NBPhO 2026

Difficulty hard

Prerequisites

• Pseudo-forces in a rotating frame (centrifugal, Coriolis)
• Linear ODE: $\ddot x + \omega_0^2 x = F\sin(\omega t)$ and its particular/homogeneous split
• Lyapunov stability of a fixed point or periodic orbit
• Decomposing a rotating vector $\vec E(t)$ into components along and perpendicular to a co-rotating axis

Learning objectives

• Write the bead's equation of motion along the rod and recognise the negative-stiffness term $-\omega^2 x$ from centrifugal force
• Find a synchronised particular solution $x_p = A\sin(\omega t)$ to a forced linear ODE and convert it to a lab-frame trajectory
• Identify the geometric reason — the inscribed-angle theorem — that the bead orbits its circle at angular rate $2\omega$
• Use $\omega_0^2 = k/m - \omega^2$ to translate spring stiffness into stability of the homogeneous mode and to locate the parametric resonance $k = 2m\omega^2$
• In a rotating-field problem, change to a frame co-rotating with one ingredient (rod, field) to reduce a multi-frequency system to a low-dimensional one

Watch out for

• It is not enough to find a particular circular solution: stability of the trajectory is decided by the homogeneous equation. In part (i) the homogeneous mode grows exponentially, so the circle is mathematically a solution but physically unobservable without the stabilising spring.
• When adding the spring, two conditions must hold simultaneously — $k > m\omega^2$ for the homogeneous mode to oscillate rather than diverge, and $k \neq 2m\omega^2$ to avoid driving the now-genuine oscillator at its own natural frequency. Quoting only one condition loses points.
• The radius formula $R = mg/[2|2m\omega^2 - k|]$ has $|\cdot|$ for a reason: above the resonance ($k > 2m\omega^2$) the steady circle sits below the pivot, not above. Forgetting the absolute value gives a negative radius.
• In part (iv) the figure shows a closed rosette, which is only possible when $\Omega/\omega$ is rational. Read off the number of petals (one petal per cycle of the slow envelope) before assuming a frequency ratio.