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The centre of percussionCOP is not a unique point on
0 a7 F$ \, z- @! A( @( Uthe barrel since it is defined in terms of a conjugate point in/ b/ V; H3 ~/ u; o4 Q, |" X: Q- W
2,6,7
# l1 V5 H, Y& c8 E5 G3 @. P" h7 n( }the handle. Each point in the handle is associated with a1 c' \) ^8 h+ T# x' F8 _! Z
different COP in the barrel. When a ball strikes a freely# B6 ]2 [6 v6 o5 H
suspended bat, the bat will translate in the direction of the7 E5 w! \+ o- `" h
incident ball and it will also rotate about the centre of mass.
+ ]0 Z5 |6 G0 i4 u) l1 |" hThe combined effects of translation and rotation result in a
3 v* d; A. b4 t/ jshift of the actual axis of rotation, away from the centre of! c; L/ c9 r. z1 A! g0 ~1 T7 u
mass toward the handle. The axis passes through the conju-1 l+ Q: o6 h% A$ E9 u g2 q) y; Y
gate point for that particular impact. If b is the distance from/ o, V( n1 o: ~7 ~0 j; B7 e
the impact point to the centre of mass, and x is the distance
9 W5 \# q2 I& N# m4 G1 |: R% S# Efrom the centre of mass to the conjugate point, then xIcm/(Mb) where Icm is the moment of inertiaof the bat
$ A/ ?4 h+ M- q& cabout its centre of mass and M is the mass of the bat. Con-
% d+ b- e+ W" H* k1 zsequently, if the bat is pivoted about an axis through the& Z7 e6 i' {& O+ E! z. O# E J
conjugate point, and if the ball strikes the corresponding
, [4 Y* j5 O9 e, \5 R' ~. UCOP, then there is no reaction force at the pivot point. It is6 j0 [. l$ h+ j9 P1 ?( X" ^
therefore reasonable to expect that every point under the
$ N" w, ]% {; I G& bhands will have its own COP and that an impact at any given( E* ?8 y+ w& C- P0 l# M
COP will generate zero reaction force at the corresponding1 ?) x" n& g' S7 K- b" H
8
5 i9 U" B; D5 q# A0 sconjugate point under the hands. Brancazio has suggested, f( K# L$ w0 }5 g
that the sweet spot zone might therefore encompass all cor-% D+ W+ y$ v) x/ C8 i
responding COP points along the barrel. There is, however,
% k+ c- d% ?; ]% Yan uncertainty in this argument since a hand-held bat is not
" m0 H3 j5 Z. n; d D2 Efreely suspended, nor is it pivoted about a single axis, and
% Y1 v5 M( ]+ \7 C6 Ymay behave differently as a result of the reaction forces ex-
4 o' X6 d' W, k, s) ~+ J9 }$ t) \erted by the two hands. Furthermore, the axis of rotation may
$ _$ d5 }4 ?& i1 l1 R3 Reven shift as a function of time since the ball remains in2 `8 X- h& M0 }8 g8 V
contact with the bat for about 1.5 ms, the transmission time
5 T( h' J& ]; q, ?" j' C Y& Rof the pulse from the impact point to the hands is about 1 ms,/ {! r( q- K$ `5 h* i
and the impact forces on the hands last for about 20 ms as a, w0 m! [5 k( N% i6 }
result of the induced vibrations and the induced motion of
" X5 W& ]& X( D% _ J$ G \) kthe bat. |
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