Coupled oscillators

Coupled oscillators are one of the most common physical systems in nature. For example, atoms in a crystal can be modeled as coupled oscillators. Moreover, the Taylor expansion of any potential around its minimum gives a term that produces oscillations, so an oscillatory motion is inherent to natural systems, much more than a linear motion. In this problem, you will completely describe the simplest coupled system of two masses.

Two equal masses m=1 kgm = 1~\mbox{kg} are connected with a spring and then connected with walls. Initially, the masses are at rest. Then the first mass is displaced to the right for d=10 cmd = 10~\mbox{cm} and released.

What's the absolute value of the distance from the equilibrium position ((in cm)\text{cm}) of the second mass after t=5 s?t' = 5~\mbox{s}?

Details and Assumptions:

  • All the springs have elasticity coefficient k=16 N/m.k = 16~\mbox{N/m}.
  • Ignore gravity and any kind of friction.
  • You don't have to do extensive calculations to deduce the normal mode frequencies of the system.

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