【正文】 2) When mass M and the location of centerof mass CM changes, batI changes, which is the dominant factor deciding the swing speed. Analysis of corked bat and wood bat [5][6] Table 43 swing period T bat mass M Bat length S CM position d coefficient of restitution BBCOR initial velocity inv swing speed batv ball massballm wood cm cork cm The hitting effect reflects by above physical parameters: Apply these values in formula (411) and (413) 1) Calculate respectively： smBBS /)1( ? smBBS /)2( ? Thus )2()1( BBSBBS ? . )1(BBS represents the maximum battedballspeed of a corked bat, and )2(BBS represents the maximum battedballspeed of a wood bat. Team 8038 Page 16 of 20 Conclusion: when the swing speed ( batv ) and the initial velocity of the ball ( inv ) remain the same, the initial velocity of the wood bat is higher than the corked bat. That is to say that the corked bat does not enhance the “sweet spot” effect. Hence, it is not the reason why Major League Baseball prohibits “corking”. But the change of the momentofinertia caused by the material can explain the prohibit. 2) Calculation of the momentofinertia of two different bats: 2*142)1( cmgI bat ? , 2* 59)2( cmgI bat ? Thus )2()1( batbat II ? . )1(batI is the momentofinertia of the corked bat, and )2(batI is the momentofinertia of the wood bat. Conclusion: the mass and the momentofinertia of the bat reduces after corking, so then swing speed gets faster, which means the professional players are able to watch the ball travel an additional 56 feet before having to mit to a swing. It makes the game unfair to increase the hitting accuracy by corking the bat. Reas。 Step 3: Calculating the distance The Euclidean distance of the positive ideal position is: ? ?? ?? )( ** xxd The Euclidean distance of the negative ideal position is: ? ?? ?? )( ** xxd Step 4: Seeking the integrated optimal region The integrated evaluation index of the evaluation object is: ????? dd db… …………………………（ 45） 2) Optimization positioning Considering bat material physical attributes of normal wood, when the period is sT ? and the vibration frequency is 520?f HZ, the ideal “sweet zone” extent can be drawn as ? ?cmcm 0 4 , . As this consequence showed, the “sweet spot” cannot be at the end the bat. This conclusion can also be verified by the model for problem II. Verifying the “sweet spot” is not at the end of the bat 1) Analyzed from the hitting effect According to Formula 411 and Table 42, the maximum battedballspeed of Team 8038 Page 13 of 20 the “sweet spot” can be calculated as smBBS sweet /? , and the maximum battedballspeed of the bat end can be calculated as smBBS end /? . It is obvious that the “sweet spot” is not at the end of the bat. 2) Analyzed from the energy According to the definition of “sweet spot” and the method of locating the “sweet spot”, energy loss should be minimized in order to transfer the maximum energy to the ball. When considering the “sweet spot” region from angle of torque, the position for maximum torque is no doubt at the end of the bat. But this position is also the maximum rebounded point according to the theory of force interaction. Rebound wastes the energy which originally could send the ball further. To sum up the above points: it can be proved that the “sweet spot” is not at the end of the bat by studying the quantitative relationship of the hitting effect and the inference of the energy transformation. Modeling and Solution to Problem II Model Preparation 1) Introduction to corked bat[5][6]: Fig 47 As shown in Fig 47, Corking a bat the traditional way is a relatively easy thing to do. You just drill a hole in the end of the bat, about 1inch in diameter, and about 10inches deep. You fill the hole with cork, super balls, or styrofoam if you leave the hole empty the bat sounds quite different, enough to give you away. Then you glue a wooden plug, like a 1inch dowel, in to the end. Finally you sand the end to cover the evidence. Some sources suggest smearing a bit of glue on the end of the bat and sprinkling sawdust over it so help camouflage the work you have done. 2) Situation studied： Situation of the best hitting effect: vertical collision occurs between the bat and the ball, and the energy loss of the collision is less than 10% and more than 90% of the momentum transfers from the bat to the ball (the hitting point is the “sweet spot”). Team 8038 Page 14 of 20 3) Analysis of COR After the collision the ball rebounded backwards and the bat rotated about its pivot. The ratio of ball speeds (outgoing / ining) is termed the collision efficiency, Ae . A kinematic factor k , which is essentially the effective mass of the bat, is defined as batballI zmk 2? ………………………………………………………… （ 46） where natI is the momentofinertia of the bat as measured about the pivot point on the handle, and z is the distance from the pivot point where the ball hits the bat. Once the kinematic factor k has been determined and the collision efficiency Ae has been measured, the BBCOR is calculated from kkeBBC OR A ??? )1( ………………………………………… （ 47） 4) Physical parameters vary with the material: The hitting effect of the “sweet spot” varies with the different bat material. It is related with the mass of the ball M , the centerofmass (CM ), the location of the centerofmass d , the location of COP L , the coefficient of re