Determine the period of the oscillations

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Homework Help Overview

The discussion revolves around a problem involving a long cylindrical rod floating in a fluid, which is displaced from its equilibrium position and is expected to exhibit simple harmonic motion (SHM). Participants are tasked with demonstrating this behavior and determining the period of oscillations, focusing on the forces acting on the rod, including buoyancy and gravitational force.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to analyze the forces acting on the rod and how they relate to acceleration and displacement. Some participants question how to demonstrate that acceleration is proportional to the negative of the displacement, while others seek clarification on the steps necessary for the mathematical derivation.

Discussion Status

The discussion is ongoing, with participants exploring the relationship between force, displacement, and acceleration. Some guidance has been offered regarding the need to prove a specific relationship (F = -kx) to establish SHM, but there is no explicit consensus on the steps to take or the mathematical approach to use.

Contextual Notes

Participants are navigating the complexities of the problem without complete information on the mathematical steps required for the write-up. There is an emphasis on understanding the underlying physics and mathematical relationships involved in SHM.

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1) A long cylindrical rod of radius, r is weighted on one end so that it floats upright in a fluid having a density. 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>>

Start with the forces: mg and the "restoring" force, bouyancy. Bouyance is dependent on displacement.

mg- restoring force = mass*acceleration

where acceleration is d"x/dx".

BUT, how does that lead me to showing the rod will execute simple harmonic motion? and how do i go about determining period of oscillations from there?
 
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show that acceleration is proportional to the negative of the displacement
 
?

i'm not sure how to go about that.. show it is proportional to negative displacement? Are there are any steps for this type of mathematical writeup? any ideas or help would be appreciated. thanks again.
 
If you can prove the thatF= -kx
where F is the resaultant force on the rod x is displacement and k is a constant, then this is a SHM.
 

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