The Science Behind Baseball
By Alejandro Madariaga, Tereka Harris, and Leon D. Huderson
Jacksonville University
February 21, 2010
Introduction
The Annual Mathematical Contest in Modeling (MCM) is a multi-day mathematics
competition held annually in USA, during the first or second weekend in February. It is
distinguished from other major mathematical competitions such as Putnam by its strong focus on
research, originality, teamwork, communication and justification of results. At the beginning of
the contest, teams have a choice between two problems. Problem A involves a system that
requires the use of continuous mathematics, and thus often involves concepts from geometry,
physics, or engineering. Problem B involves a system that requires the use of discrete
mathematics. The Teams have 96 hours to research and submit their solutions in the form of a
research paper. During this time, they may consult any available references, but may not discuss
their problem with anyone outside their teams. Several guides containing advice and
recommendations for teams and/or advisors have been published online or in print. Around one
thousand international teams of three undergraduates compete to produce original mathematical
papers in response to one of two modeling problems. Initially, participation was largely from the
United States, however in recent years international participation has grown significantly,
particularly from the People's Republic of China, so that in 2007 teams from the United States
comprised only 24% of total participation. After the competition, all papers are judged and
placed into the following categories:
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Unsuccessful Participant
Successful Participant (approximately 50% of teams)
Honorable Mention (approximately 30% of teams)
Meritorious Winner (usually 10 to 15% of teams)
Outstanding Winner (usually 1 or 2% of teams)
Each year the Outstanding Winner papers are published in The UMAP Journal.
So for the 2010 MCM contest the team decided to go for Problem A which was The Sweet Spot.
Explain the “sweet spot” on a baseball bat.
Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is
transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation
based on torque might seem to identify the end of the bat as the sweet spot, but this is known to
be empirically incorrect. Develop a model that helps explain this empirical finding.
Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and
filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect.
Augment your model to confirm or deny this effect. Does this explain why Major League
Baseball prohibits “corking”?
Does the material out of which the bat is constructed matter? That is, does this model predict
different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major
League Baseball prohibits metal bats?
Our Solution
Locating the “Sweet Spot”
There are many definition of the term for “sweet spot” on a baseball bat. It could
be the place where the maximum in the speed of the ball is reached where the bat strikes the ball.
The definition chosen by the team is that of the spot in the bat where the ball hits and there is no
force exerted on the batters hand. This spot can also be called the center of percussion (COP). If
the collision of the bat and the ball happens at this spot, then there is no reaction impulse
produced so there would be no shock to the hand of the batter. The equation used to find this so
called “sweet spot” is:
Where is the length from the pivot position to the end of the bat and is the location of
the sweet spot. To get a better understanding on how we found this equation, take a look at the
following image:
is the pivot point. This is the place where the batter chooses to hold the bat and it
depends on what is comfortable for that player. is the center of mass and we assumed it is half
from to the end of the bat. is the place at which the ball hits the bat. is the length from to
the end of the bat, and are the lengths from the pivot point to the center of mass and where
the ball hits, respectively. We are assuming the center of mass is half way down the bat from the
pivot point, therefore
is the force on to the bat in the y direction and
is the
force in the x direction and is the angular velocity of the bat.
We are looking for the spot at which there is no force exerted onto the batter’s
hand, so
would equal zero. We started this an equation of the moment for rotation about our
fixed point P, the pivot point.
is the moment of inertia about P and
is the angular acceleration. From this we got
We assume the center of mass was half way down the bat from the pivot point, so
.
Also, the sum of the moments can be expressed by the following equation
And, the angular acceleration can be expressed as,
Where
is the linear acceleration, in the x direction, of the center of mass of the bat.
Newton’s second law tells us that
We are looking for the spot at which
. So when setting
, we get
Now that we have these equations, by substituting we have the following
If we define the sweet spot as the place where the ball hits with no force transferred to the
player’s hand, then the sweet spot is located down the bat in the place where the player holds
the bat.
Maximum Power
One way to measure the power transferred onto the ball is by looking at the collision
efficiency ( ). This is simply the ratio of the final speed of the ball to the initial speed of the
ball.
Where is the final velocity of the ball, and
the bat. If we solve this equation for
we would get
and
is the speed of the ball and that of
Notice from this equation, the speed of the ball is less effective than the speed of the bat.
This is so because
so how fast the player swings the bat has a higher effect on the
speed of the ball after it has been hit that the speed at which the pitcher throws the ball. To have
maximum power, the batter must swing the ball as hard as he or she can, and this depends on the
mass of the bat and the strength of the player.
Depending on where the ball hits with respect to the sweet spot of the bat will change the
response the player feels. If the ball hits at the sweet spot then the player will feel no force. If the
ball hits below the center of mass but before the sweet spot then the player should feel a jolt in
one direction. If the ball hits somewhere beyond the sweet spot, towards the end of the bat, then
the player should experience a jolt in the opposite direction. Our model above showed us that the
sweet spot is not located at the end of the bat, but rather down the bat.
Corking
By corking a bat, the bat will weigh less. A lower mass would result in a lower force onto
the ball, but a lighter weight bats means it’s easier for the batter to swing the bat.
tells us that the speed of the swing of the bat has a higher effect on the speed of the
ball after it has been hit. So, if by the ball being lighter, the player can swing it harder, this may
result in a higher speed as it travels out into the field, after it has been hit.
The effect of corking a bat is just a myth. A corked bat has a slightly less mass by means
of drilling out the center of a wood bat and replacing it with cork. The player can shave about
1.5 ounces off of the weight of his bat but since the reduction of the weight the center-of-mass of
the bat would shift slightly towards the handle end of the bat. This would mean that the moment
of inertia of the bat would decrease and it would be easier to swing.
tells us that the speed of the swing of the bat has a higher effect on the speed of the ball after it
has been hit and research has shown that faster bat swing speed results in faster batted-ball
speed, though the change in ball speed would be minimal for most players. Less mass means a
less effective collision. So, if by the bat being lighter, the player can swing it harder, this may
result in a higher speed as it travels out into the field, after it has been hit. But lowering the mass
and the moment-of-inertia may increase the bat swing speed; however the lower mass means that
the collision between bat and ball is less effective. If the swing speed is kept constant, a heavier
bat will always propel the ball faster and farther. So removing mass from the bat will actually
reduce the batted-ball speed. The ratio of the bat composition would be affected, thus changing
how the bat reacts. That’s why the rule 6.06 from the national baseball rules refers only to bats
that are "altered or tampered with in such a way to improve the distance factor or cause an
unusual reaction on the baseball. This includes bats that are filled, flat-surfaced, nailed,
hollowed, grooved or covered with a substance such as paraffin, wax, etc." The corking of the
bat could also weaken a bat which could break easier.
Different Material
Throughout the years baseball bats have been made from different materials. From the
beginning the wood of choice was oak and hickory. Later on white ash was used, but because of
its rarity more bats are being made out of birch and hard maple. Metal bats have gone through
similar changes. Older metal bats are normally composed of aluminum while the newer ones
tend to be made out of composite metals. The major league prohibits the use of metal bats
because of the advantage a player would get. Since the weight would decrease it would increase
the speed of the bat, thus increasing the speed of the ball after being hit. Also this can be a safety
issue since the ball has more speed, which gives the ball a greater impact force, making it more
possible for players to be injured from catching or being hit by the ball.
A big question which seems to have many theories asks one thing: does the material
which composes the bat hold a significant effect on the performance of the player during the
swing? The constant problem is that while many studies have been performed to conclude
whether it has a big effect or not, it is agreed that the results are not without flaw simply because
it would be difficult to reproduce the conditions of the swings as they are performed in a game
scenario. All of the tests are performed under experimental conditions as described in either the
ASTM or NCAA method. Dr. Alan M. Nathan did his study on the differences between
aluminum and wood bats in a baseball swing using the NCAA data and methods to determine the
nature of both. He found that the aluminum bats perform better because the moment of inertia
(MOI) is smaller due to the decreased mass of the aluminum bats in comparison to the wooden
bats and that this would produce a faster swing with the aluminum bats. He also stated that two
bats of different composition would perform identically in testing settings would be different in
the playing field, but that even being of two different materials that they would perform
identically if they shared the same MOI. He gave the following reasons for how this could
happen:
1.
Make the MOI restriction on aluminum bats comparable to wood bats (i.e., 11,000
rather than 9700 for 34” bats). This would be quite easy to do in that bat manufacturers would
shift weight from the knob to the barrel cap to increase the MOI.
2.
Require that aluminum bats pass the laboratory test of hit ball speed less than 97
mph with a bat speed higher than 66 mph, with the exact speed depending on the MOI of the bat.
Of course, one would need a prescription for the relationship between bat speed and MOI, which
one could obtain from the data of Fleisig or of Crisco and Greenwald or from additional testing.
3.
Use the same 66 mph bat speed as in the NCAA test but a “sliding scale” for the
hit ball speed. That is, the upper limit on hit ball speed, currently set at 97 mph for all bats,
would be less than 97 mph for bats with a lower MOI. Once again, one would need to obtain data
(or a reasonable simulation of the ball-bat collision) to determine how to set the scale. (Nathan)
Of the mentioned above, the first options is the most viable. This seems to be a trend
typically when discussing the work of L.V. Smith who made his own unofficial method in order
to garner results. He found that the hollow nature of the aluminum bats coupled with the
decreased coefficient of restitution (COR) makes for a higher velocity and lower energy loss due
to longer contact during collision between the baseball and the bat and greater impulse to the
ball. (Smith)
People may wonder if the velocity and energy exchange of a wooden bat can be held to
the same standard with an increased MOI for aluminum bats. The answer is that while it is true
aluminum bats could be modified to perform like their wooden counterparts; many people from
the little leagues to the pros view the damages, injuries, and even death as too much to bear. A
student during a Police Athletic League game in New Jersey received a hit to the chest from a
swing of an aluminum bat. There are other cases where serious injury had been reported from
baseballs hit with swings from aluminum bats. These cases have made arguments which have
led to the banning of these instruments from all levels of play. When the New York judicial
system held a decision to ban aluminum bats from high schools, it was met with opposing
viewpoints, mainly because of the fear of such dangers. (Passan)
Summary
The term “sweet spot,” when talking about a baseball bat, could be described as the place
on the bat where the ball can achieve its maximum speed when struck. The team has defined the
“sweet spot” as the spot in the bat where the ball hits and there is no force exerted on the batters
hand. Through research and calculation, a model was found to explain how it works. The team
has also shown through research and calculation that a corked bat does not performed up to the
myth’s standards. The team has also shown that the changes in materials which the bats are made
of have made an effect on the speed and distance which the ball travels. This has led to the
banning of aluminum bats in many leagues because of its increased speed.