Question

12. The motion of an overdamped harmonic oscillator subject to an impulsive force is described by the equation $\ddot{x} + 2\gamma \dot{x} + p^2 x = \delta(t)$ for a function $x(t)$ and constants $\gamma > p > 0$. Given that $x = 0$ for $t < 0$, show by Fourier transform methods that for $t > 0$ \begin{equation} x(t) = \frac{e^{-\gamma t}}{\sqrt{\gamma^2 - p^2}} \sinh(\sqrt{\gamma^2 - p^2}t) \end{equation}

          12. The motion of an overdamped harmonic oscillator subject to an impulsive force is described by the equation $\ddot{x} + 2\gamma \dot{x} + p^2 x = \delta(t)$ for a function $x(t)$ and constants $\gamma > p > 0$. Given that $x = 0$ for $t < 0$, show by Fourier transform methods that for $t > 0$ \begin{equation*} x(t) = \frac{e^{-\gamma t}}{\sqrt{\gamma^2 - p^2}} \sinh(\sqrt{\gamma^2 - p^2}t) \end{equation*}

12. The motion of an overdamped harmonic oscillator subject to an impulsive force is described by the equation ẍ + 2γẋ + p^2 x = δ(t) for a function x(t) and constants γ > p > 0. Given that x = 0 for t < 0, show by Fourier transform methods that for t > 0
x(t) = (e^-γ t)/(√(γ^2 - p^2))sinh(√(γ^2 - p^2)t)

Added by Jennifer S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Surendra Kumar

10/14/2021

Step 1

Step 1: Take the Fourier transform of the given differential equation x^(¨)+2\gamma x^(˙)+p^(2)x=\delta (t) to obtain an algebraic equation in the frequency domain. Show more…

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The motion of an overdamped harmonic oscillator subject to an impulsive force is described by the equation x^(¨)+2gamma x^(˙)+p^(2)x=delta (t) for a function x(t) and constants gamma >p>0. Given that x=0 for t<0, show by Fourier transform methods that for t>0 x(t)=(e^(-gamma t))/(sqrt(gamma ^(2)-p^(2)))sinh(sqrt(gamma ^(2)-p^(2))t) 12. The motion of an overdamped harmonic oscillator subject to an impulsive force is described by the equation i+ 2i + px = &(t) for a function x(t) and constants > p > 0. Given that x =0 for t < 0, show by Fourier transform methods that for O<7 -Y c(t) sinh (V2-p2 t)