2% higher than a solid bat--pretty negligible.
A collision can be elastic--energy is conserved--or inelastic. A bat hitting a ball is pretty inelastic: the ball and bat are both deformed before springing back into shape, energy is lost in the sound of the crack of the bat, etc. The equations of an inelastic collision are really ugly, though, and I have no real way to estimate how inelastic the collision is, so I've analyzed it as a elastic collision.
In one-dimension, for 1=ball, 2=bat, v positive, you get that the rebound speed of the ball after it's hit, v_1-prime, is a simple function of the mass of the bat and ball and their initial speeds:
v_1-prime = (m_2 - m_1) * v_1 + 2 * m_2 * v_2
---------------------------------
m_1 + m_2
With a corked bat, you just substitute .9m_2 for m_2 and 1.12v_2 for v_2 to get the rebound speed of the ball off a corked bat. The rebound speed will determine how far the ball goes.
In the limit of the ball weighing much less than the bat, you get that a ball gains twice the speed of the bat over its pitched speed when a slugger hits it. That means that a corked bat gives the ball about 12% more improvement in speed than a solid bat.
In the real world, a bat (35 oz.) weights about seven times as much as a ball (5 oz.). For a swing faster than about 1/9th as fast as a pitched ball, a corked bat is an improvement over a solid bat, and it seems pretty reasonable to say that a 90 mph pitched ball is not nine times faster than a batter's swing.
However, you can see that I've made a lot of assumptions (in five pages of equations). Tweak these numbers around and you can easily get into a regime where a corked bat makes no difference or even hurts you.
Of course, this is a theoretical, back-of-the-envelope calculation. It would be pretty trivial to set Mr. Sosa up in a batting cage with some bats and measure his hits. You'd need several hundred, at least, to account for variation in pitches and swings, but in the end, you'd get a good, solid measure of whether a corked bat is better.
Posted by Greg at June 10, 2003 1:59 PM
I've got pointers to studies that come to each conclusion here. One thing that should be noted, though, and it's something that Robert Adair (author of The Physics of Baseball) stresses, is that a small improvement can make a big difference. Two percent may not sound like much, but if it turns a 400-foot warning-track fly ball out into a 408-foot home run, it is a very big difference.
IMHO, the main argument against bat corking is the high likelihood you'll get caught. Hollowed out bats are sure to break sooner or later. I believe this was a case of brain freeze rather than maliciousness on Sosa's part, but even then he should have known better.
I think you're solving the wrong problem. Or the wrong part of the problem, anyway...
If the swing is 12% faster, the batter can start his swing later. As far as the hitter's timing is concerned, we've just taken a respectable 90MPH fastball and turned it into a batting practice 80MPH - but one that goes farther when hit. That extra .04s to watch the flight of the ball means that the swing is going to be more accurate.
The hitter gets it on the screws more often.
That said, 12% seems the wrong order of magnitude. I'd expect between 1-3%.
12% is based on what is probably an overestimate in the change of moment of inertia, because I calculated I (moment of inertia) with an axis at the base of the bat, and the axis is probably really displaced another L/3 or so. That'll reduce the change in I for corking, which will reduce the improvement in swing speed.
Sosa said in at least one discussion of the incident that he uses corked bats in home-run demonstrations for charities--that's his explanation for how a corked bat got among his "real" bats. So he has some (non-blind) experimental evidence that corking helps him knock them out.
It has been noted that the collision between a baseball bat and a baseball is incredibly inefficient, because you've got a round ball hitting a round bat at a very narrow point of contact; if the bat were slightly less elastic, would the deformation of the bat make the collision more effective?
(I wrote that paragraph before reading this article which makes basically the same argument.)
[Kevin's slightly less elastic bat]
Call that theory #3: the softer bat, by extending the length of the collision, increases the elasticity of the collision, so that more energy goes into the rebound. That could be a significant effect, since as I read more, I grow to suspect that the collision is very inelastic.
Again, the best way to really determine this is to stick Sosa and a couple other sluggers in a batting cages for a few hundred swats each with each kind of bat. (The best experiment to date using a constant velocity swing of the bat, which seems wildly suboptimal.)
Great post. This is what's so wonderful about physics geeks.
Heh. Now I'm glad I said "You should blog this" at lunch the other day. :)
Again, the best way to really determine this is to stick Sosa and a couple other sluggers in a batting cages for a few hundred swats each with each kind of bat.
I think it would be a better experiment if we designed a machine to do this rather than deal with human variations, expecially with the number of swings we would need for a statistically significant sample size. I really wouldn't want to have to take fatigue, etc. into consideration.
I'm not sure if I agree that the softer bat would increase the elasticity of the collision. Wouldn't you lose energy in the deformation of the bat?
If you're really interested, I could probably get you some reasonable information on inelastic collisions from my materials textbooks.
Maybe Sammy could hit hundreds of balls with and without his corked bat at one of the many "charity exhibitions" he holds in Wrigley Field each year.
Rick:
While that would be nice, I'd rather put him in a batting cage where we can control the pitches better.
Elizabeth:
Apparently, the collision is very inefficient because it's a round ball hitting a round bat; the contact time is very small. Increasing the contact time may improve the energy transfer more than you lose in additional deformation.
I know, from a PBS special that I saw, that a (wooden) baseball bat will primarily vibrate in a 2-node mode when excited by the impact of a baseball. The label location, which indicates the location of the "sweet spot" of the bat, indicates the location of one of the nodes (the other is down where the hands go). Hitting the ball at the node results in the most effective transfer of energy from the bat to the ball, and the least amount of bat vibration (those who have played baseball can probably confirm the relatively smooth sensation of hitting a ball right on the sweet spot).
My idea: Adding cork to the bat increases the internal damping, and reduces the vibratory amplitude when a ball impacts outside the sweet spot. This in turn reduces the sensitivity of the bat to the location of the ball impact (relative to the node), and thus widens the effective sweet spot. Having a wider sweet spot makes it easier to hit the ball a long way, and that's why we see better/longer hits from corked bats.