What Are the Cartesian Coordinates of a Rotating Pendulum?

[Kulkid]

Homework Statement

[/B]
Find the Cartesian coordinates (x, y, z) of a fixed reference frame expressed in terms of the coordinates (x', y' , z) of a rotating frame, which rotates with the horizontal rod HR. Choose the x' -axis to point along the horizontal rod in the direction OA.
Use this to find the Cartesian coordinates (x, y, z) of the pendulum expressed in terms of θ and t

2. Homework Equations

The Attempt at a Solution

I tried this:
$x = x' cos(\omega t)$
$y = y' cos(\omega t)$
$z = z$

$x' = a\hat{i}$
$y' = b sin(\theta)\hat{j}$
$z = -b cos(\theta)\hat{k}$

So: $\vec{r} = (acos(\omega t), bsin(\theta)cos(\omega t), -bcos(\theta))$

Im later supposed to get a Lagrangian that looks like this:
$L =1/2mb^2\dot{\theta^2}+mab\omega cos(\theta)\dot{\theta}+1/2mb^2 \omega^2sin^2(\theta)+mgbcos(\theta)$
but its not quite working. Any tips?

 

[Orodruin]

Yout primed x and y coordinates should depend on both unprimed x and y coordinates in order to represent a rotation. If not, your coordinate transformation is singular at $t = 1/\omega$ (and represents a time dependent rescaling, not a rotation).