Oscillations and Waves

Forced, Damped oscillations and Resonance

  • The oscillations of a body whose amplitude goes on decreasing with time are defined as damped oscillations where F = −bV where 'b' is damping constant.
  • In forced oscillations the amplitude of oscillations decreases exponentially due to damping force like frictional force , viscous force  and  hysteresis etc.

Amplitude A = xm e−γt γ = b/2m

  • The oscillations in which a body oscillates under the influence of an external force are known as forced oscillations.

F = −bv −Kx + Fm cos w mt

  • When the frequency of the external periodic forces is equal to the natural frequency of oscillator the amplitude increases to maximum value the phenomenon is called resonance

\tt T = 2 \pi \sqrt{\frac{M}{K}}

View the Topic in this video From 0:26 To 12:05

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1. The displacement of the oscillator is given by
x(t) = Aebt/2m cos (ω't + Φ)
where, ω' the angular frequency of the damped oscillator is given by
\omega' = \sqrt{\frac{k}{m} - \frac{b^{2}}{4m^{2}}}

2. The mechanical energy E of the damped oscillator is given by
E(t) = \frac{1}{2}kA^{2} e^{-bt/m}