A flat-bottomed cylindrical test tube is weighted with steel shot so that it floats vertically in a beaker of water. The cross-sectional area of the tube is AAA, the density of water is ρ\rhoρ, and the acceleration due to gravity is ggg. In equilibrium, the submerged length of the tube is ddd. When the tube is pushed downwards through a displacement yyy and released, the net restoring force FFF acting on the tube is given by
F=Aρgy F = A\rho g y F=AρgyShow that the tube undergoes simple harmonic motion (SHM) when it is released.
The time period TTT of the vertical oscillations is given by
T=2πdg T = 2\pi\sqrt{\frac{d}{g}} T=2πgdFor a test tube with a submerged depth of d=12 cmd = 12\text{ cm}d=12 cm that is pushed down by 6.0 mm6.0\text{ mm}6.0 mm 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 marine survey vessel floating in the ocean can be modeled by the floating tube. The vessel undergoes vertical heave oscillations in response to waves. The vessel is moving steadily at 6.4 m s−16.4\text{ m s}^{-1}6.4 m s−1 relative to the seabed in the same direction as continuous waves of wavelength 110 m110\text{ m}110 m and speed 13.2 m s−113.2\text{ m s}^{-1}13.2 m s−1. The natural frequency of heave oscillations of the vessel is 0.12 Hz0.12\text{ Hz}0.12 Hz. The captain has two options to minimize the heave oscillations for a critical scientific measurement: * Option A: Stop the vessel's engines and drift (remaining stationary relative to the seabed). * Option B: Continue to sail at 6.4 m s−16.4\text{ m s}^{-1}6.4 m s−1 in the same direction as the waves. Deduce which is the better option. Support your answer with calculations of the wave encounter frequencies.
