Recent content by brettng

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    Textbook 'The Physics of Waves': Varying Force Amplitude

    Thank you for your reply TSny! In this case, do you mean that I could try the superpostion solutions with ##Ae^{-i (\omega_0 + \delta) t}## and ##Be^{-i (\omega_0 - \delta) t}##?
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    Textbook 'The Physics of Waves': Varying Force Amplitude

    Reference textbook “The Physics of Waves” in MIT website: https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/ Chapter 2 - Problem 2.3 [Page 52] (see screen capture below) Question: In Problem 2.3, I have proved the hint equation, and it leads to the complex...
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    Textbook 'The Physics of Waves': Linearity of Forced Oscillator

    Thank you so much for your help!!
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    Textbook 'The Physics of Waves': Linearity of Forced Oscillator

    So, your demonstration is based on dimensional analysis; while my logical flow follows linearity. Am I understand correctly?
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    Textbook 'The Physics of Waves': Lifetime of SHM Oscillators

    Thank you so much for your help!!!!!!
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    Textbook 'The Physics of Waves': Calculating Work Done by Force

    Thank you so much for your help!!!!!!
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    Textbook 'The Physics of Waves': Reason to force us to consider complex solution for harmonic motion?

    Thank you so much for your help!!!!!!
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    Textbook 'The Physics of Waves': Linearity of Forced Oscillator

    May I confirm if I can have the logical flow as follows: 1) Set ##z(t) = F_0 u(t)## 2) The equation of motion becomes $$F_0 \left( u'' + \Gamma u’ + \omega_0^2 u\right) = \frac {F_0 e^{-i \omega_d t}} {m}$$ $$\left( u'' + \Gamma u’ + \omega_0^2 u\right) = \frac {e^{-i \omega_d t}} {m}$$ 3)...
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    Textbook 'The Physics of Waves': Linearity of Forced Oscillator

    Reference textbook “The Physics of Waves” in MIT website: https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/ Chapter 2 - Problem 2.2 [Page 51] (see attached file) Question: In the content of Page 43 (see attached file), it also states that the amplitude of...
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    Textbook 'The Physics of Waves': Lifetime of SHM Oscillators

    I understand that lifetime means that the oscillation amplitude reduces by a factor of 2. So, why can’t we still consider the “lifetime” of overdamped and critically damped oscillators by using the same definition?
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    Textbook 'The Physics of Waves': Lifetime of SHM Oscillators

    Reference textbook “The Physics of Waves” in MIT website: https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/ Chapter 2 - Section 2.3.2 [Page 47] (see attached file) Question: In the content, it states that the lifetime of the state in free oscillation is of...
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    Textbook 'The Physics of Waves': Calculating Work Done by Force

    Reference textbook “The Physics of Waves” in MIT website: https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/ Chapter 2 - Section 2.3.1 [Page 45] (see attached file) Question: In the content, it states that we need to use real force and real displacement...
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    Textbook 'The Physics of Waves': Derive Complex Solution Form for SHM

    Thank you so much for your help!!!!!!
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    Textbook 'The Physics of Waves': Derive Complex Solution Form for SHM

    Could you elaborate more about the 1st equal sign? Because to me, I am thinking a ##t##-##a##-##z## 3D geometry (similar to ##x##-##y##-##z## plane), that ##\frac{\partial z(t+a)}{\partial a}\Biggl|_{a=0}## means the “slope in ##a## direction” of graph ##z(t+a)## evaluated at ##a=0##. However...
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    Textbook 'The Physics of Waves': Derive Complex Solution Form for SHM

    Thank you for your reply. But how do we usually write the mathematics explicitly? Differentiate ##z(t+a)=h(a)z(t)## with respect to ##a##, and set ##a=0##. $$\frac {\partial z(t+0)} {\partial a}=\frac {dh(0)} {da}z(t)$$ $$\frac {\partial z(t)} {\partial a}=Hz(t)$$ At this stage, the derivative...