Unwind the paper. a large 16.0-kg roll of paper with radius r = 18.0 cm rests against the wall and is held in place by a bracket attached to a rod through the center of the roll. the rod turns without friction in the bracket, and the moment of inertia of the paper and rod about the axis is 0.260 kgm2 . the other end of the bracket is attached by a frictionless hinge to the wall such that the bracket makes an angle of 30.0◦ with the wall. the weight of the bracket is negligible. the coefficient of kinetic friction between the paper and the wall is µk = 0.25. a constant vertical force f = 60.0 n is applied to the paper, and the paper begins to unroll. what is the initial magnitude of (a) the force that the rod exerts on the paper as it unrolls; (b) the angular acceleration of the roll? (initial because we are calculating this for when the radius is 18 cm and moment of inertia is 0.260 kgm2 .

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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