# Consider a floating cylindrical buoy with radius r, height

**Chapter , Problem 10**

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Consider a floating cylindrical buoy with radius r, height h, and uniform density 5 0:5 (recall that the density of water is 1 g=cm3). The buoy is initially suspended at rest with its bottom at the top surface of the water and is released at time t D 0. Thereafter it is acted on by two forces: a downward gravitational force equal to its weight mg D r2hg and (by Archimedes principle of buoyancy) an upward force equal to the weight r2xg of water displaced, where x D x.t / is the depth of the bottom of the buoy beneath the surface at time t (Fig. 3.4.12). Assume that friction is negligible. Conclude that the buoy undergoes simple harmonic motion around its equilibrium position xe D h with period p D 2ph=g. Compute p and the amplitude of the motion if D 0:5 g=cm3, h D 200 cm, and g D 980 cm=s2. W

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