- #1
e_mts
- 7
- 3
- Homework Statement
- For each case find the complex representation for all three quantities – displacement, velocity, and acceleration – in the form: ##Q(t) = Qe^{−i(ωt+φ)}##
- Relevant Equations
- ##e ^ {i \omega t} = Cos(\omega t) + iSin(\omega t)##
More in the attached picture
I've been trying to continue my education by self-teaching during quarantine (since I can't really go to college right now) with the MIT Opencourseware courses. I landed on one section that's got me stuck for a while which is the second part of this problem (I managed to finish the first part without any issues):
For problem d, I tried both working with ##Q(t)=Qe^{-i(\omega t + \phi)}## to make it look like d but I got as far as this (by setting ##\phi=0##): $$Q(t)=QCos(\omega t) - QiSin(\omega t)$$ which can't be simplified to look like d (as far as I can tell) because ##\beta## is a real constant (might be wrong here).
The other thing I tried to do was working with d to make it look like ##Q(t)=Qe^{-i(\omega t + \phi)}## but that leaves me with: $$\alpha \frac{e^{i\omega t}+e^{-i\omega t}}{2}-\beta \frac{e^{i\omega t}-e^{-i\omega t}}{2i}$$ after trying to simplify that I landed on: $$\frac{1}{2i}(e^{i\omega t}(\alpha i - \beta)+e^{-i\omega t}(\alpha i +\beta))$$ which doesn't look at all like what I need to get.
I'd like to get some help for this problem because I'm not even sure about what to study to start figuring out how to figure it out. It would be very appreciated, thank you.
For problem d, I tried both working with ##Q(t)=Qe^{-i(\omega t + \phi)}## to make it look like d but I got as far as this (by setting ##\phi=0##): $$Q(t)=QCos(\omega t) - QiSin(\omega t)$$ which can't be simplified to look like d (as far as I can tell) because ##\beta## is a real constant (might be wrong here).
The other thing I tried to do was working with d to make it look like ##Q(t)=Qe^{-i(\omega t + \phi)}## but that leaves me with: $$\alpha \frac{e^{i\omega t}+e^{-i\omega t}}{2}-\beta \frac{e^{i\omega t}-e^{-i\omega t}}{2i}$$ after trying to simplify that I landed on: $$\frac{1}{2i}(e^{i\omega t}(\alpha i - \beta)+e^{-i\omega t}(\alpha i +\beta))$$ which doesn't look at all like what I need to get.
I'd like to get some help for this problem because I'm not even sure about what to study to start figuring out how to figure it out. It would be very appreciated, thank you.