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# Supplementary Problems for Topics III

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Supplementary Problems for Topics III Solutions 1 An empty steel drum of mass M height h and cross section area A is floating partially submerged as shown in water The bottom of the drum is at depth d below the surface Express all answers in terms of the given quantities g and the density of water a What is d b Now an external force of magnitude F is exerted on the top of the drum pushing it further down into the water by an extra distance x0 Express F in terms of x0 and the other quantities d c Prove that if the external force is suddenly removed the drum will execute simple harmonic motion in vertical oscillations How does the net force depend on the vertical displacement x of the drum from its equilibrium position d Find the angular frequency of those oscillations a The buoyant force must equal the weight of the drum so Mg Ad g or d M A b The applied force must be balanced by the extra buoyant force so F Ax0 g c If the drum is distance x deeper than its equilibrium depth the extra buoyant force is Axg upward opposite to the displacement Thus we have a restoring force opposite to the displacement proportional to the amount of the displacement That is what give rise to SHM d The standard form for the restoring force is F M 2 x Since we have F Agx we see that 2 Ag M 2 An aspirator is a device that mixes small amounts of a fluid such as liquid fertilizer with water so that it can be sprayed over a large area Shown is such a device Water from a hose on the left flows through a constriction where a vertical tube introduces the fluid at negligible speed into the rapid v0 a A water flow The height through which the fluid must rise is h as h shown If the fluid water mixture exits into the air at speed v0 what must be the minimum ratio A a of the cross section areas of the two regions Assume the fluid and water have the same density Find the water pressure in the constriction a Use Bernoulli comparing the point in the constriction with the point where the water enters the

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