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The centre of percussionCOP is not a unique point on, o9 a4 ^0 [0 |2 U% k
the barrel since it is defined in terms of a conjugate point in9 r' f0 d9 ~: Y9 a* _
2,6,7
7 k9 n" l" u* L: J4 N3 dthe handle. Each point in the handle is associated with a# p1 t+ o* ~+ i5 b# e# A
different COP in the barrel. When a ball strikes a freely y/ v3 L! ]/ o) V/ o( c T/ m
suspended bat, the bat will translate in the direction of the; _ o; ]8 A2 r$ _6 ~5 {; g- ~! [4 t
incident ball and it will also rotate about the centre of mass.
* K# g6 V5 A& z o! x# b" kThe combined effects of translation and rotation result in a' P9 V# n; m' K( y! N
shift of the actual axis of rotation, away from the centre of
# {5 b0 {& q: j5 Smass toward the handle. The axis passes through the conju-
5 }( |8 p7 _2 H5 {gate point for that particular impact. If b is the distance from
9 G& Q! `0 m) T/ zthe impact point to the centre of mass, and x is the distance
5 u: D( r9 Q) n- {" L& u8 Sfrom the centre of mass to the conjugate point, then xIcm/(Mb) where Icm is the moment of inertiaof the bat
1 m4 D8 r. Q8 v& q, X/ Tabout its centre of mass and M is the mass of the bat. Con-4 M2 C% K2 i( W
sequently, if the bat is pivoted about an axis through the1 O6 E1 Y) t1 }" {
conjugate point, and if the ball strikes the corresponding
: E+ N% d4 g: F! q$ k- YCOP, then there is no reaction force at the pivot point. It is: A0 ?1 |9 U; f; n# Q
therefore reasonable to expect that every point under the
$ h( E# @, M% O% shands will have its own COP and that an impact at any given$ ?. a9 J5 s$ z+ l) B& ~ f, G
COP will generate zero reaction force at the corresponding9 d( @, { `9 u. X1 q- t
8; X1 o1 k, [. b; c C) W
conjugate point under the hands. Brancazio has suggested* N+ T$ [1 Y) Z% p
that the sweet spot zone might therefore encompass all cor-2 z$ ], B# ^ {; Q4 V- X) z: Y
responding COP points along the barrel. There is, however,% K. t& N. N4 _3 u/ Y/ j
an uncertainty in this argument since a hand-held bat is not
0 [5 }. E2 @1 p( rfreely suspended, nor is it pivoted about a single axis, and
$ @. h0 ^: C* \7 @. c8 Qmay behave differently as a result of the reaction forces ex-, _5 t. S* K1 H( K; c
erted by the two hands. Furthermore, the axis of rotation may
8 Z2 X& n* Z" n! \+ g. geven shift as a function of time since the ball remains in. v' v4 E8 i* r7 K1 h
contact with the bat for about 1.5 ms, the transmission time
5 G9 v8 S. c4 a- S5 p( ?of the pulse from the impact point to the hands is about 1 ms,2 s, X- \2 Q( F# R( K. p. Y
and the impact forces on the hands last for about 20 ms as a: x7 ^/ u; |, k. P. S- U& O
result of the induced vibrations and the induced motion of
" U7 w7 E8 N3 J7 \6 X/ i1 x5 F( a5 D9 bthe bat. |
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