How Do Coupled Adiabatic Pendula Behave?

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The discussion focuses on the behavior of coupled adiabatic pendula, where one pendulum is adiabatically pulled up with a varying frequency while being connected to a normal pendulum via a spring. Participants are tasked with deriving the equations of motion for this coupled system and determining the instantaneous frequency supported by it at any given time. The concept of an adiabatic pendulum is clarified, emphasizing its unique characteristics compared to standard pendulums. The interaction between the two pendula, influenced by the spring constant, is a key aspect of the analysis. Understanding these dynamics is essential for solving the homework problems effectively.
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Homework Statement
An adiabatic pendulum (right) is coupled via a spring with spring contant κ to a normal non-variable pendulum. The pendula have equal mass m and, initially, equal length l . The right pendulum is adiabatically pulled up with frequency ω(t)

1. Derive the equations of motion for this system
2. Which instantaneous frequency Ω(t) does the coupled system support at a given time t?
Relevant Equations
-
Homework Statement: An adiabatic pendulum (right) is coupled via a spring with spring contant κ to a normal non-variable pendulum. The pendula have equal mass m and, initially, equal length l . The right pendulum is adiabatically pulled up with frequency ω(t)

1. Derive the equations of motion for this system
2. Which instantaneous frequency Ω(t) does the coupled system support at a given time t?
Homework Equations: -

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What is an adiabatic pendulum?
 
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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