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Consider a mass m at the end of an ideal spring with spring constant k (so the spring force is Fspring = -kx). In introductory physics you found that the mass underwent oscillatory motion with angular frequency wo = accurate by adding in the effects of damping, a type of friction force (it removes mechanical energy from the system). The damping force is proportional to the velocity, Fdamping = -2myv, where y 2 O is the damping coefficient. Therefore, Vk/m > 0. Now let's make the physical system a little more physically -mwzx – 2myv. (la) та — Our problem is ironically made simply by first complexifying the problem. That is, we the real position x E R with a "complex position" z E Z such that Re z = x, as we did with the undamped harmonic oscillator in lecture. We intuitively know or can guess what our damped spring will behave :- it should oscillate with a decreasing amplitude. Therefore, we will make an ansatz (a guess) at what the complex position will look like, like - z(t) = Aet (1b) where 2 = w + iT is a complex number with units of [N] = s1, with real and imaginary parts w and I and A = Aoe?9o is some initial complex amplitude. (a) Express z(t) = Ae²t in polar form r(t)e*®(t), with r(t) and ¢(t) real functions expressed in terms of the real constants w, T, Ao, and p0. Using your physical intuition (think about the actual motion of a damped oscillator or think about energy), what condition must I' satisfy for our solution to be physically reasonable?

Consider a mass m at the end of an ideal spring with spring constant k (so the spring force is Fspring = -kx). In introductory physics you found that the mass underwent oscillatory motion with angular frequency wo = accurate by adding in the effects of damping, a type of friction force (it removes mechanical energy from the system). The damping force is proportional to the velocity, Fdamping = -2myv, where y 2 O is the damping coefficient. Therefore, Vk/m > 0. Now let's make the physical system a little more physically -mwzx – 2myv. (la) та — Our problem is ironically made simply by first complexifying the problem. That is, we the real position x E R with a "complex position" z E Z such that Re z = x, as we did with the undamped harmonic oscillator in lecture. We intuitively know or can guess what our damped spring will behave :- it should oscillate with a decreasing amplitude. Therefore, we will make an ansatz (a guess) at what the complex position will look like, like - z(t) = Aet (1b) where 2 = w + iT is a complex number with units of [N] = s1, with real and imaginary parts w and I and A = Aoe?9o is some initial complex amplitude. (a) Express z(t) = Ae²t in polar form r(t)e*®(t), with r(t) and ¢(t) real functions expressed in terms of the real constants w, T, Ao, and p0. Using your physical intuition (think about the actual motion of a damped oscillator or think about energy), what condition must I' satisfy for our solution to be physically reasonable?

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Consider a mass m at the end of an ideal spring with spring constant k (so the spring force is Fspring = -kx). In introductory physics you found that the mass underwent oscillatory motion with angular frequency wo = accurate by adding in the effects of damping, a type of friction force (it removes mechanical energy from the system). The damping force is proportional to the velocity, Fdamping = -2myv, where y 2 O is the damping coefficient. Therefore, Vk/m > 0. Now let's make the physical system a little more physically -mwzx – 2myv. (la) та — Our problem is ironically made simply by first complexifying the problem. That is, we the real position x E R with a "complex position" z E Z such that Re z = x, as we did with the undamped harmonic oscillator in lecture. We intuitively know or can guess what our damped spring will behave :- it should oscillate with a decreasing amplitude. Therefore, we will make an ansatz (a guess) at what the complex position will look like, like - z(t) = Aet (1b) where 2 = w + iT is a complex number with units of [N] = s1, with real and imaginary parts w and I and A = Aoe?9o is some initial complex amplitude. (a) Express z(t) = Ae²t in polar form r(t)e*®(t), with r(t) and ¢(t) real functions expressed in terms of the real constants w, T, Ao, and p0. Using your physical intuition (think about the actual motion of a damped oscillator or think about energy), what condition must I' satisfy for our solution to be physically reasonable?
Transcribed Image Text:Consider a mass m at the end of an ideal spring with spring constant k (so the spring force is Fspring = -kx). In introductory physics you found that the mass underwent oscillatory motion with angular frequency wo = accurate by adding in the effects of damping, a type of friction force (it removes mechanical energy from the system). The damping force is proportional to the velocity, Fdamping = -2myv, where y 2 O is the damping coefficient. Therefore, Vk/m > 0. Now let's make the physical system a little more physically -mwzx – 2myv. (la) та — Our problem is ironically made simply by first complexifying the problem. That is, we the real position x E R with a "complex position" z E Z such that Re z = x, as we did with the undamped harmonic oscillator in lecture. We intuitively know or can guess what our damped spring will behave :- it should oscillate with a decreasing amplitude. Therefore, we will make an ansatz (a guess) at what the complex position will look like, like - z(t) = Aet (1b) where 2 = w + iT is a complex number with units of [N] = s1, with real and imaginary parts w and I and A = Aoe?9o is some initial complex amplitude. (a) Express z(t) = Ae²t in polar form r(t)e*®(t), with r(t) and ¢(t) real functions expressed in terms of the real constants w, T, Ao, and p0. Using your physical intuition (think about the actual motion of a damped oscillator or think about energy), what condition must I' satisfy for our solution to be physically reasonable?
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