Consider the equations x a(θ-sin θ ) and z#a(1+cos0) where a is a constant. the z(x) curve is called a cycloid, and is the shape traced out by a point on the rim of a wheel that rolls on a flat surface. most often, both equations have a minus sign, which corresponds to the cusps (the "pointy corners') of the cycloid being at the bottom; however, we are interested in the upside-down version of the cycloid as described by the above form of the equations. consider a bead of mass m that slides without friction on a curved wire that follows one period of an "upside-down" cycloid. gravity acts along the z axis. (i) write down the lagrangian in terms of 0. (i) find the equation of motion (eom) for the bead. ) show that d2u/dt2+gu/4a -o, where u - cos( /2), leads to the same eom as obtained in part (i) above. (iv) show that the bead can oscillate, and that the motion in terms of 0, assuming we are not dealing with the limit of small oscillations, is not simple harmonic motion. (v) find the period for these oscillations

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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