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The centre of percussionCOP is not a unique point on
5 I" A7 E) j3 {0 ?the barrel since it is defined in terms of a conjugate point in9 P* U2 y1 }1 z- G1 r& w" a
2,6,77 S/ q* V3 J2 l
the handle. Each point in the handle is associated with a
* D `0 n' k! G8 N% q, Edifferent COP in the barrel. When a ball strikes a freely2 }* w' f6 ~& q4 q1 B$ O
suspended bat, the bat will translate in the direction of the# j6 Y( U' E6 N/ Y. n- P( u( i
incident ball and it will also rotate about the centre of mass.4 Q k* S7 u- q# L* }
The combined effects of translation and rotation result in a
4 {+ r2 l8 g$ n- V6 `/ G: ?. Z+ qshift of the actual axis of rotation, away from the centre of
9 R& C) X: s ^/ Umass toward the handle. The axis passes through the conju-! c p! @; a$ A; y- [
gate point for that particular impact. If b is the distance from
4 a/ ]& _/ w1 Z( R1 h5 jthe impact point to the centre of mass, and x is the distance
/ J+ Q9 U# A2 E5 R. C. Jfrom the centre of mass to the conjugate point, then xIcm/(Mb) where Icm is the moment of inertiaof the bat
/ f6 `4 A( O8 V1 P; ~! e9 |: b+ habout its centre of mass and M is the mass of the bat. Con-
7 w( v$ M1 ~# a, E6 Msequently, if the bat is pivoted about an axis through the: h( S3 F9 l* a, p/ q2 D% }7 s
conjugate point, and if the ball strikes the corresponding8 k2 W* O. o+ R) Q
COP, then there is no reaction force at the pivot point. It is# D: ^ M" f( Y0 x- }9 a5 t; y
therefore reasonable to expect that every point under the4 @9 e# f" C' u" l* A8 x+ M
hands will have its own COP and that an impact at any given
* v8 l& \ f9 i3 t9 E3 nCOP will generate zero reaction force at the corresponding& A) {/ [& [) J: W+ v
8; z- n! c# \+ P( _
conjugate point under the hands. Brancazio has suggested4 j: ?, n2 F: R- g" r; B3 ?7 M
that the sweet spot zone might therefore encompass all cor-8 _; D5 c9 T8 r# H+ q( z- A1 j
responding COP points along the barrel. There is, however,! F1 N3 |, n" b( _! V( M1 |: P
an uncertainty in this argument since a hand-held bat is not4 v( R7 c; |/ _# D
freely suspended, nor is it pivoted about a single axis, and
: L5 P- Q( K' j3 `, S Fmay behave differently as a result of the reaction forces ex-( U" J9 [: [) l' S
erted by the two hands. Furthermore, the axis of rotation may
# B2 T0 h0 S9 ?. g; Q) Q ^even shift as a function of time since the ball remains in
: Z4 \ h/ e! Qcontact with the bat for about 1.5 ms, the transmission time7 C9 Z' @3 b' z
of the pulse from the impact point to the hands is about 1 ms,$ f b) Q/ r3 I( \
and the impact forces on the hands last for about 20 ms as a
' y* B# }& C7 ~( _0 k$ k& a( sresult of the induced vibrations and the induced motion of
5 U- H5 d1 `. o$ g- ?the bat. |
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