 Solution

The particle moves on a horizontal circle of radiusIf $T$ is the tension in the light rod, vertical and radial balance giveDividing eliminates both $T$ and $m$, yielding the rotating conical pendulum with an offset pivot conditionThe ratio under the square root has units $T^{-2}$. The answer is independent of $m$; increasing $L$ increases the orbit radius and lowers the angular velocity required at fixed ${\varphi }_1$.