Coupled oscillators

Classical Mechanics Level 5

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 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 \(m = 1~\mbox{kg}\) 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 \(d = 10~\mbox{cm}\) and released. What's the absolute value of the distance from the equilibrium position in cm of the second mass after \(t' = 5~\mbox{s}\)?

Details and assumptions

  • All the springs have elasticity coefficient \(k = 16~\mbox{N/m}\)
  • 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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