1. ## A pendulum question

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

2. ## 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

3. ## Re: A pendulum question

The problem is I do not understand where they have got those equations from.

4. ## Re: A pendulum question

If you understand part B then you understand the derivation of the differential equation:

$\ddot \theta + 2 \lambda \dot \theta + \omega^2 \theta = 0$

In Part C they have simply substituted $\theta = e^{\alpha t}$:

$\frac {d^2(e^{\alpha t})}{dt^2} + 2 \lambda \frac {d(e^{\alpha t})}{dt} + \omega^2 e^{\alpha t} = 0$

Do the differentiation and you get

$\alpha^2 e^{\alpha t} + 2 \lambda \alpha e^{\alpha t} + \omega^2 e^{\alpha t}=0$

Now divide through by $e^{\alpha t}$ and you get their equation:

$\alpha^2 + 2 \lambda \alpha + \omega^2 = 0$

Is this helpful in getting you started in part C?

5. ## 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?