
4 Attachment(s)
A pendulum question
Attachment 28136Attachment 28137Attachment 28138
Hello I have included pictures of the question I would like some help with and also the solution I have been given. I am fine up until part c. However I really do not understand how to do any part of c. So if there is anyone out there who thinks they could make it easier to understand your help would be really appreciated.
Thank you

Re: A pendulum question
I guess the real question is do you know how to solve the differential equation? The rest of the solution is based entirely on that.
Dan

Re: A pendulum question
The problem is I do not understand where they have got those equations from.

Re: A pendulum question
If you understand part B then you understand the derivation of the differential equation:
$\displaystyle \ddot \theta + 2 \lambda \dot \theta + \omega^2 \theta = 0$
In Part C they have simply substituted $\displaystyle \theta = e^{\alpha t}$:
$\displaystyle \frac {d^2(e^{\alpha t})}{dt^2} + 2 \lambda \frac {d(e^{\alpha t})}{dt} + \omega^2 e^{\alpha t} = 0$
Do the differentiation and you get
$\displaystyle \alpha^2 e^{\alpha t} + 2 \lambda \alpha e^{\alpha t} + \omega^2 e^{\alpha t}=0$
Now divide through by $\displaystyle e^{\alpha t} $ and you get their equation:
$\displaystyle \alpha^2 + 2 \lambda \alpha + \omega^2 = 0$
Is this helpful in getting you started in part C?

Re: A pendulum question
Thank you! That is very helpful.
Firstly where did they get that substitution from?
Also how did they find the equation for theta in part i of C?