A large cylindrical spar buoy, used for offshore wind telemetry, floats vertically in seawater. The cross-sectional area of the buoy is AAA, the density of seawater is ρs\rho_sρs, and the acceleration due to gravity is ggg. In equilibrium, the draft (submerged length) of the buoy is HHH. When the buoy is displaced vertically through a displacement zzz and released, the net restoring force FFF acting on the buoy is given by
F=−Aρsgz F = -A\rho_s g z F=−AρsgzShow that the buoy undergoes simple harmonic motion (SHM) when it is released.
The time period TTT of the vertical heave oscillations is given by
T=2πHg T = 2\pi\sqrt{\frac{H}{g}} T=2πgHFor a spar buoy with an equilibrium draft of H=16 mH = 16\text{ m}H=16 m that is pushed down by 0.50 m0.50\text{ m}0.50 m and released, calculate its maximum acceleration. Use g=9.81 m s−2g = 9.81\text{ m s}^{-2}g=9.81 m s−2.
Explain what is meant by resonance.
A research catamaran floating in the ocean can be modeled by a similar floating system. The vessel undergoes vertical heave oscillations in response to wave swell. The natural frequency of heave oscillations of the catamaran is 0.25 Hz0.25\text{ Hz}0.25 Hz. Continuous waves of wavelength 80 m80\text{ m}80 m propagate across the sea surface at a speed of 15.0 m s−115.0\text{ m s}^{-1}15.0 m s−1. The captain has two options to minimize the heave oscillations for a critical sonar measurement:
Deduce which is the better option. Support your answer with calculations of the wave encounter frequencies.
