When you hit a baseball with a bat, the bat flexes and then vibrates. We can model this vibration as a transverse standing wave. The modes of this standing wave are similar to the modes of a stretched string, but with one important difference: The ends of the bat are antinodes instead of nodes, because the ends of the bat are free to move. The modes thus look like the modes of a stretched string with antinodes replacing nodes and nodes replacing antinodes. If the ball hits the bat near an antinode of a standing-wave mode, the bat will start oscillating in this mode. The batter holds the bat at one end, which is also an antinode, so a large vibration of the bat causes an unpleasant vibration in the batter's hands. This can be avoided if the ball hits the bat at what players call the "sweet spot," which is a node of the standing-wave pattern. The first standing-wave mode of a vibrating bat is the m= 2 mode. Estimate the approximate distance of the sweet spot (as a fraction of the bat's length L) from the end of the bat. Express your answer in terms of L.
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