A cube of density ρc floats in a liquid of density ρ1 as shown in the figure, At rest, an amount

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A cube of density ρc floats in a liquid of density ρ1 as...

A cube of density $\rho_{c}$ floats in a liquid of density $\rho_{1}$ as shown in the figure, At rest, an amount $h$ of the cube's height is submerged in liquid. If the cube is pushed down, it bobs up and down like a spring and oscillates about its equilibrium position. Show that the frequency of its oscillations is given by $f=(2 \pi)^{-1} \sqrt{g / h}$.

Key Concepts

Restoring Force in Oscillatory Systems

When the cube is displaced from its equilibrium position, the change in the displaced volume alters the buoyant force. The resulting net force acts as a restoring force, which for small displacements is directly proportional to the displacement, analogous to the force in a spring system which obeys Hooke’s law.

Small Oscillation Approximation

This approximation assumes that the displacements from the equilibrium position are sufficiently small so that the restoring force remains linearly proportional to the displacement. This linearization is key in simplifying the analysis and directly applying the equations of simple harmonic motion to derive the frequency of oscillations.

Simple Harmonic Motion (SHM)

Given that the restoring force is proportional to displacement, the system can be modeled as undergoing simple harmonic motion. This allows the oscillatory behavior to be described by standard SHM equations, leading to the derivation of an oscillation frequency based on the effective spring constant and mass.

Equilibrium of Floating Bodies

At equilibrium, the weight of the object is exactly balanced by the buoyant force. For the cube, the equilibrium condition determines that a specific fraction of its height, h, is submerged, making this state the reference point for analyzing any perturbations.

Buoyancy (Archimedes' Principle)

This principle states that any object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. In the floating cube problem, it is this buoyant force that balances the weight of the cube at equilibrium and defines the submerged depth.

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