I'm trying to find the acceleration vector of a simple pendulum. The vector is labeled a in this image from Wikipedia:
Trying to solve it, I've split the acceleration into perpendicular components: radial (along the string) and tangential (along the velocity vector). I've found that:
$a_\textrm{tangential} = d \ddot\theta$
$a_\textrm{radial} = d \dot\theta^2 + g \cos\theta$
where d is the length of the rod and g the acceleration from gravity. However, combining these two causes an awful expression, which makes me think that I've got something wrong:
$|a| = \sqrt{ a_\textrm{tangential}^2 + a_\textrm{radial}^2 } = \sqrt{ d^2 \ddot\theta^2 + ( d \dot\theta^2 + g \cos\theta )^2 }$
What is the actual magnitude of the acceleration vector of a simple pendulum?
My goal is to find an expression for the acceleration magnitude ($\sqrt{a_x^2+a_y^2+a_z^2}$) as measured by a three-axis accelerometer inside the bob.
Edit: I just realised that since $\ddot\theta = -\frac{g}{d}\sin\theta$, we can simplify:
$|a| = \sqrt{ d^2 \ddot\theta^2 + ( d \dot\theta^2 + g \cos\theta )^2 } = \sqrt{ g^2 + d^2\dot\theta^4 + 2dg\dot\theta^2 \cos\theta }$