A long, cylindrical rod of radius r is weighted on one end...

A long, cylindrical rod of radius $r$ is weighted on one end so that it floats upright in a fluid having a density $\rho .$ It is pushed down a distance $x$ from its equilibrium position and released. Show that the rod will execute simple harmonic motion if the resistive effects of the fluid are negligible and determine the period of the oscillations.

Key Concepts

Oscillation Period Determination

Deriving the period involves relating the restoring force (derived from the buoyant force's variation with displacement) to the mass of the object using Newton's second law. This leads to a second-order differential equation typical of harmonic oscillators, from which the angular frequency and period can be calculated.

Restoring Force and Its Proportionality

For an object to undergo simple harmonic motion, the net force acting on it must be directly proportional to the displacement from its equilibrium position and directed towards that position. In this context, the change in buoyant force as the object is pushed down creates a restoring force that obeys this linear relation.

Simple Harmonic Motion (SHM)

SHM is characterized by an acceleration proportional to and opposite in direction to the displacement from equilibrium. This fundamental concept allows the system—when resistive effects are negligible—to oscillate in a predictable periodic manner, making it possible to derive properties such as the period of oscillation.

Buoyancy and Archimedes' Principle

This concept explains that a floating object is subject to an upward buoyant force equal to the weight of the displaced fluid. It is fundamental when analyzing the equilibrium and restoring forces acting on objects immersed in a fluid.

Equilibrium and Stability

Understanding the equilibrium position of a floating body involves recognizing that the upward buoyant force and the downward gravitational force are balanced. When the object is displaced from this equilibrium, a restoring force arises, driving the object back toward equilibrium, which is essential for the analysis of oscillatory behavior.

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