The Physics of Baseball

Batters know from experience that there is a sweet spot on the bat, about 17 cm from the end of the barrel, where the shock of the impact, felt by the hands, is reduced to such an extent that the batter is almost unaware of the collision. At other impact points, the impact is usually felt as a painful sting or jarring of the hands and forearm, particularly if the impact occurs at a point well removed from the sweet spot.
The sweet spot of a bat exists partly because bat vibrations are not excited significantly at that spot and partly because the spot is close to the centre of percussion. These spots are described in the tennis racquet page. A bat is similar to racquet but there are some obvious differences. For a tennis racquet only the fundamental mode of vibration is excited by the impact since the impact duration, about 5 msec, is too long to excite higher frequency modes of the racquet. The impact on a baseball bat is shorter (since the bat and ball are harder or stiffer), about 1.5 msec. As a result, the fundamental and second vibration modes are both excited with about the same amplitude. These modes are shown in the diagram below. Hence, there are two vibration nodes in the barrel. An impact at the fundamental node will not excite that mode, but it will excite the second mode. Similarly, an impact at the node of the second mode will not excite the second mode but it will excite the fundamental mode. The ideal spot to hit the ball is half way between the two nodes since both modes will be excited but only with small amplitude. This spot is also close to the centre of percussion.