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Vertical springs in series or parallel

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Prove that when two springs are attached one at the end of the other, the coefficient of the final spring becomes 1 / (1/k1 + 1/k2 ) where k1 and k2 are the coefficient of the two individual springs, the direction of the springs is VERTICAL not horizontal. Then consider two systems of springs, one in which a mass m is attached to the end of two springs which are placed one at the end of the other, another in which m is attached to the two springs in parallel (each spring has coefficient respectively k1 and k2). Find the frequency of oscillations of mass m in the two cases.

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Solution Summary

This solution is provided in 180 words. It uses equations for total displacement to find the equivalent coefficient and frequency of oscillations in the system.

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1. Two springs in Series (attached one at the end of the other):

F = Keq*x where Keq = Equivalent Coefficient

Apply a force of F to the springs in series.

F1 = k1*x1
F2 = ...

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