Buoyancy as a restoring force Imagine a cylinder of length

Chapter , Problem 50

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Buoyancy as a restoring force Imagine a cylinder of length L and cross-sectional area A floating partially submerged in a calm lake (see figure). Assume that the density of the water is rw = 1000 kg>m3 and the density of the cylinder is r 6 1000 kg>m3. Archimedes principle states that the cylinder T experiences an upward buoyant force equal to the weight of the water displaced by the cylinder. Equilibrium y _ 0 y(t) L a. As shown in the figure, assume that when floating at rest a fraction y*>L of the cylinder is submerged. Note that the weight of the cylinder is mg = rALg and the weight of the displaced water is y*rwAg. Conclude that the fraction of the cylinder that is submerged is the ratios of the densities; that is, y* L = r rw . b. Let y = 0 correspond to the level of the bottom of the cylinder at equilibrium. Now suppose that the cylinder is pushed down from its equilibrium position and released. Let y1t2 be the position of the bottom of the cylinder t seconds after it is released, where y increases in the downward direction. Applying Newtons second law my_ = Fext, explain why the buoyant force is Fext = -rwAyg (in addition to the buoyant force that maintains the equilibrium). c. Conclude that the cylinder undergoes undamped oscillations governed by the equation y_ = -v0 2 y, where v0 2 = rwg rL . d. What is the period of the oscillations? How does the period vary with the length of the cylinder? Is the period greater when r _ rw or when r V rw? Explain your answer. e. According to this model, what is the period of the oscillation when r = rw? Describe this situation physically.