Why Do mω²xa and 2kxa Both Act in Normal Modes Oscillations?

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The discussion revolves around the confusion regarding the forces acting in normal mode oscillations, specifically questioning the roles of mω²xa and 2kxa. It is clarified that both forces are necessary because they account for different aspects of the system's dynamics. The contributor acknowledges a misunderstanding and attributes it to fatigue, indicating that the complexities of the equations can be challenging. The mention of gravity suggests that it plays a significant role in the overall behavior of the oscillating system. Understanding these forces is crucial for properly analyzing normal mode oscillations.
LCSphysicist
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Homework Statement
All below
Relevant Equations
All below
1594179721591.png


I am not sure if i get this part of a book i am using:

1594179748679.png


Why are the mwo²xa acting too? Is not 2kxa enough?
 
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Ignore it, i made a confusion :P need to sleep.
 
LCSphysicist said:
Homework Statement:: All below
Relevant Equations:: All below

View attachment 266042

I am not sure if i get this part of a book i am using:

View attachment 266043

Why are the mwo²xa acting too? Is not 2kxa enough?
It is due to gravity.
 
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The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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