Mechanical and Electrical Oscillations

Oscillation of a Mass-Spring System Compared with Electrical Oscillation in an L-C Circuit

Mass-Spring System

  • Kinetic energy = \frac12mv_x^2
  • Potential energy = \frac12kx^2
  • \frac12mv_x^2+\frac12kx^2=\frac12kA^2
  • v_x = \pm\sqrt{\frac km}\sqrt{A^2-x^2}
  • v_x=\frac{dx}{dt}
  • \omega=\sqrt{\frac km}
  • x=A\cos(\omega t+\phi)

Inductor-Capacitor Circuit

  • Magnetic energy = \frac12Li^2
  • Electrical energy = \frac{q^2}{2C}
  • \frac12Li^2+\frac{q^2}{2C}=\frac{Q}{2C}
  • i = \pm\sqrt{\frac1{LC}}\sqrt{Q^2-q^2}
  • i=\frac{dq}{dt}
  • \omega=\sqrt{\frac1{LC}}
  • q=Q\cos(\omega t+\phi)
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