Rotating conical pendulum with an offset pivot
= Rotating conical pendulum with an offset pivot
A pendulum of length $a$ whose pivot rotates at radius $L$ can remain at angle $\phi$ from the downward vertical when
$$
\omega^2=\frac{g\tan\phi}{L+a\sin\phi}.
$$
If the connector is a spring of extension $x$, replace $a$ in the radius by $a+x$ and use vertical balance to determine $x$.