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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answer: 31, 5556. 952 seconds

bacteria that live on the ocean floor are sustained by phytoplankton. phytoplankton, microscopic marine photosynthetic organisms, have a vastly significant role to play not only in the marine food web of which they’re part of, but also on a more global scale. despite their infinitely small size in comparison to other marine organisms, these tiny creatures occupy an immensely important ecological niche: they are the foundation of the marine food web, and as primary producers, play key roles in supporting all other organisms in the marine environment, as well as in the regulation of the earth’s climate through the sequestration of carbon, oxygen production, and other related processes. phytoplankton account for roughly half of all global primary productivity; therefore, their significance extends far beyond the marine environment alone. there is an intriguing sense of irony in the realization that these tiny living beings that often live out their existence unnoticed and undetected by the rest of the world, have such a far-reaching impact on the lives of virtually all other living organisms on the planet, particularly on those in the marine environment. the world that we have become accustomed to has been and is continuously shaped by the workings of these miniscule yet vital “plants of the sea”.

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