Simple harmonic motion in an accelerating car

[EddiePhys]

There's a pendulum attached to a car accelerating with $A$. I know I can find it's time period using the "effective" g method, but I want to find it from first principles.

My attempt:
$tan\theta = A/g$
Now I displace it by $\alpha$ giving $mgsin(\theta+\alpha)-mAcos(\theta+\alpha) = ma$

Expanding this and using small angle approximations:

$gsin\theta+g\alpha cos\theta-Acos\theta + A\alpha sin\theta = a$

Where did I go wrong?

 

[Ibix]

Can you simplify your last equation with your first? Does the result look like $\ddot\alpha\propto-\alpha$? If so, is the constant of proportionality what you expect from doing it the easy way? Is its sign right?

 

[nasu]

You also need to express a as a function of α.

 

[EddiePhys]

Can you simplify your last equation with your first? Does the result look like $\ddot\alpha\propto-\alpha$? If so, is the constant of proportionality what you expect from doing it the easy way? Is its sign right?

I got it. Thanks!