A 950 -kg cylindrical can buoy floats vertically in salt...

A 950 -kg cylindrical can buoy floats vertically in salt water. The diameter of the buoy is 0.900 $\mathrm{m}$ . (a) Calculate the additional distance the buoy will sink when a $70.0-\mathrm{kg}$ man stands on top. (Use the expression derived in part (b) of Problem $14.66 .$ ) (b) Calculate the period of the resulting vertical SHM when the man dives off. (Use the expression derived in part (c) of Problem 14.66 , and as in that problem, you can ignore the damping due to fluid friction.)

Key Concepts

Restoring Force and Hooke's Law

The variation in buoyant force with displacement in a fluid can be linearized for small displacements, resulting in a restoring force that acts proportionally to the displacement. This behavior mirrors Hooke's law for springs, allowing the derivation of an effective spring constant and subsequently the calculation of the period of oscillations.

Simple Harmonic Motion (SHM) in Floating Bodies

When a floating object is perturbed from its equilibrium, it can perform small oscillations that resemble simple harmonic motion. The restoring force in these cases is provided by the change in buoyancy, and the motion can be approximated as SHM under conditions of small displacement.

Buoyancy and Equilibrium

Buoyancy refers to the upward force exerted by a fluid that opposes the weight of an immersed object. A floating object reaches equilibrium when its weight is exactly balanced by the buoyant force, meaning the weight of fluid displaced equals the object's weight. Any additional load shifts this equilibrium by increasing the volume of fluid displaced.

Archimedes' Principle

This principle holds that any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the displaced fluid. It is fundamental to understanding why objects float or sink and how additional weight affects the submerged volume of a floating body.

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