Recent content by brettng

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}##?

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...

Thank you so much for your help!!

So, your demonstration is based on dimensional analysis; while my logical flow follows linearity. Am I understand correctly?

Thank you so much for your help!!!!!!

Thank you so much for your help!!!!!!

Thank you so much for your help!!!!!!

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)...

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...

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?

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...

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...

Thank you so much for your help!!!!!!

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...

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...