Period for harmonic motion (horizontal)

AI Thread Summary
The discussion focuses on determining the relationship between the period of harmonic motion (T) for a cantilevered rod with a hanging mass and its length (x). Participants explore relevant equations that describe the frequency of such a system, specifically referencing the concept of a cantilever with an end mass. The key factors influencing the period include the length of the free-hanging portion of the rod and the mass attached to it. The conversation highlights the need for understanding the dynamics of cantilever beams in relation to harmonic motion. Overall, the thread seeks to clarify the mathematical principles governing this physical scenario.
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Consider a light flexible rod placed on a horizontle table with part of the rod (length say "x") hanging freely (ie without support of the table) see attachment for clarity

A mass is also hung from the rod t one end.

Are there any equations that relate the Period T of the end of the rod to the length "x" , and the mass(hanging) ?
 

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Look up "cantilever with end mass frequency".
 

Thread 'Derivation of Hamilton's Principle: Questions'

I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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