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Multiple Spring system?
A light rod is supported by two springs fixed on ceiling. A mass hangs from centre of the rod with a third spring as shown in the diagram. Find the natural frequency of the system:
(i) When spring 1 and 2 share the weight equally
(ii) When the bar is assumed to stay horizontal all the time
Answer:
I am not getting the force equations of the springs. Any ideas?

Basic Assumptions
First, two previously known lemmas about multiple spring systems that you can assume to be true or derive for yourself:
(1) In parallel, so
(2) In series, so
Start with , ,
because these are in series, so . Since the load is split evenly between springs #1 and #2, we can assume .
(i) Assume so WLOG . So or . Since , . Solving for gives . By lemma 2,
(ii) By lemma 1, and by lemma 2,
In each instance of course,