Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 9C (Dynamics and Relativity)

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 ${\varphi }_1$.

The spring length is $a+x$, its tension is $kx$, and the orbit radius isVertical balance first givesRadial balance isUsing the vertical equation,Unlike the rigid-rod result, this depends on $m$: a larger mass stretches the spring farther and changes the radius of the orbit.