Lengthening Pendulum Homework: Working with Equation 18.1

[dykuma]

Homework Statement

Homework Equations

The Attempt at a Solution

Working with equation 18.1 i found that

However, this obviously is not the equation in 18.3.
I found a source talking about this problem, and they get a similar equation
http://file.scirp.org/pdf/JAMP_2014031310562629.pdf

I have no clue where to go with this problem. Where is the 2 d theata / dl term coming from?

 

[TSny]

The Attempt at a Solution

Working with equation 18.1 i found that

Looks like you treated $l$ as a constant. But $l$ is a function of time.

 

[dykuma]

Looks like you treated $l$ as a constant. But $l$ is a function of time.

Hmm. I was assuming that it was a constant before I made the substitution, I see the flaw in my logic there. So then I need to leave it in this form

But I am not familiar with how to treat L in this situation.

 

[TSny]

In the expression $\frac{d}{dt} \left(l^2 \cdot \frac{d\theta}{dt} \right)$ it might help to replace the time derivatives with derivatives with respect to $l$ using your relation $\frac{d}{dt} = v \frac{d}{dl}$.