A simple pendulum is suspended from the ceiling of an accelerating car

[badluckmath]

Homework Statement

Show that $\omega = \frac{\sqrt{a^{2}+g^{2} }}{l}$ for small angles

Relevant Equations

After solving the Hamilton equations, we find that $\frac{d^{2}\theta}{dt^{2}} = -gsin(\theta)/l- acos(\theta)/l$

Here is an image of the problem:

The problem consist in finding the moviment equation for the pendulum using Lagrangian and Hamiltonian equations.
I managed to get the equations , which are shown insed the blue box:

Using the hamilton equations, i finally got that the equilibrium angle $\theta_{e}$ : $$\theta_{e} = \tan^{-1}(\frac{-a}{g})$$m which is the angle where $\frac{d^{2}\theta}{dt^{2}} =0$.

Now, i got stuck. I tried to solve an EDO using the small angles aproximation, but it doesn't seems to lead me anywhere, because i don't really have information for the initial values.

 

[BvU]

You want small angles wrt $\theta_e$, so work the equation around to $\theta' = \theta -\theta_e$ in first order of $\theta'$.

 

[BvU]

Did you understand my post at all ? You want to work with $\theta'$, not with $y$ .

 

[badluckmath]

Did you understand my post at all ? You want to work with θ′, not with y .

This was another question. About your answer, i used $\mu = \theta + \theta_{e}$ and i got the right answer. Which is the same as you said.

 

[BvU]

This was another question.

And do we have a problem statement for that one ?

Side note: use \cos and \sin, just like \tan.
That way there is no confusion about e.g.
$acos(\theta)/l$ read as $\operatorname {acos}(\theta)/l$ instead of $a\cos(\theta)/l$

 

[badluckmath]

And do we have a problem statement for that one ?

Side note: use \cos and \sin, just like \tan.
That way there is no confusion about e.g.
$acos(\theta)/l$ read as $\operatorname {acos}(\theta)/l$ instead of $a\cos(\theta)/l$

It's the same as before, but now, the acceleration is upwards .

 

[BvU]

Thread now looks garbled: there was a post #3 from you that has disappeared
(not so bad, it was barely legible )