Buoyant force varies as the displacement of the rod is best...

Buoyant force varies as the displacement of the rod is best represented by (A) (B) (C) (D)Passage 3 A wire is bend at an angle $\theta . \mathrm{A}$ rod of mass $m$ can slide along the bend wire without friction as shown in Fig. $8.46$. If a soap film is maintained in the frame and frame is kept in a vertical position, the rod is in equilibrium.

Buoyant force varies as the displacement of the rod is best represented by
(A)
(B)
(C)
(D)Passage 3 A wire is bend at an angle $\theta . \mathrm{A}$ rod of mass $m$ can slide along the bend wire without friction as shown in Fig. $8.46$. If a soap film is maintained in the frame and frame is kept in a vertical position, the rod is in equilibrium.

Key Concepts

Simple Harmonic Motion (SHM)

When a system exhibits a restoring force directly proportional to its displacement from equilibrium, the motion can be modeled as simple harmonic motion. In fluid systems where buoyant force changes linearly with displacement (under small displacement approximations), the resulting motion of the object (such as the rod) may be analogous to SHM, allowing for predictions about its oscillatory behavior.

Archimedes' Principle

This principle states that any object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. Understanding this concept is essential since, as the object (or rod) is displaced within the fluid (or any medium producing a similar effect, such as a soap film), the volume of fluid displaced changes, directly affecting the buoyant force acting on it.

Restoring Force

When an object in a fluid is displaced from its equilibrium position, the change in the volume of displaced fluid causes a change in the buoyant force. This difference between the new buoyant force and the force at equilibrium acts as a restoring force that tends to bring the object back to its original position. Recognizing this property is key to analyzing how the buoyant force varies with displacement.

Small Displacement Approximation

For minute departures from equilibrium, many non?linear force variations can often be approximated as linear. In the case of buoyancy, when the displacement is small, the change in volume—and hence the buoyant force—can be assumed to vary in a nearly linear manner with displacement. This approximation simplifies the analysis, often leading to equations similar to those for simple harmonic motion.

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