Transversal oscillations of a leaded rope
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An elastic cord of negligible mass is at equilibrium, tense with a force between two fixed points and of a distance from each other.
The rope supports, regularly spaced, three weights , and of the same mass .
Neglecting the weight of the leads, each string section of length is in the initial equilibrium state is characterized by the stiffness of the string and natural length .
Small transverse movements of the leads are studied ; ordinate of lead is at time ( ).
It is assumed that its abscissa is constantly equal to .
Establish a differential system of the second order relative to the studied movement.
We seek the solution of type (all weights vibrating in phase with the same frequency).
Determine the possible values of for such movements (proper modes of the system).
The linear system is obtained :
This system has nontrivial solutions if the determinant is zero :
We then obtain three angular frequencies of the oscillations :
Simulation : Amazing pendulum wave effect
Do you know how to build this wave pendulum ?