【正文】 vot point. If a player was holding the bat in his/her hands, this would result in an impulsive force felt in the hands. ? If the impact force F strikes the bat below the centerofmass, but above the centerofpercussion, point P would experience both a translational acceleration in the direction of the force and a rotational acceleration in the opposite direction as the bat attempts to rotate about its centerofmass. The translational acceleration to the left would be greater than the rotational acceleration to the right and a player would still feel an impulsive force in the hands. ? If the impact force strikes the bat below the centerofpercussion, then point P would still experience oppositely directed translational and rotational accelerations, but now the rotational acceleration would be greater. ? If the impact force strikes the bat precisely at the centerofpercussion, then the translational acceleration and the rotational acceleration in the opposite direction exactly cancel each other. The bat would rotate about the pivot point but there would be no force felt by a player holding the bat in his/her hands. ? Define point O as the centerofpercussion（ COP） 1） Locating the COP According to physical knowledge, it can be determined by the following Team 8038 Page 8 of 20 method: Instead of being distributed throughout the entire object, let the mass of the physical pendulum M be concentrated at a single point located at a distance L from the pivot point. This point mass swinging from the end of a string is now a simple pendulum, and its period would be the same as that of the original physical pendulum if the distanceL was MdIL bat? （ 41） This locationL is known as the centerofoscillation. A solid object which oscillates about a fixed pivot point is called a physical pendulum. When displaced from its equilibrium position the force of gravity will attempt to return the object to its equilibrium position, while its inertia will cause it to overshoot. As a result of this interplay between restoring force and inertia the object will swing back and forth, repeating its cyclic motion in a constant amount of time. This time, called the period, depends on the mass of the object M , the location of the centerofmass relative to the pivot point d , the rotational inertia of the object about its pivot point 0I and the gravitational field strengthg according to MgdIT 02?? (42) 2) Analysis of the vibration:[1] Fig. 42 As showed in Fig. 42, mechanical vibration occurs when the bat hits the ball. Hands feel fortable only when the holding position lies in the balance point. The batting point is the vibration source. Define the position of the vibration source as the vibrational node. Now this vibrational node is one of the optional “sweet spots”. Solutions to the two “sweet spot” regions 1) Locating the COP[1][4] Team 8038 Page 9 of 20 ? Determining the parameters： a. mass of the bat M 。 then, develop a model to predict different behavior for wood or metal bats to find out the reason why Major League Baseball prohibits metal bats?[1][4] The mass (M) and the center of mass (CM) of the bat are different because of the material out of which the bat is constructed. The changes of the location of COP and moment of inertia ( batI ) could be inferred.[2][3] Above physical attributes influence not only the swing speed of the player (the less the moment of inertia batI is, the faster the swing speed is) but also the sweet spot effect of the ball which can be reflected by the maximum batted ball speed (BBS). The BBS of different material can be got by analyzing the material parameters that affect the moment of inertia. Then, it can be proved that the hitting effects of different bat material are different. Team 8038 Page 6 of 20 Assumptions and Symbols Model Assumptions 1) The collision discussed in this paper refers to the vertical collision on the “sweet spot”。 4) The coefficient of restitution of the bat bees smaller than before。 2) Less mass (lower moment of inertia) means faster swing speed。 second, define the regions that these optional sweet spots may appear as the “sweet zones”, and the length of each sweet zone as distance。 For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________ Team Control Number 8038 Problem Chosen A For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________ Team 8038 February 23, 2020 Summary Baseball is a popular batandball game involving both athletics and wisdom. There are strict restrictions on the material, size and manufacture of the bat. It is vital important to transfer the maximum energy to the ball in order to give it the fastest batted speed during the hitting process. Firstly, this paper locates the centerofpercussion (COP) and the viberational node based on the single pendulum theory and the analysis of bat vibration. With the help of the synthesizing optimization approach, a mathematical model is developed to execute the optimized positioning for the “sweet spot”, and the best hitting spot turns out not to be at the end of the bat. Secondly, based on the basic model hypothesis, taking the physical and material attributes of the bat as parameters, the moment of inertia and the highest batted ball speed (BBS) of the “sweet spot” are evaluated using different parameter values, which enables a quan