A damped harmonic oscillator of mass m it released at time t=0 and displaced by a distance xo. Show motion is described by the following second order differential equation mẍ + λẋ + m(ω_0^2)x=0; with ω_0^2=k/m For a lightly damped (λ<< ω_0) oscillator the quality factor (Q value), is defined as Q = (Energy (E) stored in the simple harmonic system)/(energy lost(E_lost) per radian of oscillation) Show that Q = ω_0/2λ Hint: Ue the trial solution x(t) =A exp(-γt)cos(ωt-δ) Find E_lost = (1/2)kA^2=((mω_0^2)/2)(A^2)exp(-γt) A radian corresponds to the time difference t = 1/ω_0
A damped harmonic oscillator of mass m it released at time t=0 and displaced by a distance xo. Show motion is described by the following second order differential equation mẍ + λẋ + m(ω_0^2)x=0; with ω_0^2=k/m For a lightly damped (λ<< ω_0) oscillator the quality factor (Q value), is defined as Q = (Energy (E) stored in the simple harmonic system)/(energy lost(E_lost) per radian of oscillation) Show that Q = ω_0/2λ Hint: Ue the trial solution x(t) =A exp(-γt)cos(ωt-δ) Find E_lost = (1/2)kA^2=((mω_0^2)/2)(A^2)exp(-γt) A radian corresponds to the time difference t = 1/ω_0
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A damped harmonic oscillator of mass m it released at time t=0 and displaced by a distance xo. Show motion is described by the following second order differential equation
mẍ + λẋ + m(ω_0^2)x=0; with ω_0^2=k/m
For a lightly damped (λ<< ω_0) oscillator the quality factor (Q value), is defined as Q = (Energy (E) stored in the simple harmonic system)/(energy lost(E_lost) per radian of oscillation)
Show that Q = ω_0/2λ
Hint: Ue the trial solution x(t) =A exp(-γt)cos(ωt-δ)
Find E_lost = (1/2)kA^2=((mω_0^2)/2)(A^2)exp(-γt)
A radian corresponds to the time difference t = 1/ω_0
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