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Physics LibreTexts

5.5: Exercises

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Exercise 5.5.1

In Section 5.2, we encountered the complex frequencies ω±=iγ±ω20γ2. For fixed ω0 and ω0>γ (under-damping), prove that ω± lie along a circular arc in the complex plane.

Exercise 5.5.2

Derive the general solution for the critically damped harmonic oscillator, Eq. (5.3.16), by following these steps:

  1. Consider the complex ODE, in the under-damped regime ω0>γ. We saw in Section 5.3 that the general solution has the form z(t)=ψ+exp[(γiω20γ2)t]+ψexp[(γ+iω20γ2)t] for some complex parameters ψ+ and ψ. Define the positive parameter ε=ω20γ2. Re-write z(t) in terms of γ and ε (i.e., eliminating ω0).

  2. The expression for z(t) is presently parameterized by the independent parameters ψ+, ψ, ε, and γ. We are free to re-define the parameters, by taking α=ψ++ψβ=iε(ψ+ψ). Using these equations, express z(t) using a new set of independent complex parameters, one of which is ε. Explicitly identify the other independent parameters, and state whether they are real or complex.

  3. Expand the exponentials in z(t) in terms of the parameter ε. Then show that in the limit ε0, z(t) reduces to the critically-damped general solution (5.3.16).

Exercise 5.5.3

Repeat the above derivation for the critically-damped solution, but starting from the over-damped regime γ>ω0.

Exercise 5.5.4

Let z(t) be a complex function of a real input t, which obeys the differential equation dzdt=i(ω1iγ)z(t), where ω1 and γ are real. Find the general solution for z(t), and hence show that z(t) satisfies the damped oscillator equation [d2dt2+2γddt+ω20]z(t)=0 for some ω20. Finally, show that this harmonic oscillator is always under-damped.

Answer

The general solution is z(t)=Aexp[i(ω1iγ)t]. It can be verified by direct substitution that this is a solution to the differential equation. It contains one free parameter, and the differential equation is first-order, so it must be a general solution. Next, d2zdt2+2γdzdt=(i)2(ω1iγ)2z(t)2iγ(ω1iγ)z(t)=[ω21+γ2+2iγω12iγω12γ2)]z(t)=(ω21+γ2)z(t). Hence, z(t) obeys a damped harmonic oscillator equation with ω20=ω21+γ2. This expression for the natural frequency ensures that ω20>γ2 (assuming the parameters γ and ω1 are both real); hence, the harmonic oscillator is always under-damped.


This page titled 5.5: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

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