The Sweet Spot: A Wave Model of Baseball Bats
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The Sweet Spot: A Wave Model of Baseball Bats
Rajib Quabili
Peter Diao
Yang Mou
Princeton University
Princeton,NJ
Advisor:Robert Calderbank
Abstract
; |# X3 y- [4 N; Q! k ^We determine the sweet spot on a baseball bat. We capture the essential physics of the ball-bat impact by taking the ball to be a lossy spring and the bat to be an Euler-Bernoulli beam. To impart some intuition about the model, we begin by presenting a rigid-body model. Next, we use our full model to reconcile various correct an incorrect claims about the sweet spot found in the literature. Finally, we discuss the sweet spot and the performances of corked an aluminum bat, with a particular emphasis on hoop modes.
Introduction
& e/ @$ W/ ?+ V& w( s5 y' |# qAlthough a hitter might expect a model of the bat-baseball collision to yield insight into how the bat breaks, how the bat imparts spin on the ball, how best to swing the bat, and so on, we model only the sweet spot.
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There are at least
: V- g( e1 A/ u4 s! @ S8 v/ ]two notions of where the sweet spot should be-an inpact location on the bat that either
. minimizes the discomfort to the hands, or
. maximizes the outgoing velocity of the ball.
We focus exclusively on the second definition.
+ [+ y4 g9 _" |5 @The velocity of the ball leaving the bat is determined by
. the initial velocity and rotation of the ball,
. the relative position and orientation of the bat and ball, and
. the force over time that the hitter’s hands applies on the handle.
We assume that the ball is not rotating and that its velocity at impact is perpendicular to the length of the bat. We assume that everything occurs in a single plane, and we will argue that the hands’ interaction is negligible. In the frame of reference of the center of mass of the bat, the initial conditions are completely specified by
. the angular velocity of the bat,
. the velocity of the ball, and
. the position of impact along the bat.
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The location of the sweet spot depends not on just the bat alone but also on the pitch and on the swing.
: d, j: H% E' Q( H+ b% lThe **st model for the physics involved has the sweet spot at the center of percussion[Brody1986], the impact location that minimizes discomfort to the hand. The model assumes the ball to be a rigid body for which there are conjugate points: An
9 y; p0 `9 W4 d% Y8 w x0 y; Y* nimpact at one will exactly balance the angular recoil and linear recoil at the other. By gripping at one and impacting at the other (the center of percussion), the hands experience minimal shock and the ball exits with high velocity. The center of percussion depends heavily on the moment of inertia and the lacation of the hands. We cannot accept this model because it both erroneously equates the two definitions of sweet spot and furthermore assumes incorrectly that the bat is a rigid body.
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